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1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> twisted smooth cohomology in string theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3865/#Item_18" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown"> <h3 id="context">Context</h3> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential 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href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <p>This entry contains lecture notes on</p> <ul> <li> <p><a href="#Examples"><strong>A)</strong></a> <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> smooth <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> structures appearing in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>;</p> </li> <li> <p><a href="#LocalPrequantumFieldTheory"><strong>B)</strong></a> the corresponding <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a></p> </li> </ul> <p>formulated in <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>.</p> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Examples'><strong>A)</strong> Examples of twisted smooth cohomology in string theory</a></li> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#TableOfTwists'>Overview: The Table of Twists</a></li> <li><a href='#VielbeinFields'><strong>I)</strong> Geometric structure (generalized, exceptional)</a></li> <ul> <li><a href='#ClassOfTheTangentBundle'>The class of the tangent bundle</a></li> <li><a href='#SmoothModuliStacks'>Smooth moduli stacks</a></li> <li><a href='#ReductionOfTheStructureGroup'>Reduction of the structure group</a></li> <li><a href='#ModuliSpaceOfOrhtogonalStructures'>Moduli space of orthogonal structures: twisted cohomology</a></li> <li><a href='#moduli_stack_of_orthogonal_structures'>Moduli stack of orthogonal structures</a></li> <li><a href='#SpinConnection'>Differential refinement: Spin connection</a></li> <li><a href='#PullbackOfOrthogonalStructures'>Pullback of orthogonal structures</a></li> <li><a href='#generalized_vielbein_fields_type_ii_geometry_generalized_cy_and_uduality'>Generalized vielbein fields: type II geometry, generalized CY and U-duality</a></li> </ul> <li><a href='#SpinStringFivebraneStructures'><strong>II)</strong> Spin-, String- and Fivebrane-structure</a></li> <ul> <li><a href='#WhiteheadTower'>Whitehead tower</a></li> <li><a href='#NecessityOfSmoothRefinement'>Necessity of smooth refinement</a></li> <li><a href='#SmoothRefinement'>Smooth refinement</a></li> <li><a href='#SmoothWhitehead'>Smooth Whitehead tower</a></li> <li><a href='#DifferentialRefinement'>Differential refinement</a></li> <li><a href='#DifferentialWhitehead'>Differential Whitehead tower</a></li> </ul> <li><a href='#Anomalies'><strong>Interlude)</strong> Anomaly line bundle on smooth moduli stacks of fields</a></li> <ul> <li><a href='#higher_gauge_fields_in_the_presence_of_magnetic_charge_current'>Higher gauge fields in the presence of magnetic charge current</a></li> <li><a href='#gauge_transformations'>Gauge transformations</a></li> <li><a href='#magnetic_current_induced_by_background_fields'>Magnetic current induced by background fields</a></li> <li><a href='#infinitesimal_moduli_stacks_brst_complexes'>Infinitesimal moduli stacks: BRST complexes</a></li> <li><a href='#extended_higher_chernsimonstype_functionals'>Extended higher Chern-Simons-type functionals</a></li> <li><a href='#GaugeInteractionAndChargeAnomaly'>Gauge interaction and the charge anomaly</a></li> </ul> <li><a href='#TwistedK'><strong>III)</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Fivebrane</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Fivebrane^c</annotation></semantics></math>-structure</a></li> <ul> <li><a href='#HigherUnitaryTwistedCovers'>Higher unitary-twisted covers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math></a></li> <li><a href='#BFieldInK'><strong>a)</strong> Twisted differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structure: Freed-Witten mechanism</a></li> <li><a href='#SuGraFluxQuantization'><strong>b)</strong> Twisted differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">String^{\mathbf{c}_2}</annotation></semantics></math>-structure: M-theory flux quantization</a></li> <li><a href='#TwistedStringStructure'><strong>c)</strong> Twisted differential String-structure – Green-Schwarz mechanism</a></li> <li><a href='#d_twisted_differential_fivebrane_structure__dual_greenschwarz_mechanism'><strong>d)</strong> Twisted differential Fivebrane structure – dual Green-Schwarz mechanism</a></li> </ul> <li><a href='#HigherOrientifold'><strong>IV)</strong> Higher orientifold structure</a></li> <ul> <li><a href='#orientifolds'>Orientifolds</a></li> <li><a href='#HWCompactifications'>Hořava-Witten compactifications</a></li> </ul> <li><a href='#further_twists'>Further twists</a></li> <ul> <li><a href='#TwistedSuperBundle'>Twisted super bundle</a></li> <li><a href='#RelativeFields'>Relative fields</a></li> <li><a href='#twisted_tmf'>Twisted tmf</a></li> <li><a href='#twisted_morava_ktheory'>Twisted Morava K-theory</a></li> </ul> </ul> <li><a href='#LocalPrequantumFieldTheory'><strong>B)</strong> Local boundary prequantum field theory from twisted smooth cohomology</a></li> <ul> <li><a href='#cohesive_contexts_for_equivariant_differential_twisted_cohomology'>Cohesive contexts for equivariant differential twisted cohomology</a></li> <li><a href='#cohesive_contexts_for_stable_twisted_cohomology'>Cohesive contexts for stable twisted cohomology</a></li> <ul> <li><a href='#tangent_topos'>Tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos</a></li> <li><a href='#tangent_cohesive_homotopy_theory'>Tangent cohesive homotopy theory</a></li> <li><a href='#CohesiveAndDifferentialRefinement'>Cohesive and differential refinement in tangent cohesion</a></li> </ul> <li><a href='#local_prequantum_field_theory'>Local prequantum field theory</a></li> <li><a href='#motivic_quantization_of_twisted_fields'>Motivic quantization of twisted fields</a></li> <ul> <li><a href='#linearization_by_twisted_cohomology_spectra'>Linearization by twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology spectra</a></li> <li><a href='#ExpositionCohomologicalQuantization'>Cohomological quantization by pull-push</a></li> </ul> </ul> <li><a href='#GeneralTheory'>General theory</a></li> <ul> <li><a href='#homotopy_type_theory'>Homotopy type theory</a></li> <ul> <li><a href='#Cohomology'>Cohomology</a></li> <li><a href='#PrincipalBundles'>Principal bundles</a></li> <li><a href='#TwistedCohomology'>Twisted cohomology</a></li> <li><a href='#AssociatedAndTwistedBundles'>Associated and twisted bundles</a></li> </ul> <li><a href='#cohesive_homotopy_type_theory'>Cohesive homotopy type theory</a></li> <ul> <li><a href='#geometric_realization'>Geometric realization</a></li> <li><a href='#differential_cohomology'>Differential cohomology</a></li> <li><a href='#chernweil_theory'>Chern-Weil theory</a></li> </ul> <li><a href='#differential_homotopytype_theory'>Differential homotopy-type theory</a></li> <ul> <li><a href='#de_rham_stack'>de Rham stack</a></li> <li><a href='#tale_stacks__orbifolds'>Étale stacks / orbifolds</a></li> </ul> </ul> <li><a href='#RelatedEntries'>Related entries</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="Examples"><strong>A)</strong> Examples of twisted smooth cohomology in string theory</h2> <blockquote> <p>This section originates in notes prepared along with a lecture series (<a href="#SchreiberLect">Schreiber ESI 2012</a>) at the Erwin-Schrödinger institute in 2012.</p> </blockquote> <p>This section discusses four classes of examples of twisted smooth cohomology in string theory</p> <ol> <li> <p><em><a href="#VielbeinFields">Geometric structure (generalized, exceptional)</a></em></p> </li> <li> <p><em><a href="#SpinStringFivebraneStructures">Spin-, String-, and Fivebrane structure</a></em></p> </li> <li> <p><em><a href="#TwistedK"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Fivebrane</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Fivebrane^c</annotation></semantics></math>-structure</a></em></p> </li> <li> <p><em><a href="#HigherOrientifold">Higher orientifold structure</a></em></p> </li> </ol> <h3 id="idea">Idea</h3> <p>The <a class="existingWikiWord" href="/nlab/show/background+gauge+fields">background gauge fields</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> appearing in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> are mathematically described by <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> and <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential</a> refinements of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>. A famous example is the <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> that describes the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>-twisted <a class="existingWikiWord" href="/nlab/show/Yang-Mills+fields">Yang-Mills fields</a> over <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>. But there are many more classes of examples of twisted / smooth / differential cohomology appearing throughout string theory.</p> <p>This page provides a survey of and introduction to such examples, organized along a <em><a href="#TableOfTwists">Table of twists</a></em>, that indicates how all of these are instances a single pattern. For further reading and more details see the list of <em><a href="#References">references</a></em> below.</p> <p>We start with an introduction to the general notion of twisted smooth cohomology by way of the simple but instructive class of examples of</p> <ul> <li><strong>I)</strong> <em><a href="#VielbeinFields">Geometric structure (generalized, exceptional)</a></em>,</li> </ul> <p>via <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+groups">reduction of structure groups</a>, which, simple as it is, serves as a blueprint for all of the examples to follow, and which we use to introduce the general machinery. It also serves to highlight the need and use of <em>smooth</em> cohomology in addition to both <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">ordinary topological/homotopical</a> as well as <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>.</p> <p>(The mathematically inclined reader wishing to see a more formal development of the general theory behind the discussion here should look at the section <em><a href="#GeneralTheory">General theory</a></em> below for pointers.)</p> <p>Then we proceed in direct analogy, but now with ordinary <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> generalized to the genuine <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, and discuss aspects of the main classes of examples of twisted smooth cohomology appearing there. First we indicate how higher spin structures as such lead to higher smooth homotopy theory:</p> <ul> <li>II) <em><a href="#SpinStringFivebraneStructures">Spin-, String-, and Fivebrane structure</a></em>.</li> </ul> <p>Then we roughly indicate the relation between higher gauge fields and <a class="existingWikiWord" href="/nlab/show/quantum+anomalies">quantum anomalies</a>:</p> <ul> <li>Interlude) <em><a href="#Anomalies">Anomaly line bundle on smooth moduli stacks of fields</a></em>.</li> </ul> <p>Finally we put the pieces together and scan through various situations appearing in string theory with their anomaly structure and discuss the smooth moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks of anomaly-free field configurations / of twisted smooth cocycles:</p> <ul> <li>III) <em><a href="#TwistedK"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math>-, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Fivebrane</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Fivebrane^c</annotation></semantics></math>-structure</a></em> .</li> </ul> <p>There are various further examples. As an outlook we indicate aspects of</p> <ul> <li>IV) <em><a href="#HigherOrientifold">Higher orientifold structure</a></em>.</li> </ul> <h3 id="TableOfTwists">Overview: The Table of Twists</h3> <p>The following sections discuss classes of examples of <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted smooth structures</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>. All these examples are governed by the the same general pattern of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> refined to <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth cohomology</a> (a gentle explanation/example follows <a href="#VielbeinFields">in a moment</a>, for formal details see further <a href="#TwistedCohomology">below</a>). They are specified by a <em>universal local coefficient bundle</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F &\to& E \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>which exists in a context of <em><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">geometric homotopy types</a></em>: an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">higher topos</a> to be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} \coloneqq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>/smooth <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>.</p> <p>Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> – a higher <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a coefficient object equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">E \to \mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated bundle</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a> over the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>.</p> </li> </ul> <p>Given such, and given a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we have:</p> <ul> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{B}G</annotation></semantics></math> determines a <em>twisting bundle</em> or <em>twisting <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ϕ</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \phi</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mover><mi>ϕ</mi><mo stretchy="false">^</mo></mover></mover></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\hat \phi}{\to}&& E \\ & {}_{\mathllap{\phi}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+infinity-bundle">twisted bundle</a>, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-twisted background gauge field.</p> </li> </ul> <p>The following table lists examples of such local coefficient bundles and tabulates the correspondings twisting fields and twisted fields. This is to be read as an extended table of contents. Explanations are in the sections to follow.</p> <table><thead><tr><th>class of examples</th><th>universal local coefficient bundle</th><th>twisting bundle</th><th>twisting field</th><th>twisted bundle</th><th>twisted field</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a href="#GeneralTheory">0)</a></strong> <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">general pattern</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ F &\to& E \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/higher+gauge+field">gauge field</a> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a>-sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-bundle</td><td style="text-align: left;">twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\Omega F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> sector</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mstyle mathvariant="bold"><mi>curv</mi></mstyle></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \flat \mathbf{B}^n U(1) &\to& \mathbf{B}^n U(1) \\ && \downarrow^{\mathbf{curv}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham</a> <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher abelian gauge field</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>F</mi> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>E</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ F_{conn} &\to& E_{conn} \\ && \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ && \mathbf{B}G_{conn} }</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/higher+gauge+field">gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-bundle</td><td style="text-align: left;">twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\Omega F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></td></tr> <tr><td style="text-align: left;"><strong><a href="#VielbeinFields">I)</a></strong> <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+group">reduction of structure group</a>/<a class="existingWikiWord" href="/nlab/show/G-structure">geometric structure</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ GL(n)/O(n) &\to& \mathbf{B} O(n) \\ && \downarrow \\ && \mathbf{B} GL(n) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> structure / <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ GL(n)/O(n) &\to& \mathbf{B} O(n)_{conn} \\ && \downarrow \\ && \mathbf{B} GL(n)_{conn} }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>d</mi><mo>,</mo><mn>2</mn><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SU</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>d</mi><mo>,</mo><mn>2</mn><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ O(2d,2d)/SU(d,d) &\to& \mathbf{B} SU(d,d) \\ && \downarrow \\ && \mathbf{B} O(2d,2d) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifolds">generalized Calabi-Yau manifolds</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ O(n)\backslash O(n,n)/O(n) &\to& \mathbf{B} O(n) \times O(n) \\ && \downarrow \\ && \mathbf{B} O(n,n) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></td><td style="text-align: left;">DFT <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>\</mo><mi>O</mi><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ SU(3)\backslash O(6,6) / SU(3) &\to& \mathbf{B} SU(3) \\ && \downarrow \\ && \mathbf{B} O(6,6) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">d = 6</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">comactifications</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mrow><mn>7</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SU</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mrow><mn>7</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ E_{7(7)}/SU(7) &\to& \mathbf{B} SU(7) \\ && \downarrow \\ && \mathbf{B}E_{7(7)} }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/exceptional+tangent+bundle">exceptional tangent bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E7">E7</a>-<a href="supergravity#UDuality">U-duality</a> moduli (<a class="existingWikiWord" href="/nlab/show/split+real+form">split real form</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">d = 7</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d supergravity</a> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactifications</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mi>n</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>H</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ E_{n(n)}/H_n &\to& \mathbf{B}H_n \\ && \downarrow \\ && \mathbf{B}E_{n(n)} }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/exceptional+tangent+bundle">exceptional tangent bundle</a></td><td style="text-align: left;"><a href="supergravity#UDuality">U-duality</a> moduli</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactification</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d supergravity</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>11</mn><mo>−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">d = 11-n</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>P</mi><mi>U</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mstyle mathvariant="bold"><mi>dd</mi></mstyle></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B} U &\to& \mathbf{B} P U \\ && \downarrow^{\mathbf{dd}} \\ && \mathbf{B}^2 U(1) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle</a>/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted unitary bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mn>8</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>a</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}String(E_8) &\to& \mathbf{B} E_8 \\ && \downarrow^{\mathrlap{\mathbf{a}}} \\ && \mathbf{B}^3 U(1)} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a> / <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted String(E8)</a>-2-form gauge field</td></tr> <tr><td style="text-align: left;"><strong><a href="#SpinStringFivebraneStructures">II)</a></strong> <a href="spin+structure#Higher">higher spin structures</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B} Spin &\to& \mathbf{B} SO \\ && \downarrow^{\mathrlap{\mathbf{w}_2}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}String &\to& \mathbf{B} Spin \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1}} \\ && \mathbf{B}^3 U(1) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a>/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted smooth string structure</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}String_{conn} &\to& \mathbf{B} Spin_{conn} \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\hat \mathbf{p}_1}} \\ && \mathbf{B}^3 U(1)_{conn} }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a> <a class="existingWikiWord" href="/nlab/show/current">magnetic current</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>+<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}Fivebrane &\to& \mathbf{B} String \\ && \downarrow^{\mathrlap{\tfrac{1}{6}\mathbf{p}_2}} \\ && \mathbf{B}^7 U(1) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 7-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string">string</a> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted smooth fivebrane structure</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Fivebrane</mi> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}Fivebrane_{conn} &\to& \mathbf{B} String_{conn} \\ && \downarrow^{\mathrlap{\tfrac{1}{6}\hat \mathbf{p}_2}} \\ && \mathbf{B}^7 U(1)_{conn} }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 7-bundle with connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string">string</a> <a class="existingWikiWord" href="/nlab/show/current">electric current</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></td><td style="text-align: left;">dual <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>+<a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></td></tr> <tr><td style="text-align: left;"><strong><a href="#TwistedK">III)</a></strong> <a href="spin^c+structure#Higher">higher spin^c-structures</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>SO</mi><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>w</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>c</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}Spin^c &\to& \mathbf{B} (SO \times U(1)) \\ && \downarrow^{\mathrlap{ w_2 - c_1}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msup><mi>Spin</mi> <mi>c</mi></msup><msup><mo stretchy="false">)</mo> <mstyle mathvariant="bold"><mi>dd</mi></mstyle></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>PU</mi><mo>×</mo><mi>SO</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle mathvariant="bold"><mi>dd</mi></mstyle><mo>−</mo><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}(Spin^c)^{\mathbf{dd}} &\to& \mathbf{B} (PU \times SO) \\ && \downarrow^{\mathrlap{ \mathbf{dd} - \mathbf{W}_3 }} \\ && \mathbf{B}^2 U(1) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly">Freed-Witten anomaly</a> for <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a> on <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>String</mi> <mstyle mathvariant="bold"><mi>a</mi></mstyle></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>−</mo><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}String^{\mathbf{a}} &\to& \mathbf{B} Spin \times E_8 \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - 2 \mathbf{a}}} \\ && \mathbf{B}^3 U(1) }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a>/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted smooth string^c structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>+<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>+<a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></td></tr> <tr><td style="text-align: left;"><strong><a href="#HigherOrientifold">IV)</a></strong> Giraud-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbes">∞-gerbes</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}^2 U(1) &\to& \mathbf{B}Aut(\mathbf{B}U(1)) \\ && \downarrow \\ && \mathbf{B}\mathbb{Z}_2 }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+cover">double cover</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jandl+gerbe">Jandl gerbe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbf{B}^3 U(1) &\to& \mathbf{B}Aut(\mathbf{B}^2 U(1)) \\ && \downarrow \\ && \mathbf{B}\mathbb{Z}_2 }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+cover">double cover</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten orientifold</a></td></tr> </tbody></table> <h3 id="VielbeinFields"><strong>I)</strong> Geometric structure (generalized, exceptional)</h3> <p>The ordinary notion of <em><a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a></em> in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> (equivalently: <em><a class="existingWikiWord" href="/nlab/show/soldering+form">soldering form</a></em> or <em><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></em>) turns out to be a simple special case of the general notion of twisted smooth cohomology that we are concerned with here. Viewed from this perspective it already contains the seeds of all of the more sophisticated examples to be considered below. Therefore we discuss this case here as a warmup, such as to introduce the general theory by way of example.</p> <h4 id="ClassOfTheTangentBundle">The class of the tangent bundle</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>By definition this means that there is an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></munderover><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{ \mathbb{R}^n \underoverset{\simeq}{\phi_i^{-1}}{\to} U_i \hookrightarrow X\} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/coordinate+charts">coordinate charts</a>. On each overlap <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j</annotation></semantics></math> of two charts, the <a class="existingWikiWord" href="/nlab/show/partial+derivatives">partial derivatives</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/coordinate+transformations">coordinate transformations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>j</mi></msub><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \phi_j\circ \phi_i^{-1} : U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n </annotation></semantics></math></div> <p>form the <a class="existingWikiWord" href="/nlab/show/Jacobian+matrix">Jacobian matrix</a> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup><msub><mrow></mrow> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>[</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><msub><mi>ϕ</mi> <mi>j</mi></msub><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo><mo>]</mo></mrow><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><msub><mi>GL</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> ((\lambda_{i j})^{\mu}{}_{\mu}) \coloneqq \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu) \right] : U_i \cap U_j \to GL_n </annotation></semantics></math></div> <p>with values in invertible <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a>, hence in the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>. By construction (by the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a>), these functions satisfy on triple overlaps of coordinate charts the <a class="existingWikiWord" href="/nlab/show/matrix+product">matrix product</a> equations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup><msub><mrow></mrow> <mi>λ</mi></msub><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mi>λ</mi></msup><msub><mrow></mrow> <mi>ν</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup><msub><mrow></mrow> <mi>ν</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu} = (\lambda_{i k})^\mu{}_{\nu} \,, </annotation></semantics></math></div> <p>(here and in the following sums over an index appearing upstairs and downstairs are explicit)</p> <p>hence the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>⋅</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k} </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>,</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(U_i \cap U_j \cap U_k, GL(n))</annotation></semantics></math> of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-valued functions on the chart overlaps.</p> <p>This is the <em><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> condition</em> for a smooth <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cocycle</a> in degree 1 with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> (precisely: with coefficients in the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> ). We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>smooth</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(\lambda_{i j})] \in H^1_{smooth}(X, GL_n) \,. </annotation></semantics></math></div> <p>It is useful to reformulate this statement in the language of <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>/<a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>.</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> itself is a Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mover><mo>→</mo><mo>→</mo></mover><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X \stackrel{\to}{\to} X)</annotation></semantics></math> with trivial morphism structure;</p> </li> <li> <p>from the atlas <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> we get the corresponding <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>for</mi><mspace width="thinmathspace"></mspace><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C(\{U_i\}) = (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i) = \left\{ \array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\; for\, x \in U_i \cap U_j \cap U_k \right\} \,, </annotation></semantics></math></div> <p>whose objects are the points in the atlas, with morphisms identifying lifts of a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to different charts of the atlas;</p> </li> <li> <p>any <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> induces its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>G</mi><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G = \left( G \stackrel{\to}{\to} * \right) \,. </annotation></semantics></math></div></li> </ul> <p>The above situation is neatly encoded in the existence of a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of Lie groupoids of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>λ</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ C(\{U_i\}) &\stackrel{\lambda}{\to}& \mathbf{B} GL(n). \\ {}^{\mathllap{\simeq}}\downarrow \\ X } \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the left morphism is <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>-wisse (around small enough <a class="existingWikiWord" href="/nlab/show/neighbourhoods">neighbourhoods</a> of each point) an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a> (we make this more precise in a moment);</p> </li> <li> <p>the horizontal functor has as components the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{i j}</annotation></semantics></math> and its functoriality is the cocycle condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>⋅</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}</annotation></semantics></math>.</p> </li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/natural+transformation">transformation</a> of smooth functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mn>1</mn></msub><mo>⇒</mo><msub><mi>λ</mi> <mn>2</mn></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda_1 \Rightarrow \lambda_2 : C(\{U_i\}) \to \mathbf{B} GL(n)</annotation></semantics></math> is precisely a <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a> between two such cocycles.</p> <h4 id="SmoothModuliStacks">Smooth moduli stacks</h4> <p>We want to think of such a diagram as being directly a morphism of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> T X \; : \; X \to \mathbf{B} GL(n) \;\; \in \mathbf{H} </annotation></semantics></math></div> <p>in a suitable context <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, such that this may be regarded as a smooth refinement of the underlying <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> class of a map into the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL(n)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \to B GL(n) \in Top \,. </annotation></semantics></math></div> <p>Evidently, for this we need to turn the <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>-wise <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C(\{U_i\}) \to X</annotation></semantics></math> into an actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. This is a <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">non-abelian/non-stable</a> generalization of what happens in the construction of a <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a>, for instance in the theory of <a class="existingWikiWord" href="/nlab/show/topological+string">topological branes</a>.</p> <p>To make this precise, first notice that every Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>→</mo></mover><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (A_1 \stackrel{\to}{\to} A_0)</annotation></semantics></math> yields on each smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> a groupoid of maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>U</mi><mo>↦</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A : U \mapsto (C^\infty(U,A_1) \stackrel{\to}{\to} C^\infty(U,A_0)) \,, </annotation></semantics></math></div> <p>the groupoid of <em>smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-families</em> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Moreover, for every smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U_1 \to U_2</annotation></semantics></math> there is an evident restriction map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(U_2) \to A(U_1)</annotation></semantics></math> and so this yields a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of groupoids, hence a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Func(SmthMfd^{op}, Grpd)</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> says that thinking of Lie groupoids as presheaves of ordinary groupoids this way does not lose information — and <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> say that it is generally a good idea.</p> <p>Let therefore</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>Grpd</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mi>stalkwise</mi><mspace width="thinmathspace"></mspace><mi>h</mi><mo>.</mo><mi>e</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \coloneqq Func(SmthMfd^{op}, Grpd)[\{stalkwise\, h.e\}^{-1}] </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">localization</a> of groupoid-valued presheaves that universally turns <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> into actual homotopy equivalences: if a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\eta : A \to B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(SmthMfd^{op}, Grpd)</annotation></semantics></math> is such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</a> and each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x \in U</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_x \subset U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta(U_x) : A(U_x) \to B(U_x)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/homotopy+inverse">homotopy inverse</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>We call this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> the <em><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a></em> of <em><a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth groupoids</a></em> or of <em><a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth stacks</a></em>.</p> <p>Discussed there are tools for describing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> concretely. For the moment we only need to know that</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C(\{U_i\}) \to X</annotation></semantics></math> of every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> has a <a class="existingWikiWord" href="/nlab/show/homotopy+inverse">homotopy inverse</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, as already used;</p> </li> <li> <p>if the cover is <em><a class="existingWikiWord" href="/nlab/show/good+open+cover">good</a></em> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a Lie group, then every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is presented by a <a class="existingWikiWord" href="/nlab/show/anafunctor">zig-zag</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}G</annotation></semantics></math>.</p> </li> </ol> <p>Then we have</p> <ol> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \mathbf{B} GL(n)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is precisely a smooth real vector bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a homotopy between two such morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is precisely a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between the two vector bundles.</p> </li> </ol> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}GL(n)</annotation></semantics></math> regarded as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></em> of real vector bundles.</p> <p>Of course there is a “smaller” Lie groupoid that also classifies real vector bundles, but whose gauge transformations are restricted to be <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>-valued functions. Passing to this “smaller” Lie groupoid is what the choice of vielbein accomplishes, to which we now turn.</p> <h4 id="ReductionOfTheStructureGroup">Reduction of the structure group</h4> <p>Consider the defining inclusion of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> into the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> O(n) \hookrightarrow GL(n) \,. </annotation></semantics></math></div> <p>We may understand this inclusion geometrically in terms of the canonical <a class="existingWikiWord" href="/nlab/show/metric">metric</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>. We may also understand it purely <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theoretically</a> as the inclusion of the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>. This makes it manifest that the inclusion is trivial at the level of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (it is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> of the underlying <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>) and hence <em>only</em> encodes geometric information.</p> <p>The inclusion induces a corresponding inclusion (<a class="existingWikiWord" href="/nlab/show/truncated+object+of+an+%28infinity%2C1%29-category">0-truncated morphism</a>) of moduli stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{Orth} : \mathbf{B} O(n) \to \mathbf{B} GL(n) \;\;\; \in \mathbf{H} </annotation></semantics></math></div> <p>simply by regarding it as a morphism of Lie groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (O(n) \stackrel{\to}{\to} * ) \to (GL(n) \stackrel{\to}{\to} * ) </annotation></semantics></math></div> <p>in the evident way.</p> <p>Now we can say what a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>/<a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is:</p> <p>A choice of <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> is a factorization of the above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-valued cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Orth</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Orth}</annotation></semantics></math>, up to a smooth <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, hence a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>λ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mrow><msup><mi>E</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>Orth</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{h}{\to}&& \mathbf{B} O(n) \\ & {}_{\mathllap{\lambda}}\searrow &\swArrow_{\mathrlap{E^{-1}}}& \swarrow_{\mathrlap{\mathbf{Orth}}} \\ && \mathbf{B}GL(n) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>This consists of two pieces of data</p> <ul> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is (by the same reasoning as for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> above) a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>-valued 1-cocycle – a collection of <em>orthogonal transition functions</em> – hence on each overlap of coordinate patches a smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow></mrow> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((h_{i j}){}^a{}_b) : U_i \cap U_j \to O(n) </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>⋅</mo><msub><mi>h</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>h</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> h_{i j} \cdot h_{j k} = h_{i k} </annotation></semantics></math></div> <p>on all triple overlaps of coordinate charts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k</annotation></semantics></math>;</p> </li> <li> <p>the homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is on each chart a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mi>a</mi></msup><msub><mrow></mrow> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_i = ((E_i)^a{}_\mu) : U_i \to GL(n) </annotation></semantics></math></div> <ul> <li> <p>such that on each overlap of coordinate charts it intertwines the transition functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> of the tangent bundle with the new orthogonal transition functions, meaning that the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mi>a</mi></msup><msub><mrow></mrow> <mi>μ</mi></msub><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup><msub><mrow></mrow> <mi>ν</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>j</mi></msub><msup><mo stretchy="false">)</mo> <mi>b</mi></msup><msub><mrow></mrow> <mi>ν</mi></msub></mrow><annotation encoding="application/x-tex"> (E_i)^a{}_{\mu} (\lambda_{i j})^{\mu}{}_\nu = (h_{i j})^a{}_b (E_j)^b{}_\nu </annotation></semantics></math></div> <p>holds. This exhibits the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>:</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>E</mi> <mi>i</mi></msub></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>E</mi> <mi>j</mi></msub></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ * &\stackrel{E_i}{\to}& * \\ {}^{\mathllap{\lambda_{i j}}}\downarrow && \downarrow^{\mathrlap{h_{i j}}} \\ * &\stackrel{E_j}{\to}& * } </annotation></semantics></math></div></li> </ul> <p>The component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> defines an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, or its <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>. The component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is the corresponding <strong><a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a></strong>. It exhibits an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>:</mo><mi>T</mi><mi>X</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>V</mi></mrow><annotation encoding="application/x-tex"> E : T X \stackrel{\simeq}{\to} V </annotation></semantics></math></div> <p>between a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V \to X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/structure+group">structure group</a> explicitly being the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>, and the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> itself, hence it exhibits the <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>.</p> <h4 id="ModuliSpaceOfOrhtogonalStructures">Moduli space of orthogonal structures: twisted cohomology</h4> <p>We consider now the space of choices of vielbein fields on a given tangent bundle, hence the <em><a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a></em> or <em><a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></em> of <a class="existingWikiWord" href="/nlab/show/orthogonal+structures">orthogonal structures</a>/<a class="existingWikiWord" href="/nlab/show/Riemannian+metrics">Riemannian metrics</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>This is usefully discussed in terms of the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n)</annotation></semantics></math>. One finds that the homotopy fiber is the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n) \backslash GL(n)</annotation></semantics></math>.</p> <p>This means that there is a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ GL(n)/O(n) &\to& \mathbf{B}O(n) \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& \mathbf{B} GL(n) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)/O(n)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> with the property of sitting in such a diagram.</p> <p>We may think of this <em><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></em> as being a bundle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}GL(n)</annotation></semantics></math> with typical fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)/O(n)</annotation></semantics></math>. As such, it is the smooth <a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated bundle</a> to the smooth <a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal GL(n)-bundle</a> induced by the canonical <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)\backslash GL(n)</annotation></semantics></math>.</p> <p>One basic properties of homotopy pullbacks is that they are preserved by forming <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,-)</annotation></semantics></math> out of any other object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. This means that also</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(X,GL(n)/O(n)) &\to& \mathbf{H}(X,\mathbf{B}O(n)) &\to& \mathbf{H}(X, \mathbf{B}GL(n)) } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>. This in turn says that orthogonal structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that the underlying tangent bundle is trivializable, are given by smooth functions into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)/O(n)</annotation></semantics></math>.</p> <p>This means that if the tangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> is trivializable, then the coset space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)\backslash GL(n)</annotation></semantics></math> is the moduli space for vielbein fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mspace width="thickmathspace"></mspace><mi>trivializable</mi><mo stretchy="false">)</mo><mo>⇒</mo><mi>SpaceOfVielbeinFieldsOn</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (T X \; trivializable) \Rightarrow SpaceOfVielbeinFieldsOn(T X) = \mathbf{H}(X, O(n)\backslash GL(n)) = C^\infty(X, O(n)\backslash GL(n)) \,. </annotation></semantics></math></div> <p>However, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> is not trivial, then this is true only locally: there is then an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> such that restricted to each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> the moduli space of vielbein fields is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(U_i, GL(n)/ O(n))</annotation></semantics></math>, but globally these now glue together in a non-trivial way as encoded by the tangent bundle: we may say that</p> <p>the tangent bundle <em>twists</em> the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to GL(n)/O(n)</annotation></semantics></math>. If we think of an ordinary such function as a cocycle in degree-0 cohomology, then this means that a vielbein is a cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>-_<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>_ relative to the <em>twisting local coefficient bundle</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>.</p> <p>We can make this more manifest by writing equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ O(n)\backslash GL(n) &\to& (O(n)\backslash GL(n)) // GL(n) \\ && \downarrow \\ && \mathbf{B}GL(n) } \,, </annotation></semantics></math></div> <p>where now on the right we have inserted the <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> resolution of the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> as provided by the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a>: this is the morphism out of the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> of the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)\backslash GL(n)</annotation></semantics></math>.</p> <p>The pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>T</mi><mi>X</mi><msub><mo>×</mo> <mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>T</mi><mi>X</mi></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ T X \times_{GL(n)} (O(n)\backslash GL(n)) &\to& O(n)\backslash GL(n) // GL(n) \\ \downarrow && \downarrow \\ X &\stackrel{T X}{\to}& \mathbf{B}GL(n) } </annotation></semantics></math></div> <p>gives the non-linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>-associated bundle whose space of sections is the “twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>\</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)\backslash GL(n)</annotation></semantics></math>-0-cohomology”, hence the space of inequivalent vielbein fields.</p> <h4 id="moduli_stack_of_orthogonal_structures">Moduli stack of orthogonal structures</h4> <p>The above says that the space of vielbein fields is the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/over-%28infinity%2C1%29-topos">slice (2,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}GL(n)}</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Orth} : \mathbf{B}O \to \mathbf{B}GL(n)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><msub><mi>Struc</mi> <mi>TX</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Orth} Struc_{TX}(X) \coloneqq \mathbf{Orth}(T X) \coloneqq \mathbf{H}_{/\mathbf{B}GL(n)}(T X, \mathbf{Orth}) \,. </annotation></semantics></math></div> <p>But also this space of choices of vielbein fields has a smooth structure, it is itself a smooth <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>. This is obtained by forming the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in the slice over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}GL(n)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28infinity%2C1%29-category">locally cartesian closed (2,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mrow><mi>T</mi><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">[</mo><mi>T</mi><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><msub><mo stretchy="false">]</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{OrthStruc}_{T X}(X) \coloneqq [T X, \mathbf{Orth}]_{\mathbf{B}GL(n)} \,. </annotation></semantics></math></div> <p>For more on this see also the discussion at <em><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></em>.</p> <h4 id="SpinConnection">Differential refinement: Spin connection</h4> <p>We may further refine this discussion to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> to get genuine <em>differential</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>-structures.</p> <p>Recall that the moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is presented in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> by the presheaf of groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>:</mo><mi>U</mi><mo>↦</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} G : U \mapsto (C^\infty(U,G) \stackrel{\to}{\to} *) \,. </annotation></semantics></math></div> <p>We may think of this for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> as being the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-gauge transformations acting on the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle is a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> valued form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \Omega^1(U, \mathfrak{g})</annotation></semantics></math>. Accordingly, the presheaf of groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>:</mo><mi>U</mi><mo>↦</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G_{conn} : U \mapsto (C^\infty(U,G) \times \Omega^1(U, \mathfrak{g}) \stackrel{\to}{\to} \Omega^1(U, \mathfrak{g}) ) </annotation></semantics></math></div> <p>is that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-connections and gauge transformations between them: the <em><a class="existingWikiWord" href="/nlab/show/groupoid+of+Lie-algebra+valued+forms">groupoid of Lie-algebra valued forms</a></em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. As an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">SmoothGrpd</a> this the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi><msub><mi>Bund</mi> <mo>∇</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \mathbf{B}G_{conn}) \simeq G Bund_\nabla(X) \,. </annotation></semantics></math></div> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Orth</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Orth}</annotation></semantics></math> has an evident differential refinement to a morphism between such differentially refined moduli stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Orth</mi></mstyle> <mi>conn</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Orth}_{conn} : \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn} </annotation></semantics></math></div> <p>by acting on the differential forms with the induced inclusion of the <a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a> into the <a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{o}(n) \hookrightarrow \mathfrak{gl}(n)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of this differential refinement turns out to be the same moduli space as before</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>Orth</mi></mstyle> <mi>conn</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ GL(n)/ O(n) &\to& \mathbf{B} O(n)_{conn} \\ && \downarrow^{\mathrlap{\mathbf{Orth}_{conn}}} \\ && \mathbf{B} GL(n)_{conn} } \,, </annotation></semantics></math></div> <p>so that the moduli space of “differential vielbein fields” is the same as that of plain vielbein fields. But we nevertheless do gain differential information: consider an <a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a> on the tangent bundle, which is now given by a morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the moduli stack</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>T</mi><mi>X</mi></mrow></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_{T X} : X \to \mathbf{B}GL(n) \,. </annotation></semantics></math></div> <p>This is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> which locally on an atlas is given by the <a class="existingWikiWord" href="/nlab/show/Christoffel+symbols">Christoffel symbols</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>Γ</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mi>μ</mi></msub><mrow></mrow><msup><mrow></mrow> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_i = ((\Gamma_i)_\mu{}{}^{\alpha}{}_\beta) \in \Omega^1(U_i, \mathfrak{gl}(n)) \,. </annotation></semantics></math></div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>T</mi><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\nabla_{T X}</annotation></semantics></math>-twisted differential cocycle is now a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mi>V</mi></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>O</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mrow><mi>T</mi><mi>X</mi></mrow></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mrow><msup><mi>E</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub></mtd> <mtd><msub><mo>↙</mo> <mover><mstyle mathvariant="bold"><mi>Orth</mi></mstyle><mo stretchy="false">^</mo></mover></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\nabla_{V}}{\to}&& \mathbf{B}O_{conn} \\ & {}_{\mathllap{\nabla_{T X}}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\hat {\mathbf{Orth}}} \\ && \mathbf{B}GL(n)_{conn} } \,. </annotation></semantics></math></div> <p>In components over the atlas, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>V</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_V</annotation></semantics></math> is a “<a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>” given by local 1-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\omega_i \in \Omega^1(U_i, \mathfrak{o}(n))\}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>E</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>dR</mi></msub><msubsup><mi>E</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><msub><mi>E</mi> <mi>i</mi></msub><mi>Γ</mi><msubsup><mi>E</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex"> \omega_i = E_i d_{dR} E_i^{-1} + E_i \Gamma E_i^{-1} </annotation></semantics></math></div> <p>and the vielbeing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> now exhibits on each chart <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> the familiar relation between the components of the spin connection and the Christoffel-symbols:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><msup><mrow></mrow> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>=</mo><msubsup><mi>E</mi> <mi>i</mi> <mi>a</mi></msubsup><msub><mrow></mrow> <mi>ν</mi></msub><msub><mi>d</mi> <mi>dR</mi></msub><msubsup><mi>E</mi> <mi>i</mi> <mi>ν</mi></msubsup><msub><mrow></mrow> <mi>b</mi></msub><mo>+</mo><msubsup><mi>E</mi> <mi>i</mi> <mi>a</mi></msubsup><msub><mrow></mrow> <mi>ν</mi></msub><msub><mi>Γ</mi> <mi>i</mi></msub><msup><mrow></mrow> <mi>ν</mi></msup><msub><mrow></mrow> <mi>λ</mi></msub><msubsup><mi>E</mi> <mi>i</mi> <mi>λ</mi></msubsup><msub><mrow></mrow> <mi>b</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega{}^a{}_b = E_i^a{}_\nu d_{dR} E_i^\nu{}_b + E_i^a{}_\nu \Gamma_i {}^\nu{}_\lambda E_i^\lambda{}_b \,. </annotation></semantics></math></div> <h4 id="PullbackOfOrthogonalStructures">Pullback of orthogonal structures</h4> <p>It is a familiar fact that many fields in physics “naturally pull back”. For instance a <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> on a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbb{C}</annotation></semantics></math>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> between spactimes, there is the corresponding pullback function/field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>ϕ</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">f^* \phi : Y \to \mathbb{C}</annotation></semantics></math>.</p> <p>Similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{B}G_{conn}</annotation></semantics></math> a gauge field, as discussed above, it has naturally a pullback along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> given simply by forming the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>ϕ</mi><mo>:</mo><mi>Y</mi><mover><mo>→</mo><mi>f</mi></mover><mi>X</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">f^* \phi : Y \stackrel{f}{\to} X \stackrel{\phi}{\to} \mathbf{B}G_{conn}</annotation></semantics></math>.</p> <p>But the situation is a little different for <em>twisted</em> fields such as orthogonal structures/Riemannian metrics. If we think of a Riemannian metric as given by a non-degenerate rank-2 <a class="existingWikiWord" href="/nlab/show/tensor">tensor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then the problem is that, while its pullback along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> will always be a rank-2 tensor, it is not in general non-degenerate anymore – unless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>.</p> <p>This is also nicely formulated in the language used above, in a way that has a useful generalization when we come to higher twisted structures below: since the metric is encoded not just in a plain morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mrow><annotation encoding="application/x-tex">h : X \to \mathbf{B}O</annotation></semantics></math>, but one that fits into a triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>T</mi><mi>X</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>E</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>orth</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X && \stackrel{h}{\to} && \mathbf{B} O \\ & {}_{\mathllap{T X}}\searrow & \swArrow_{E}& \swarrow_{\mathrlap{orth}} \\ && \mathbf{B} GL } </annotation></semantics></math></div> <p>a simple precomposition with just a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> is not the right operation to send this triangle based on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to one based on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <p>But since this triangle is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>orth</mi></mstyle></mrow><annotation encoding="application/x-tex">(h,E) : T X \to \mathbf{orth}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}GL}</annotation></semantics></math>, it is clear that it <em>does</em> pull back precisely along refinements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> to a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}GL}</annotation></semantics></math>.</p> <p>Such a refinement is a commuting triangle of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>T</mi><mi>Y</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>T</mi><mi>X</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y &&\stackrel{f}{\to}&& X \\ & {}_{\mathllap{T Y}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{T X}} \\ && \mathbf{B} GL } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. But this is evidently the same as an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>Y</mi><mo>≃</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T Y \simeq f^* T X</annotation></semantics></math> between the tangent bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and the pullback of the tangent bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. And this exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>.</p> <h4 id="generalized_vielbein_fields_type_ii_geometry_generalized_cy_and_uduality">Generalized vielbein fields: type II geometry, generalized CY and U-duality</h4> <p>The above discussion of ordinary <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> fields is just a special case of an analogous discussion for general <a class="existingWikiWord" href="/nlab/show/reduction+of+structure+groups">reduction of structure groups</a>, giving rise to <a class="existingWikiWord" href="/nlab/show/generalized+vielbein">generalized vielbein</a> fields. Many geometric structures in string theory arise in this way, as indicated in the <a href="#TableOfTwists">table of twists</a>.</p> <p>As one more out of these examples, we discuss in the above language of twisted smooth cohomology how a <em><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></em> of <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> is the <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> of the <a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a> along the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d) \times O(d) \to O(d,d)</annotation></semantics></math>.</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathrm{O}(d) \times \mathrm{O}(d) \to \mathrm{O}(d,d) </annotation></semantics></math></div> <p>of those <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal transformations</a>, that preserve the positive definite part or the negative definite part of the <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a> of signature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d,d)</annotation></semantics></math>, respectively.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(d,d)</annotation></semantics></math> is presented as the group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi><mo>×</mo><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d \times 2d</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/matrix">matrices</a> that preserve the <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a> given by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi><mo>×</mo><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d \times 2d</annotation></semantics></math>-matrix</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>≔</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><msub><mi mathvariant="normal">id</mi> <mi>d</mi></msub></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">id</mi> <mi>d</mi></msub></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \eta \coloneqq \left( \array{ 0 & \mathrm{id}_d \\ \mathrm{id}_d & 0 } \right) </annotation></semantics></math></div> <p>then this inclusion sends a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>A</mi> <mo>−</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_+, A_-)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math>-matrices to the matrix</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>A</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>↦</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>A</mi> <mo>−</mo></msub></mtd> <mtd><msub><mi>A</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>A</mi> <mo>−</mo></msub></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>A</mi> <mo>−</mo></msub></mtd> <mtd><msub><mi>A</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>A</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (A_+ , A_-) \mapsto \frac{1}{\sqrt{2}} \left( \array{ A_+ + A_- & A_+ - A_- \\ A_+ - A_- & A_+ + A_- } \right) \,. </annotation></semantics></math></div> <p>This induces the corresponding morphism of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a>, which we denote</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{TypeII} : \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d)) \to \mathbf{B} \mathrm{O}(d,d) \,. </annotation></semantics></math></div> <p>Forming the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> now yields the local coefficient bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>\</mo><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ O(d) \backslash O(d,d) / O(d) &\to& \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d)) \\ && \downarrow^{\mathrlap{\mathbf{TypeII}}} \\ && \mathbf{B} \mathrm{O}(d,d) } \,, </annotation></semantics></math></div> <p>There is also a canonical embedding</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">GL</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>↪</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathrm{GL}(d) \hookrightarrow \mathrm{O}(d,d) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>.</p> <p>In the above matrix presentation this is given by sending</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>↦</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> a \mapsto \left( \array{ a & 0 \\ 0 & a^{-T} } \right) \,, </annotation></semantics></math></div> <p>where in the bottom right corner we have the <a class="existingWikiWord" href="/nlab/show/transpose+matrix">transpose</a> of the inverse matrix of the invertble matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>.</p> <p>Under this inclusion, the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> defines an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(d,d)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo>:</mo><mi>X</mi><mover><mo>→</mo><mrow><mi>T</mi><mi>X</mi></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">GL</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T X \oplus T^* X : X \stackrel{T X}{\to} \mathbf{B}\mathrm{GL}(d) \stackrel{}{\to} \mathbf{B} \mathrm{O}(d,d) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> canonically associated to this composite cocycle may canonically be identified with the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">T X \oplus T^* X</annotation></semantics></math>, and so we will refer to this cocycle by these symbols, as indicated. This is also called the <strong><a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Therefore we may canonically consider the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">T X \oplus T^* X</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{TypeII}</annotation></semantics></math>-structures, according to the general notion of <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a>.</p> <p>A <strong>type II generalized vielbein</strong> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mover><mo stretchy="false">(</mo><mo>˜</mo></mover><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>E</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\widetilde(T X \oplus T^* X)}{\to}&& \mathbf{B}(O(n) \times O(n)) \\ & {}_{\mathllap{T X \oplus T^* X}}\searrow &\swArrow_{E}& \swarrow_{\mathrlap{\mathbf{TypeII}}} \\ && \mathbf{B} O(n,n) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, hence a cocycle in the smooth <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle><mi>Struc</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E \in \mathbf{TypeII}Struc(X) \coloneqq \mathbf{H}_{/\mathbf{B} O(n,n)}(T X \oplus T^* X, \mathbf{TypeII}) \,. </annotation></semantics></math></div> <div class="un_prop"> <h6 id="proposition__definition">Proposition / Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>TypeII</mi></mstyle><mi mathvariant="normal">Struc</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{TypeII}\mathrm{Struc}(X)</annotation></semantics></math> is that of “generalized vielbein fields” on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, as considered for instance around equation (2.24) of (<a href="type+II+geometry#GMPW">GMPW</a>) (there only locally, though).</p> <p>In particular, its set of equivalence classes is the set of type-II generalized geometry structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Over a local <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>≃</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^d \simeq U_i \hookrightarrow X</annotation></semantics></math>, the most general such generalized vielbein (hence the most general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(d,d)</annotation></semantics></math>-valued function) may be parameterized as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msub><mi>e</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>e</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mo>+</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo>−</mo><msubsup><mi>e</mi> <mo>−</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo><mi>B</mi></mtd> <mtd><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mo>+</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo>−</mo><msubsup><mi>e</mi> <mo>−</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mi>e</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>e</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mo>+</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo>+</mo><msubsup><mi>e</mi> <mo>−</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo><mi>B</mi></mtd> <mtd><mo stretchy="false">(</mo><msubsup><mi>e</mi> <mo>+</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo>+</mo><msubsup><mi>e</mi> <mo>−</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> E = \frac{1}{2} \left( \array{ (e_+ + e_-) + (e_+^{-T} - e_-^{-T})B & (e_+^{-T} - e_-^{-T}) \\ (e_+ - e_-) - (e_+^{-T} + e_-^{-T})B & (e_+^{-T} + e_-^{-T}) } \right) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>e</mi> <mo>−</mo></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_+, e_- \in C^\infty(U_i, \mathrm{O}(d))</annotation></semantics></math> are thought of as two <a class="existingWikiWord" href="/nlab/show/vielbein">ordinary vielbein</a> fields, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is any smooth skew-symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math>-matrix valued function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>≃</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}^d \simeq U_i</annotation></semantics></math>.</p> <p>By an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(d) \times \mathrm{O}(d)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> this can always be brought into a form where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mo>+</mo></msub><mo>=</mo><msub><mi>e</mi> <mo>−</mo></msub><mo>=</mo><mo>:</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>e</mi></mrow><annotation encoding="application/x-tex">e_+ = e_- =: \tfrac{1}{2}e</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>e</mi></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msup><mi>B</mi></mtd> <mtd><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E = \left( \array{ e & 0 \\ - e^{-T}B & e^{-T} } \right) \,. </annotation></semantics></math></div> <p>The corresponding “generalized metric” over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>T</mi></msup><mi>E</mi><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msup><mi>e</mi> <mi>T</mi></msup></mtd> <mtd><mi>B</mi><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>e</mi></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msup><mi>B</mi></mtd> <mtd><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi></mrow></msup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>g</mi><mo>−</mo><mi>B</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>B</mi></mtd> <mtd><mi>B</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>B</mi></mtd> <mtd><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> E^T E = \left( \array{ e^T & B e^{-1} \\ 0 & e^{-1} } \right) \left( \array{ e & 0 \\ - e^{-T}B & e^{-T} } \right) = \left( \array{ g - B g^{-1} B & B g^{-1} \\ - g^{-1} B & g^{-1} } \right) \,, </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>≔</mo><msup><mi>e</mi> <mi>T</mi></msup><mi>e</mi></mrow><annotation encoding="application/x-tex"> g \coloneqq e^T e </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/metric">metric</a> (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>q</mi></msup><mo>≃</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}^q \simeq U_i</annotation></semantics></math> a smooth function with values in symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math>-matrices) given by the <a class="existingWikiWord" href="/nlab/show/vielbein">ordinary vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>.</p> </div> <h3 id="SpinStringFivebraneStructures"><strong>II)</strong> Spin-, String- and Fivebrane-structure</h3> <p>Above we have seen (<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian structure</a> given by lifts through the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} O(n) \to \mathbf{B} GL(n)</annotation></semantics></math>. Now we consider further lifts, through the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}O</annotation></semantics></math>. This encodes <em><a href="spin+structure#Higher">higher spin structures</a></em>.</p> <p>Where a <em><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></em> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is necessary to cancel a <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> of the <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>/<a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>, so the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a> <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> requires, in the absence of a <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>, a “<a href="#spin+structure#Higher">higher spin structure</a>”, called a <em><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a></em>. Further up in dimension, <em><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></em> in the absence of the gauge field involves a <em><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></em>.</p> <p>In the presence of a nontrivial <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a>, string structures are generalized to <em><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted string structure</a></em>, which in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> are part of the <em><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></em>. These we discuss <a href="#TwistedK">below</a>.</p> <h4 id="WhiteheadTower">Whitehead tower</h4> <p>The nature of higher spin structures is governed by what is called the <em><a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a></em> of the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <a class="existingWikiWord" href="/nlab/show/BO"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>O</mi> </mrow> <annotation encoding="application/x-tex">B O</annotation> </semantics> </math></a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>, where in each stage a <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> is removed <em>from below</em>. This is <a class="existingWikiWord" href="/nlab/show/duality">dual</a> to the <em><a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a></em>, where in each stage a homotopy group is added from above.</p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">B O</annotation></semantics></math> start out as</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">k =</annotation></semantics></math></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>O</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">\pi_k(B O) = </annotation></semantics></math></td><td style="text-align: left;">*</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td></tr> </tbody></table> <p>The Whitehead tower of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">B O</annotation></semantics></math> starts out as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mi>Whitehead</mi><mi>tower</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>B</mi><mi>Fivebrane</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>second</mi><mi>frac</mi><mi>Pontr</mi><mo>.</mo><mi>class</mi></mtd> <mtd><mi>B</mi><mi>String</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mn>8</mn></msup><mi>ℤ</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>first</mi><mi>frac</mi><mi>Pontr</mi><mo>.</mo><mi>class</mi></mtd> <mtd><mi>B</mi><mi>Spin</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>second</mi><mi>SW</mi><mi>class</mi></mtd> <mtd><mi>B</mi><mi>S</mi><mi>O</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>first</mi><mi>SW</mi><mi>class</mi></mtd> <mtd><mi>B</mi><mi>O</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>8</mn></mrow></msub><mi>B</mi><mi>O</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>4</mn></mrow></msub><mi>B</mi><mi>O</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mi>B</mi><mi>O</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>1</mn></mrow></msub><mi>B</mi><mi>O</mi><mo>≃</mo><mi>B</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mi>Postnikov</mi><mi>tower</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & Whitehead tower \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ second frac Pontr. class & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ first frac Pontr. class & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ second SW class & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ first SW class & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & Postnikov tower \\ & \downarrow \\ & B GL(n) } </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the stages are the <a class="existingWikiWord" href="/nlab/show/deloopings">deloopings</a> of</p> <p>… <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> classes are the <a class="existingWikiWord" href="/nlab/show/universal+characteristic+classes">universal characteristic classes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/first+Stiefel-Whitney+class">first Stiefel-Whitney class</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">w_1</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin+class">first fractional Pontryagin class</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}p_1</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin+class">second fractional Pontryagin class</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\tfrac{1}{6}p_2</annotation></semantics></math></p> </li> </ul> </li> <li> <p>every possible square in the above is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> square (using the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>).</p> </li> </ul> <p>For instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2</annotation></semantics></math> can be identified as such by representing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mi>B</mi><mi>O</mi><mo>≃</mo><mi>BO</mi><msub><mo stretchy="false">/</mo> <mrow><msub><mo>∼</mo> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">B O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n}</annotation></semantics></math> by a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> (see at <a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a>) between <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> so that then the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> (as discussed there) is given by an ordinary pullback. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sSet(S^2,-)</annotation></semantics></math> can be applied and preserves the pullback as well as the homotopy pullback, hence sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mi>B</mi><mi>O</mi></mrow><annotation encoding="application/x-tex"> B O \to \tau_{\leq 2} B O</annotation></semantics></math> to an isomorphism on connected components. This identifies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>SO</mi><mo>→</mo><msup><mi>B</mi> <mn>2</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">B SO \to B^2 \mathbb{Z}</annotation></semantics></math> as being an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on the second <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>. Therefore, by the <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a>, it is also an isomorphism on the <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2(-,\mathbb{Z}_2)</annotation></semantics></math>. Analogously for the other characteristic maps.</p> <p>In summary, more concisely, the tower is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>Fivebrane</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>B</mi> <mn>8</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>B</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>O</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>B</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mi>GL</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots \\ \downarrow \\ B Fivebrane \\ \downarrow \\ B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z} \\ \downarrow \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z} \\ \downarrow \\ B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2 \\ \downarrow \\ B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ B GL } \,, </annotation></semantics></math></div> <p>where each “hook” is a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> says that</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to lifting an <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">T X : X \to B O</annotation></semantics></math> to an <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> structure is the homotopy class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>w</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>O</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[w_1(T X)] \in H^1(B O, \mathbb{Z}_2)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mover><mo>→</mo><mrow><mi>T</mi><mi>X</mi></mrow></mover><mi>B</mi><mi>O</mi><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow></mover><mi>B</mi><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_1(T X) : X \stackrel{T X}{\to} B O \stackrel{w_1}{\to} B \mathbb{Z}_2</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/first+Stiefel-Whitney+class">first Stiefel-Whitney class</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to lifting an <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>SO</mi></mrow><annotation encoding="application/x-tex">T X : X \to B SO</annotation></semantics></math> to an <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> is the <a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mover><mo>→</mo><mrow></mrow></mover><mi>B</mi><mi>S</mi><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover><msup><mi>B</mi> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2(T X) : X \stackrel{}{\to} B S \stackrel{w_2}{\to} B^2 \mathbb{Z}_2</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to lifting a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi><mi>Spin</mi></mrow><annotation encoding="application/x-tex">X \to B Spin</annotation></semantics></math> to an <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a> is the <a class="existingWikiWord" href="/nlab/show/first+fractional+Pontryagin+class">first fractional Pontryagin class</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to lifting a <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi><mi>String</mi></mrow><annotation encoding="application/x-tex">X \to B String</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a> is the <a class="existingWikiWord" href="/nlab/show/second+fractional+Pontryagin+class">second fractional Pontryagin class</a>.</p> </li> </ul> <h4 id="NecessityOfSmoothRefinement">Necessity of smooth refinement</h4> <p>We will below consider a <em>smooth refinement</em> of the above Whitehead tower. Before we do so, here a few words on why we need to do this.</p> <p>One way to state the general problem is:</p> <ol> <li> <p><em>The <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of, say, the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> is <strong>not</strong> a</em><a class="existingWikiWord" href="/nlab/show/fine+moduli+space">fine moduli space</a><em>.</em></p> <p>Because, while homotopy classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>Spin</mi><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Maps(X, B Spin)_\sim</annotation></semantics></math> of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi><mi>Spin</mi></mrow><annotation encoding="application/x-tex">X \to B Spin</annotation></semantics></math> are in bijection with equivalence classes of <a class="existingWikiWord" href="/nlab/show/spin+bundles">spin bundles</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the homotopy classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Ω</mi><mi>B</mi><mi>Spin</mi><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub><mo>=</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Spin</mi><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Maps(X, \Omega B Spin)_\sim = Maps(X, Spin)_\sim</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> from the trivial map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo><mo>→</mo><mi>B</mi><mi>Spin</mi></mrow><annotation encoding="application/x-tex">X \to * \to B Spin</annotation></semantics></math> are <em>not</em> in general in bijection with the <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> of the trivial spin bundle: the latter form the set of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Spin</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X,Spin)</annotation></semantics></math>, not just the homotopy classes of these.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Maps(X, B(-))_\sim</annotation></semantics></math> does not give the right <a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a>; hence speaking about <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> in terms of just bare (as opposed to geometric) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> does not yield an admissible starting point for <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> (by <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Maps(X, B(-))_\sim</annotation></semantics></math> cannot distinguish a group from its <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a>, such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>↪</mo><msub><mi>GL</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">O \hookrightarrow GL_n</annotation></semantics></math>, and hence cannot see <a class="existingWikiWord" href="/nlab/show/vielbeins">vielbeins</a>, not <a class="existingWikiWord" href="/nlab/show/generalized+vielbeins">generalized vielbeins</a>, not <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>:</p> </li> </ol> <p>in terms of classifying spaces the entire discussion <a href="#VielbeinFields">of vielbein fields above</a> would collapse;</p> <p>higher analogs of this problem include for instance that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Maps(X, B(-))_\sim</annotation></semantics></math> cannot distinguish over 10-dimensional spacetime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> from a <a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a>;</p> <ol> <li>eventually an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on the space of fields is to be constructed as a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, or in fact as a smooth <em><a class="existingWikiWord" href="/nlab/show/flat+section">flat section</a></em> of a smooth <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a> on the space of fields – which requires some smooth structure on that space.</li> </ol> <p>These problems are all fixed by refining <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>Spin</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">B Spin \in \infty Grpd</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>∈</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} Spin \in Smooth \infty Grpd</annotation></semantics></math>.</p> <h4 id="SmoothRefinement">Smooth refinement</h4> <p>We considered <a href="#SmoothModuliStacks">above</a> the smooth refinement of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> to a <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>. While that works well, one can see on general grounds that this cannot provide a smooth refinement of the higher stages of the Whitehead tower, if one asks the refinement to preserve <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> theory. The problem is that a smooth stack is necessarily a smooth <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy 1-type</a> (even if its <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> is a higher type! see below), while the higher stages of the smooth Whitehead tower need to be smooth <a class="existingWikiWord" href="/nlab/show/homotopy+n-types">homotopy n-types</a>/<a class="existingWikiWord" href="/nlab/show/n-groupoids">n-groupoids</a> for higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>But there is an evident refinement of the above discussion to such <em>smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-types</em>.</p> <p>To that end we first need a good model for bare homotopy types. One observes that the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> functor embeds <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> into <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan</a> <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, as precisley those which are <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a>, meaning that only their 0-cells and 1-cells are non-trivial. Accordingly, a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> which is <a class="existingWikiWord" href="/nlab/show/coskeleton">(n+1)-coskeletal</a> may be regarded as an <em><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a></em> modelling a <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>, and hence a general Kan complex as an <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></em>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Groupoids</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>N</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Categories</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>KanComplexes</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>N</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>QuasiCategories</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>SimplicialSets</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Groupoids \\ & \swarrow && \searrow^{\mathrlap{N}} \\ Categories &&& & KanComplexes \\ & {}_{\mathllap{N}}\searrow && \swarrow \\ && QuasiCategories \\ && \downarrow \\ && SimplicialSets } \,. </annotation></semantics></math></div> <p>A morphism between groupoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a> precisely if it is an <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective functor</a> and a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>. This is equivalent to it inducing an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> classes / connected components, and on <a class="existingWikiWord" href="/nlab/show/automorphism+groups">automorphism groups</a>. This in turn is equivalent to it inducing an isomorphism on the 0th and the first <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math>.</p> <p>Accordingly, we say that a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> between <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> which induces an isomorphism on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>. These can be defined for general <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, and we say a morphism between these is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> if it induces such isomorphisms.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Groupoids</mi><mo>≃</mo><mi>SimplicialSets</mi><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mi>weak</mi><mspace width="thickmathspace"></mspace><mi>homotopy</mi><mspace width="thickmathspace"></mspace><mi>equivalences</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \infty Groupoids \simeq SimplicialSets[ \{ weak\;homotopy\;equivalences\}^{-1} ] </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> obtained by <a class="existingWikiWord" href="/nlab/show/simplicial+localization">localization</a> at the weak homotopy equivalences: <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> defines a Kan complex/ <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> by the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math>, and this establishes an <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">equivalence between</a> the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of topological spaces and simplicial sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mover><munderover><mo>→</mo><mo>≃</mo><mi>Sing</mi></munderover><mover><mo>←</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top \stackrel{\stackrel{{\vert-\vert}}{\leftarrow}}{ \underoverset{\simeq}{Sing}{\to} } sSet \,. </annotation></semantics></math></div> <p>In view of this, the <a href="#WhiteheadTower">above</a> <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> can be understood entirely as taking place in Kan complexes.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>disc</mi></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msup></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 U(1)_{disc} = \left\{ \array{ && \to \\ & \nearrow && \searrow \\ * &&\Downarrow^{a \in A}&& * \\ & \searrow && \nearrow \\ && \to } \right\} </annotation></semantics></math></div> <p>corresponds to the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(A,2)</annotation></semantics></math>.</p> <p>More generally, for each <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> provides a Kan complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Xi(A_\bullet)</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a> are the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet \to B_\bullet</annotation></semantics></math> is sent to a weak homotopy equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Ξ</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Xi(A_\bullet) \to \Xi(B_\bullet)</annotation></semantics></math>. In this sense <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is a <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian</a> generalization of chain complexes with quasi-isomorphisms inverted.</p> <p>Using this, we can easily state the generalization of the definition of smooth stacks from above: we obtain a <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <em><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-stacks</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} \coloneqq </annotation></semantics></math> <em><a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a></em> by considering simplicial sets parameterized over smooth manifolds and forcing <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise weak homotopy equivalences to become homotopy equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mi>stalkwise</mi><mspace width="thinmathspace"></mspace><mi>w</mi><mo>.</mo><mi>h</mi><mo>.</mo><mi>e</mi><mo>.</mo><msup><mo stretchy="false">}</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \coloneqq Func(SmthMfd^{op}, sSet)[\{stalkwise\, w.h.e.\}^{-1}] \,. </annotation></semantics></math></div> <p>For instance there is the smooth 2-stack</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 U(1) \in \mathbf{H} </annotation></semantics></math></div> <p>given by assigning to each test space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> on the (discrete) abelian group of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \to U(1)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>:</mo><mi>U</mi><mo>↦</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mrow><mi>c</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 U(1) : U \mapsto K( C^\infty(U, U(1)), 2) = \left\{ \array{ && \to \\ & \nearrow && \searrow \\ * &&\Downarrow^{c \in C^\infty(U,U(1))}&& * \\ & \searrow && \nearrow \\ && \to } \right\} \,. </annotation></semantics></math></div> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^2 U(1)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is equivalently a zig-zag <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mi>λ</mi></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \stackrel{\lambda}{\to} \mathbf{B}^2 U(1)</annotation></semantics></math> through the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a>. This now defines in degree 2</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi></mrow></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,1) &&\to&& (x,k) } \;\;\; \mapsto \;\;\; \array{ && * \\ & \nearrow &\Downarrow^{\lambda_{i j k})(x}& \searrow \\ * &&\to&& * } </annotation></semantics></math></div> <p>smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> on triple overlaps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \lambda_{i j k } : U_i \cap U_j \cap U_k \to U(1) </annotation></semantics></math></div> <p>and the condition on quadruple overlaps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k \cap U_l</annotation></semantics></math> says that they satisfy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lambda_{i j k} \lambda_{i k l} = \lambda_{j k l} \lambda_{i j k} \,. </annotation></semantics></math></div> <p>This now identifies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mo lspace="0em" rspace="thinmathspace">lamda</mo> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\lamda_{i j k})</annotation></semantics></math> with a cocycle in degree-2 <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>smooth</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(\lambda_{i j k})] \in H^2_{smooth}(X, U(1)) \,. </annotation></semantics></math></div> <p>This classifies a smooth <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle</a> / <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>.</p> <p>Generally we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>H</mi> <mi>smooth</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \mathbf{H}(X, \mathbf{B}^n U(1)) \simeq H^n_{smooth}(X, U(1)) </annotation></semantics></math></div> <p>and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a smooth manifold</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq H^{n+1}(X, \mathbb{Z}) \,. </annotation></semantics></math></div> <p>So apparently the smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> is a <em>smooth refinement</em> of the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z},n+1)</annotation></semantics></math>.</p> <p>This is made precise as follows.</p> <p><strong>Theorem</strong> There is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>:</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mi>Π</mi></mover><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> {\vert-\vert} : Smooth\infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{{\vert-\vert}}{\to} Top </annotation></semantics></math></div> <p>which is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint</a> to the functor that assigns <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stacks">constant ∞-stacks</a>. We call this the <em><a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a></em> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>.</p> <p>We say that a choice of lift of a diagram of bare homotopy types through geometric realization is a smooth <em>geomtric refinement</em>.</p> <p>For instance one finds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>≃</mo><msup><mi>B</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> {\vert \mathbf{B}^n U(1) \vert} \simeq B^n U(1)\simeq B^{n+1}\mathbb{Z} \simeq K(\mathbb{Z}, n+1) </annotation></semantics></math></div> <p>and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> is a smooth geometric refinement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z}, n+1)</annotation></semantics></math>.</p> <p>We now apply this to the above Whitehead tower.</p> <h4 id="SmoothWhitehead">Smooth Whitehead tower</h4> <p>We state the smooth refinement of the above Whitehead tower and then explain some aspects of how it is constructed.</p> <p><strong>Theorem</strong> There is a smooth geometric refinement of the <a href="#WhiteheadTower">above Whitehead tower</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+type">bare homotopy types</a> to a tower of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth homotopy types</a>/<a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-stacks">smooth ∞-stacks</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots \\ \downarrow \\ \mathbf{B} Fivebrane \\ \downarrow \\ \mathbf{B} String &\stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 U(1) \\ \downarrow \\ \mathbf{B} Spin &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow \\ \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 \\ \downarrow \\ \mathbf{B} GL } \,. </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n+1} U(1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the smooth <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, the smooth moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-stack of smooth <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundles</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} String</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/smooth+string+2-group">smooth string 2-group</a>, the moduli 2-stack of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundles">principal 2-bundles</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Fivebrane</annotation></semantics></math> is the delooping of the <a class="existingWikiWord" href="/nlab/show/smooth+fivebrane+6-group">smooth fivebrane 6-group</a>, the smooth moduli 6-stack of smooth fivebrane-<a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal 6-bundles</a>.</p> </li> </ul> <p>and where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tfrac{1}{2} \mathbf{p}_1</annotation></semantics></math> classifies the universal <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> and hence identifies it with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}String \to \mathbf{B}Spin</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\tfrac{1}{6} \mathbf{p}_2</annotation></semantics></math> classifies the universal <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle">Chern-Simons circle 7-bundle</a> and hence identifies it with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Fivebrane \to \mathbf{B}String</annotation></semantics></math>.</p> </li> </ul> <p>This is constructed using essentially the following three tools for presenting presheaves of higher groupoids:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo>:</mo><msub><mi>Ch</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mover><mo>→</mo><mo>≃</mo></mover><mi>sAb</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> \Xi : Ch_{\geq 0} \stackrel{\simeq}{\to} sAb \to sSet </annotation></semantics></math></div> <p>and its prolongation to presheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo>:</mo><mo stretchy="false">[</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>Ch</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mo stretchy="false">[</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sAb</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Xi : [SmthMfd^{op}, Ch_{\geq 0}) \stackrel{\simeq}{\to} [SmthMfd^{op},sAb] \to [SmthMfd^{op},sSet] </annotation></semantics></math></div> <p>allows to use presheaves of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> to present presheaves of strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids with strict abelian group structure.</p> <p>For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Ξ</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1) \simeq \Xi( C^\infty(-,U(1))[n] ) </annotation></semantics></math></div> <p>is equivalent to the image under the DK correspondence of the sheaf of chain complexes which is concentrated in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> on the group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-valued functions.</p> </li> <li> <p>Some <a href="Dold-Kan+correspondence#StatementGeneral">nonabelian generalizations</a> of the Dold-Kan correspondence allow to use chain complexes of not entirely abelian groups – <em><a class="existingWikiWord" href="/nlab/show/crossed+complexes">crossed complexes</a></em> – to present a few more classes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids. Notably nonabelian 2-term chain complexes,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> G_1 \stackrel{\delta}{\to} G_0 </annotation></semantics></math></div> <p>called <em><a class="existingWikiWord" href="/nlab/show/crossed+modules">crossed modules</a></em>, due to them being equipped with a compatible action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_0 \to Aut(G_1)</annotation></semantics></math>, serve to equivalently present <a class="existingWikiWord" href="/nlab/show/strict+2-groups">strict 2-groups</a>.</p> <p>For instance, one way to construct the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> above is via the crossed module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>Ω</mi><mo stretchy="false">^</mo></mover> <mo>*</mo></msub><mi>Spin</mi><mo>→</mo><msub><mi>P</mi> <mo>*</mo></msub><mi>Spin</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\hat \Omega_* Spin \to P_* Spin)</annotation></semantics></math> induced from the <a class="existingWikiWord" href="/nlab/show/Kac-Moody+central+extension">Kac-Moody central extension</a> of the <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>.</p> <p>For a given crossed module, the corresponding moduli 2-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(G_1 \stackrel{\delta}{\to} G_0)</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/2-morphism">2-cells</a> that look like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><msub><mi>g</mi> <mn>2</mn></msub><msub><mi>g</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><mi>h</mi><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}(G_1 \to G_0) = \left\{ \array{ && * \\ & {}^{\mathllap{g_1}}\nearrow &\Downarrow^{\mathrlap{h}}& \searrow^{\mathrlap{g_2}} \\ * &&\underset{\delta(h) g_2 g_1}{\to}&& } \;\; | \;\; g_1,g_2 \in G_0, h \in G_1 \right\} \,. </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{\leq n} \exp(\mathfrak{g})</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> yields the corresponding <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. For instance the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> above is also equivalently given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo>≃</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} String \simeq \tau_{\leq 2} \exp(\mathfrak{string})</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{string}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a>.</p> </li> </ol> <p>Using these tools, the stages of the above smooth Whitehead tower are constructed as follows:</p> <ul> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{w}_1 : \mathbf{B}O \to \mathbf{B} \mathbb{Z}_2</annotation></semantics></math> is directly induced from the canonical Lie group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>→</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">O \to \mathbb{Z}_2</annotation></semantics></math>.</p> </li> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{w}_2 : \mathbf{B}SO \to \mathbf{B}^2 \mathbb{Z}_2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is presented by the <a class="existingWikiWord" href="/nlab/show/infinity-anafunctor">zig-zag</a> of crossed modules</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mn>1</mn><mo>→</mo><mi>SO</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\mathbb{Z}_2 \to Spin) &\stackrel{}{\to}& (\mathbb{Z}_2 \to 1) \\ {}^{\mathllap{\simeq}}\downarrow \\ (1 \to SO) } </annotation></semantics></math></div></li> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)</annotation></semantics></math> is constructed (<a href="#FSSa">FSSa</a>) as the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the canonical <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">L-∞ 3-cocylce</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>μ</mi> <mn>3</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔰𝔬</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mu_3 : \mathbf{B}\mathfrak{so} \to \mathbf{B}^3 \mathbb{R}</annotation></semantics></math>.</p> </li> <li> <p>similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub><mo>≃</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>μ</mi> <mn>7</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{6}\mathbf{p}_2 \simeq \exp(\mu_7)</annotation></semantics></math> is the Lie integration of a canonical 7-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>μ</mi> <mn>7</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mu_7 : \mathbf{B}\mathfrak{string} \to \mathbf{B}^7 \mathbb{R}</annotation></semantics></math> (<a href="#FSSa">FSSa</a>).</p> </li> </ul> <h4 id="DifferentialRefinement">Differential refinement</h4> <p>While the <a href="#SmoothWhitehead">above</a> smooth refinement of the Whitehead tower already improves on the <a href="#WhiteheadTower">bare Whitehead tower</a> by remembering the correct spaces of <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a>, it still only sees “<a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> sectors” of <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> and <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a>, namely the underlying <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>. We add now the refinement from <em>smooth cohomology</em> to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> such as to encode the actual <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a> themselves. This differential cohomology in turn is naturally available in terms of <em>curvature twisted flat cohomology</em> or equivalently <em>curvature-twisted <a class="existingWikiWord" href="/nlab/show/local+systems">local systems</a></em>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>In order to get a feeling for what differential refinements of higher moduli stacks are going to be like, recall two structures that we have already seen above:</p> <ol> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> the smooth moduli stack of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> from <a href="#SpinConnection">above</a> is presented by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow></mrow></munder><mover><mo>→</mo><mrow></mrow></mover></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mi>A</mi><mover><mo>→</mo><mi>g</mi></mover><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>g</mi><mo>+</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G_{conn} = (C^\infty(-,G) \times \Omega^1(-, \mathfrak{g}) \stackrel{\overset{}{\to}}{\underset{}{\to}} \Omega^1(-, \mathfrak{g})) = \left\{ A \stackrel{g}{\to} (g^{-1} A g + g^{-1} d g) | A \in \Omega^1(-,\mathfrak{g}), g \in C^\infty(-,G) \right\} \,. </annotation></semantics></math></div> <p>In the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = U(1)</annotation></semantics></math> is abelian, this is the image under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> of the length 1 complex of sheaves of abelian groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>=</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1)_{conn} = [C^\infty(-) \stackrel{d_{dR}}{\to} \Omega^1(-)] \,. </annotation></semantics></math></div></li> <li> <p>The smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> is realized as the image under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> by the chain complex of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-,U(1))</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1) \simeq [ C^\infty(-,U(1)) \to 0 \to \cdots \to 0 ] \,. </annotation></semantics></math></div></li> </ol> <p>From the look of these expressions there is already a plausible candidate for the differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math>, the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> – it should be the <em><a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>=</mo><mrow><mo>[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^{n} U(1)_{conn} = \left[ C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(- ) \right] \,. </annotation></semantics></math></div> <p>For instance a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^2 U(1)_{conn}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Funct</mi><mo stretchy="false">(</mo><msup><mi>SmthMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Funct(SmthMfd^{op}, sSet)</annotation></semantics></math> and relative to a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> given by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">C(\{U_i\}) \to \mathbf{B}^2 U(1)_{conn}</annotation></semantics></math>, which is</p> <ul> <li> <p>on each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection 2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_i \in \Omega^2(U_i)</annotation></semantics></math>;</p> </li> <li> <p>on each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j</annotation></semantics></math> a 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_{i j} \in \Omega^1(U_i \cap U_j)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>B</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B_j - B_i = d_{dR} A_{i j} </annotation></semantics></math>;</p> </li> <li> <p>on each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k</annotation></semantics></math> a smooth functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_{i j k} \in C^\infty(U_i \cap U_j \cap U_k, U(1))</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>A</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>−</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>log</mi><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{i j} + A_{j k} - A_{i k} = d_{dR} log \phi_{i j k}</annotation></semantics></math> and such that on each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k \cap U_l</annotation></semantics></math> the equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow></msub><mo>=</mo><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>l</mi></mrow></msub><msub><mi>ϕ</mi> <mrow><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phi_{i j k} \phi_{i k l} = \phi_{i j l} \phi_{j k l}</annotation></semantics></math> holds.</p> </li> </ul> <p>Tthe <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is (in the absence of various possible twists, to be discussed), such a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^2 U(1)_{conn}</annotation></semantics></math>; and the <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a> (similarly in the absence of possible twists, to be discussed below) is given by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^3 U(1)_{conn}</annotation></semantics></math>.</p> <p>Such cocycles in <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne</a> <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a> define classes in <em><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{diff}^{n+1}(X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_{diff}^{n+1}(X) = \pi_0 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \,. </annotation></semantics></math></div> <p>There is an evident morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1) </annotation></semantics></math></div> <p>which <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgets</a> the connection data. We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math> is a <em>differential refinement</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math>.</p> <p>We now want to construct a differential refinement of the above smooth Whitehead tower, hence of the smooth <a class="existingWikiWord" href="/nlab/show/universal+characteristic+classes">universal characteristic classes</a> appearing in it. To do so, we now first provide a more conceptual way to think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math>, a way to obtain this more abstractly from fundamental principles.</p> <p>The key is that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-stacks</a> comes with a canonical notion of <em><a class="existingWikiWord" href="/nlab/show/local+system">local system</a></em> or <em><a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">flat ∞-connection</a></em>, and that we can <em>twist</em> this to find a notion of <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a>-twisted and hence non-flat <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connection</a>. The notion of local systems is induced from two basic derived adjoint functors that exist on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Disc</mi></mover></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> evaluates a presheaf on the point, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math> sends an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid to the presheaf constant on that value. We form the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mo>→</mo><mi>Γ</mi></mover><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mi>Disc</mi></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \flat : \mathbf{H} \stackrel{\Gamma}{\to} \infty Grpd \stackrel{Disc}{\to} \mathbf{H} \,, </annotation></semantics></math></div> <p>to be pronounced “flat”: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> a smooth homotopy type, we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\flat A</annotation></semantics></math> the corresponding <em>flat local coefficient</em> object.</p> <p>For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≃</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>disc</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>disc</mi></msub><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma \mathbf{B}G \simeq B (G_{disc}) = K(G_{disc}, 1)</annotation></semantics></math>, and so a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \flat \mathbf{B}G</annotation></semantics></math> is equivalently a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>disc</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \mathbf{B} (G_{disc})</annotation></semantics></math>, hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>disc</mi></msub></mrow><annotation encoding="application/x-tex">G_{disc}</annotation></semantics></math>-covering space, hence a flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a>.</p> <p>Generally, we say that a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>♭</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> X \to \flat A </annotation></semantics></math></div> <p>is an <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></em> or <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-valued <a href="cohesive+%28infinity,1%29-topos+--+structures#FlatDifferentialCohomology">flat ∞-connection</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>There is a canonical forgetful morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mo>♭</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">u : \flat A \to A</annotation></semantics></math> which forgets the flat connection: this is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">Disc</mo><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Disc \dashv \Gamma)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a>. Consider the coefficient object of those flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-connections whose underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> is trivial</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≔</mo><mrow><mo>{</mo><mo>∇</mo><mo>∈</mo><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \flat_{dR} \mathbf{B}G \coloneqq \left\{ \nabla \in \flat \mathbf{B}G | (u(\nabla) \simeq * ) \right\} \,. </annotation></semantics></math></div> <p>From the example of ordinary <a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> it is familiar that flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-connections on trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles are equivalently flat <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+differential+forms">Lie algebra valued differential forms</a>. Below we will see that for general <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groups">smooth ∞-groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \flat_{dR} \mathbf{B}G</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebroid-valued+differential+form">∞-Lie algebra valued differential forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Reading the above expression in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, its <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the counit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≔</mo><mo>*</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \flat_{dR} \mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G \,. </annotation></semantics></math></div> <p>By this construction and applying the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, there is a canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>G</mi><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\theta : G \to \flat_{dR} \mathbf{B}G</annotation></semantics></math>, hence a canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued form on any cohesive <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>: this identifies as the canonical <em><a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a></em> on the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>θ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>u</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ G &\to& * \\ {}^{\mathllap{\theta}}\downarrow && \downarrow \\ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ \downarrow && \downarrow^{\mathrlap{u}} \\ * &\to& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>For the special class of cases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = \mathbf{B}^n U(1)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle (n+1)-group</a> we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>≔</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">curv_{\mathbf{B}^n U(1)} \coloneqq \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1} U(1)</annotation></semantics></math> the <em>universal <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> class</em> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>.</p> <p>Due to the existence of the further functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi : \mathbf{H} \to \infty Grpd</annotation></semantics></math> discussed <a href="#SmoothWhitehead">above</a> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\flat : \mathbf{H} \to \infty Grpd</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> and hence commutes with homotopy pullback. This in turn implies that by forming one more homotopy fiber above, we obtain the following differential version of a universal local coefficient bundle:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>curv</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \flat \mathbf{B}^n U(1) &\to& \mathbf{B}^n U(1) \\ && \downarrow^{\mathrlap{curv}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1) } \,. </annotation></semantics></math></div> <p>By the general concept of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, we see that this defines a notion of <em>curvature twisted flat differential cohomology</em> – hence of differential cohomology.</p> <p>Specifically, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_X : X \to \Omega^{n+1}_{cl}(-)</annotation></semantics></math> a closed <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">F_X</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>curv</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{curv}</annotation></semantics></math>-cohomology is equivalently a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a> with that curvature</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>F</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>∇</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>curv</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>curv</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{}{\to}&& \mathbf{B}^n U(1) \\ & {}_{\mathllap{F}}\searrow &\swArrow_{\nabla}& \swarrow_{\mathrlap{\mathbf{curv}}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1)_{conn} } \;\; \in \mathbf{H}_{\flat \mathbf{B}^{n+1}U(1)}(F_X, \mathbf{curv}) \,. </annotation></semantics></math></div> <p>For varying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>curv</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{curv}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> identifies with <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>: the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n U(1)_{conn} &\to& \Omega^{n+1}_{cl}(-) \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{curv}{\to}& \mathbf{B}^{n+1} U(1)_{conn} } </annotation></semantics></math></div> <p>is presented, under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>, by the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a>, discussed above. This exhibits ordinary differential cohomology as the curvature-twisted flat cohomology</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mi>curv</mi></mstyle><msub><mi>Struc</mi> <mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}_{diff}(X, \mathbf{B}^n U(1)) = \mathbf{curv}Struc_{\Omega^{n+1}}(X) \,. </annotation></semantics></math></div> <p>Using this geometric-homotopy-type theoretic description of ordinary differential cohomology, we obtain now a natural notion of differential refinement of smooth <a class="existingWikiWord" href="/nlab/show/universal+characteristic+classes">universal characteristic classes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{c} : \mathbf{B}G \to \mathbf{B}^{n+1} U(1)</annotation></semantics></math>. We say that a <em>differential refinement</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> fitting into a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>♭</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow></mover></mtd> <mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>curv</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \flat \mathbf{B}G &\stackrel{\flat \mathbf{c}}{\to}& \flat \mathbf{B}^{n+1} U(1) \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\stackrel{\hat \mathbf{c}}{\to}& \mathbf{B}^{n+1} U(1)_{conn} \\ \downarrow && \downarrow^{\mathrlap{curv}} \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^{n+1} U(1) } </annotation></semantics></math></div> <p>that factors the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> square of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>.</p> <h4 id="DifferentialWhitehead">Differential Whitehead tower</h4> <p><strong>Theorem</strong> (<a href="#SSSa">SSSa</a>, <a href="#FSS">FSSa</a>) There exists a smooth differential refinement of the <a href="#WhiteheadTower">Whitehead tower of BO</a> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Fivebrane</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>S</mi><msub><mi>O</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>O</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots \\ \downarrow \\ \mathbf{B} Fivebrane_{conn} \\ \downarrow \\ \mathbf{B} String_{conn} &\stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow \\ \mathbf{B} Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow \\ \mathbf{B} S O_{conn} &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow \\ \mathbf{B} O_{conn} &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 \\ \downarrow^{} \\ \mathbf{B} GL_{conn} } \,. </annotation></semantics></math></div> <p>This construction is a joint generalization of <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> and <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> and <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <p>For instance</p> <ul> <li> <p>the differential refinement of the first fractional Pontryagin class above yields the action functional</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(i S_{\tfrac{1}{2}\mathbf{p}_1}) : [\Sigma, \mathbf{B}Spin_{conn}] \stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to} [\Sigma, \mathbf{B}^{3}U(1)_{conn}] \stackrel{\exp(i \int_\Sigma(-))}{\to} U(1) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/Spin">Spin</a>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>, refined to the integrated off-shell <a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a> of the theory;</p> </li> <li> <p>the differential refinement of the second fractional Pontryagin class above yields the action functional</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(i S_{\tfrac{1}{6}\mathbf{p}_2}) : [\Sigma, \mathbf{B}String_{conn}] \stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to} [\Sigma, \mathbf{B}^{7}U(1)_{conn}] \stackrel{\exp(i \int_\Sigma(-))}{\to} U(1) </annotation></semantics></math></div> <p>of 7-dimensional Chern-Simons theory on nonabelian String 2-form fields (<a href="#FSSb">FSSb</a>)</p> </li> </ul> <p>We indicate briefly how this is constructed.</p> <p>(…)</p> <h3 id="Anomalies"><strong>Interlude)</strong> Anomaly line bundle on smooth moduli stacks of fields</h3> <p>Before coming to the description in <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a> below, we make some introductory comments on the general origin of twisted differential structures in higher gauge theory, following (<a href="#Freed">Freed</a>). We add some stacky aspects to that and explain why.</p> <p>In summary, we discuss how the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of higher gauge theory in the presence of electric and magnetic charge is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>higher</mi><mspace width="thickmathspace"></mspace><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\nabla_{higher\;gauge\;anomaly}</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, \mathbf{Fields}]</annotation></semantics></math> of field configurations on a given <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, exhibited by a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>higher</mi><mspace width="thickmathspace"></mspace><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub><mo>≔</mo><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>mag</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mover><mo>→</mo><mrow></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \nabla_{higher\;gauge\;anomaly} \coloneqq \exp(2 \pi i \int_X \hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag} ) : [X, \mathbf{Fields}] \stackrel{}{\to} \mathbf{B} U(1)_{conn} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>mag</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>dim</mi><mi>X</mi><mo>+</mo><mn>2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag} : \mathbf{Fields} \to \mathbf{B}^{dim X +2} U(1)_{conn}</annotation></semantics></math> is the differential characteristic morphism induced by the <a class="existingWikiWord" href="/nlab/show/differential+cup+product">differential cup product</a> (<a href="#FSSd">FSSd</a>) of universal electric and magnetic currents, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(2\pi i \int_X(-))</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a> refined to smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks (this is the “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>dim</mi><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(dim X)+1</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/schreiber/show/infinity-Chern-Simons+theory">infinity-Chern-Simons theory</a>” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">\hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag}</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> 1).</p> <h4 id="higher_gauge_fields_in_the_presence_of_magnetic_charge_current">Higher gauge fields in the presence of magnetic charge current</h4> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">Gauge theory</a> starts maybe with <a class="existingWikiWord" href="/nlab/show/James+Clerk+Maxwell">Maxwell</a> around 1850, who discovered, in modern language, that the <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is encoded in a closed <a class="existingWikiWord" href="/nlab/show/differential+form">differential 2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in \Omega^2_{cl}(X)</annotation></semantics></math>.</p> <p>Then in the 1930s <a class="existingWikiWord" href="/nlab/show/Paul+Dirac">Dirac</a>‘s <a href="electromagnetic+field#ChargeQuantization">famous argument</a> showed that more precisely – <em>in the absence</em> of, or <em>outside of</em> the <a class="existingWikiWord" href="/nlab/show/support">support</a> of <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <a class="existingWikiWord" href="/nlab/show/current">current</a> – this 2-form is the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a>, a 2-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <p>In view of this the <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge</a> <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of configurations of the electromagnetic field on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> form the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>τ</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^2_{diff}(X) \coloneqq \tau_0 \mathbf{H}(X, \mathbf{B}U(1)_{conn}) </annotation></semantics></math></div> <p>of differential cohomology classes in degree 2 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Dirac’s argument works outside the support of the magnetic current, where the situation is comparatively easier to handle. But the twists and anomalies that we are concerned with here arise when one completes Dirac’s argument, and generalizes the model of the electromagnetic field to exist also over parts of spacetime where the magnetic current is non-trivial (<a href="#Freed">Freed</a>). Among other things the following shows that twists by higher bundles and differential cohomology is not just something that arises in string theory, but is already present in dear-old electromagnetism.</p> <p>To see what happens in that general case, notice that the original <a class="existingWikiWord" href="/nlab/show/Maxwell+equations">Maxwell equations</a> on the <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a>/<a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form of the electromagnetic field are:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>F</mi><mo>=</mo><msub><mi>J</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">d_{dR} F = J_{mag}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <a class="existingWikiWord" href="/nlab/show/current">current</a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>⋆</mo><mi>F</mi><mo>=</mo><msub><mi>J</mi> <mi>el</mi></msub></mrow><annotation encoding="application/x-tex">d_{dR} \star F = J_{el}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> <a class="existingWikiWord" href="/nlab/show/current">current</a>)</p> </li> </ol> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a> for the given <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> (the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <ol> <li> <p>The first one is <em><a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a></em>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>mag</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">J_{mag} = 0</annotation></semantics></math> it just expresses that the cuvature 2-form is closed, which is part of the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math> is a differential cocycle, so it is satisfied by all kinematic field configurations, meaning: all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{n+1}_{diff}(X)</annotation></semantics></math>.</p> </li> <li> <p>The second is <em><a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a></em>, being the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of the system. The configurations that satisfy this form the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> (<a class="existingWikiWord" href="/nlab/show/BV-BRST+complex">BV-BRST complex</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>↪</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">P \hookrightarrow [X, \mathbf{B}U(1)_{conn}]</annotation></semantics></math> of the theory. For our purposes here this will not concern us, since the anomalies and twists are kinematic in nature, we work “off-shell”.</p> </li> </ol> <p>While for experimentally observed electromagnetism it is consistent to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>mag</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">J_{mag} = 0</annotation></semantics></math>, this is not the case for general gauge theories, notably not for <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a>, as we discuss in a moment. There the gauge field and the field of gravity induce a non-vanishing “fivebrane magnetic current”</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>mag</mi> <mrow><mi>NS</mi><mn>5</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo>,</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">J^{NS5}_{mag}(\phi_{gr}, \phi_{ga}) = \langle F_\omega \wedge F_\omega\rangle -\langle F_A \wedge F_A\rangle</annotation></semantics></math></li> </ul> <p>But for a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{n+1}_{diff}(X)</annotation></semantics></math>, the curvature is necessarily closed, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>F</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{dR} F = 0</annotation></semantics></math>. So there must be another way to refine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>F</mi><mo>=</mo><msub><mi>J</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">d F = J_{mag}</annotation></semantics></math> to differential cohomology.</p> <p>(Notice for later that the natural home of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">J_{mag}</annotation></semantics></math> is not plain <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>, but <a class="existingWikiWord" href="/nlab/show/compactly+supported+cohomology">compactly supported cohomology</a>. The equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>F</mi><mo>=</mo><msub><mi>J</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">d_{dR} F = J_{mag}</annotation></semantics></math> is a trivialization of the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">J_{mag}</annotation></semantics></math> in de Rham cohomology, but not in general a trivialization of the magnetic current as an entity living in compactly supported cohomology.)</p> <p>Consider therefore now the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>τ</mi> <mn>1</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}^{n+1}_{diff}(X) \coloneqq \tau_1 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \,, </annotation></semantics></math></div> <p>whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in degree-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math> differential cohomology: <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>-<a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are equivalence classes of <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> between these, hence equivalence classes of morphisms of higher bundles <em>with connection</em>.</p> </li> </ul> <p>This “<a class="existingWikiWord" href="/nlab/show/categorification">categorifies</a>” the cohomology set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{n+1}_{diff}(X)</annotation></semantics></math> in that the letter is its <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a>: the set of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> classes of objects.</p> <p>For instance if differential cohomology is modeled by the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> with <a class="existingWikiWord" href="/nlab/show/differential">differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo>±</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">D = d_{dR} \pm \delta</annotation></semantics></math>, then a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>α</mi><mo stretchy="false">^</mo></mover><mo>:</mo><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>→</mo><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat \alpha : \hat F_1 \to \hat F_2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{H}_{diff}^{n+1}(X)</annotation></semantics></math> is a Deligne <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mover><mi>α</mi><mo stretchy="false">^</mo></mover><mo>=</mo><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>−</mo><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">D \hat \alpha = \hat F_2 - \hat F_1</annotation></semantics></math>.</p> <p>Or in terms of our smooth moduli stacks, this is a homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><msup><mo>⇓</mo> <mover><mi>α</mi><mo stretchy="false">^</mo></mover></msup></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{\hat F_1}} \\ X &\Downarrow^{ \hat \alpha }& \mathbf{B}^n U(1)_{conn} \\ & \searrow \nearrow_{ \mathrlap{\hat F_2} } } \,. </annotation></semantics></math></div> <p>Notice that, since morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{H}^{n+1}_{diff}(X)</annotation></semantics></math> preserve the <a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">higher connection</a>, a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex"> 0 \to \hat F </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{H}^{n+1}_{diff}(X)</annotation></semantics></math> is a <em>flat section</em> of the corresponding circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-bundle, while a morphim</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex"> B \to \hat F </annotation></semantics></math></div> <p>for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \in \Omega^n(X) \hookrightarrow \mathcal{H}^{n+1}_{diff}(X)</annotation></semantics></math> is a possibly non-flat section, hence a section just of the underlying <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>: it exhibits the fact that if the underlying bundle has a section, then the connection is equivalently given by a globally defined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>(Beware of this subtlety when comparing with (<a href="#Freed">Freed</a>): a differential as on the fifth line of <a href="arxiv.org/pdf/hep-th/0011220v2.pdf#page=8">p. 8</a> there may change the curvature by an exact term, hence may not preserve the connection, in contrast to the coboundaries further below on that page and on <a href="arxiv.org/pdf/hep-th/0011220v2.pdf#page=9">p. 9</a>, which are the ones we are considering here.)</p> <p>(Another reason for considering the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{H}^{n+1}_{diff}(X)</annotation></semantics></math> is that it is needed in order to construct the <a class="existingWikiWord" href="/nlab/show/quadratic+refinement">quadratic refinement</a> of the <a class="existingWikiWord" href="/nlab/show/secondary+intersection+pairing">secondary intersection pairing</a> that defines the partition function of <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> (<a href="#HopkinsSinger">Hopkins-Singer</a>). This underlies the discussion of flux quantization <a href="#SuGraFluxQuantization">below</a>.)</p> <p>Using this, we may improve the definition of the electromagnetic field on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: take it to be a non-flat section</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mover><msub><mi>c</mi> <mi>mag</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat \mathbf{c} \stackrel{\hat F}{\to} c_{mag} \,. </annotation></semantics></math></div> <p>of a <em>magnetic charge <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo>∈</mo><msubsup><mi>ℋ</mi> <mi>diff</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat \mathbf{c} \in \mathcal{H}^{3}_{diff}(X)</annotation></semantics></math>. Equivalently, in terms of the corresponding classifying morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> this is a homotopy in a diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>c</mi> <mi>mag</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mover><mi>F</mi><mo stretchy="false">^</mo></mover></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mrow></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\to& \\ {}^{\mathllap{c_{mag}}}\downarrow &\swArrow_{\hat F}& \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ \Omega^2(-) &\stackrel{}{\hookrightarrow}& \mathbf{B}^2 U(1)_{conn} } \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math> is given by a Deligne cochain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g_{i j}, A_i)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> by a cochain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>,</mo><msub><mi>γ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>β</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c_{i j k}, \gamma_{i j}, \beta_i)</annotation></semantics></math> then this means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>D</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>δ</mi><mi>g</mi><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>A</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>log</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>c</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mspace width="thickmathspace"></mspace><mo>−</mo><msub><mi>γ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>c</mi> <mi>mag</mi></msub><mo>−</mo><msub><mi>β</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} D (g_{i j}, A_i) & \coloneqq ((\delta g)_{i j k},\; A_j - A_i + d_{dR} log g_{i j},\; d_{dR} A_i) \\ & = ( c_{i j k}^{-1},\; -\gamma_{i j}, \; c_{mag} - \beta_i ) \end{aligned} </annotation></semantics></math></div> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math> is a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+infinity-bundle">twisted bundle</a></em> with <em>twisted curvature</em> being</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≔</mo><mi>d</mi><msub><mi>A</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>β</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \coloneqq d A_i + \beta_i \,. </annotation></semantics></math></div> <p>This now correspondingly has a twisted <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a>, which is precisely so that it solves the first <a class="existingWikiWord" href="/nlab/show/Maxwell+equation">Maxwell equation</a> in the presence of magnetic current: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>F</mi><mo>=</mo><msub><mi>J</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">d_{dR} F = J_{mag}</annotation></semantics></math>.</p> <p>While we have been discussing this here for ordinary electromagnetism, this is precisely the mechanism by which also the higher cases will work: for the <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic</a> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> the analogy is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>twisted</mi><mspace width="thickmathspace"></mspace><mi>curvature</mi></mtd> <mtd></mtd> <mtd><msub><mi>d</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>potential</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mi>twist</mi></mtd></mtr> <mtr><mtd><mi>F</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>A</mi> <mi>i</mi></msub></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo></mtd> <mtd><msub><mi>β</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><mi>general</mi><mi>case</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msub><mi>H</mi> <mi>i</mi></msub></mtd> <mtd><mo>=</mo></mtd> <mtd><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>B</mi> <mi>i</mi></msub></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo></mtd> <mtd><mi>CS</mi><mo stretchy="false">(</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mi>heterotic</mi><mspace width="thickmathspace"></mspace><mi>sugra</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ twisted\;curvature & & d_{dR}(gauge\;potential) && twist \\ F &=& d_{dR} A_i &+& \beta_i && general case \\ \\ H_i &=& d_{dR} B_i &+& CS(\omega_i) - CS(A_i) && heterotic\;sugra } \,. </annotation></semantics></math></div> <p>Before we get there, we need to observe that working with the 1-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℋ</mi> <mi>diff</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{H}^{n+1}_{diff}(X)</annotation></semantics></math> is not sufficient. We discuss now that we necessarily need the full <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> and moreover its smooth refinement to the full <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth n-stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, \mathbf{B}^n U(1)_{conn}]</annotation></semantics></math> in order to capture the physics situation.</p> <h4 id="gauge_transformations">Gauge transformations</h4> <p>To see that we need the full higher groupoid, just consider the question: what is a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between twisted electromagnetic fields, that are now identified with morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mover><msub><mi>c</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">\hat \mathbf{c} \stackrel{\hat F}{\to} c_{mag}</annotation></semantics></math> as above? Clearly, for this we need the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of differential cocycles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>2</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_2 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})</annotation></semantics></math></p> <p>to next say that the equivalenc class of a gauge transformation of twisted fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">^</mo></mover><mo>:</mo><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>→</mo><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat a : \hat F_1 \to \hat F_2</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mtd> <mtd><msup><mo>⇓</mo> <mover><mi>α</mi><mo stretchy="false">^</mo></mover></msup></mtd> <mtd><msub><mi>J</mi> <mi>mag</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mi>τ</mi> <mn>2</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{\hat F_1}} \\ \hat \mathbf{c} &\Downarrow^{\hat \alpha}& J_{mag} \\ & \searrow \nearrow_{\mathrlap{\hat F_2}} } \;\;\; \in \tau_{2} \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \,. </annotation></semantics></math></div> <p>That we moreover need the full <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth n-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, \mathbf{B}^n U(1)_{conn}]</annotation></semantics></math> has several reasons, we discuss three. The third one of these is related to the higher gauge anomalies proper.</p> <h4 id="magnetic_current_induced_by_background_fields">Magnetic current induced by background fields</h4> <p>The magnetic twist <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> will depend on other field configurations that induce magnetic charge. So it is not a constant, but <em>varies with the fields</em>.</p> <p>(In fact, only this way is it a non-trivial stricture, for if both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat c</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">c_{mag}</annotation></semantics></math> are independent of the fields, the above definition of the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math>s is <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalent</a> to that of untwisted electromagnetic fields: because <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> only depend on the connected component of the base points, up to equivalence.)</p> <p>Let therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mi>bg</mi></msup></mrow><annotation encoding="application/x-tex">G^{bg}</annotation></semantics></math> be the gauge group of another <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>G</mi> <mi>conn</mi> <mi>bg</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{B}G^{bg}_{conn}</annotation></semantics></math> be its moduli stack of gauge field configurations. If a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mi>bg</mi></msup></mrow><annotation encoding="application/x-tex">G^{bg}</annotation></semantics></math>-field <em>induces</em> magnetic current, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> must depend on the fields, hence it should be a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>G</mi> <mi>conn</mi> <mi>bg</mi></msubsup><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [X, \mathbf{B}G^{bg}_{conn}] \to [X, \mathbf{B}^2 U(1)_{conn}] </annotation></semantics></math></div> <p>between these spaces of fields, and it should be a <em>smooth</em> such map. Moreover, in general this is not expected to depend specifically the specific choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, but just on the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mi>bg</mi></msup></mrow><annotation encoding="application/x-tex">G^{bg}</annotation></semantics></math>-fields in general, so it should be given just by postcompositon</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>G</mi> <mi>conn</mi> <mi>bg</mi></msubsup><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \hat \mathbf{c} : \mathbf{B}G^{bg}_{conn} \to \mathbf{B}^2 U(1)_{conn} </annotation></semantics></math></div> <p>with a universal smooth map on moduli. With that given, the above picture for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math>-twisted higher electric field becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>bg</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>G</mi> <mi>conn</mi> <mi>bg</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mover><mi>F</mi><mo stretchy="false">^</mo></mover></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{\phi_{bg}}{\to}& \mathbf{B}G^{bg}_{conn} \\ \downarrow &\swArrow_{\hat F}& \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ \Omega^2(-) &\stackrel{}{\to}& \mathbf{B}^n U(1)_{conn} } \,. </annotation></semantics></math></div> <p>We can then subsume all this and consider the smooth collection of all such twisting background fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mi>bg</mi></msup></mrow><annotation encoding="application/x-tex">\phi^{bg}</annotation></semantics></math> and twisted gauge fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math>. By general reasoning, this is given by the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> that universally completes the above diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>bg</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>G</mi> <mi>conn</mi> <mi>bg</mi></msubsup><msup><mrow></mrow> <mi>univ</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mrow><msup><mover><mi>F</mi><mo stretchy="false">^</mo></mover> <mi>univ</mi></msup></mrow></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{Fields} &\stackrel{\phi_{bg}}{\to}& \mathbf{B}G^{bg}_{conn}{}^{univ} \\ \downarrow &\swArrow_{\hat F^{univ}}& \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ \Omega^2(-) &\stackrel{}{\to}& \mathbf{B}^n U(1)_{conn} } </annotation></semantics></math></div> <h4 id="infinitesimal_moduli_stacks_brst_complexes">Infinitesimal moduli stacks: BRST complexes</h4> <p>So far we are talking about <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> and <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a> on which we are evaluating an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> (see below). Eventually one wants to <a class="existingWikiWord" href="/nlab/show/quantization">quantize</a> such a setup. There are two issues with this: first of all the action functional needs to be well-defined in the first place, we get to in the next point. But second, once we have a well-defined action functional on gauge fields, the only way to quantize this is to invove <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>: we need the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the gauge fields.</p> <p>or ordinary gauge theory this is the <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> of the smooth version <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, \mathbf{B}G_{conn}]</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>BRST</mi><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mi>Lie</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> BRST = C^\infty Lie([X, \mathbf{B}G_{conn}]) \,. </annotation></semantics></math></div> <p>Similarly for higher gauge theory it is the <a class="existingWikiWord" href="/nlab/show/L-infinity+algebroid">L-infinity algebroid</a>.</p> <h4 id="extended_higher_chernsimonstype_functionals">Extended higher Chern-Simons-type functionals</h4> <p>The anomaly line bundle to be discussed in a moment <a href="GaugeInteractionAndChargeAnomaly">below</a> is a special case of a general construction in extended “<a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a>”. So before getting to that special case, we indicate here the general pattern.</p> <p>The <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is traditionally taken to be simply a <a class="existingWikiWord" href="/nlab/show/function">function</a>, for a gven compact 3-manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_3</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>cs</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>G</mi><msub><mi>Bund</mi> <mo>∇</mo></msub><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp( i S_{cs}) : G Bund_\nabla(\Sigma_3) \to U(1) </annotation></semantics></math></div> <p>on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_3</annotation></semantics></math>. This perspective can be refined.</p> <p>First of all, since this function is <a class="existingWikiWord" href="/nlab/show/gauge+invariance">gauge invariant</a> we may think of it as being defined on the full moduli stack</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>cs</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>3</mn></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp( i S_{cs}) : [\Sigma_3, \mathbf{B}G_{conn}] \to U(1) \,. </annotation></semantics></math></div> <p>This also exhibits the smoothness of the action.</p> <p>An important construction in <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is the <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of this action functional in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub><mo>=</mo><msub><mi>Σ</mi> <mn>2</mn></msub><mo>×</mo><mi>Interval</mi></mrow><annotation encoding="application/x-tex">\Sigma_3 = \Sigma_2 \times Interval</annotation></semantics></math> , which yields a holomorphic line bundle with connection on the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> of the theory, which may be identified with the space of flat connections over a <em>2-dimensional</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math>. Generally, one expects to see a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle k-bundle with connection</a> assigned to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p><a href="#DifferentialWhitehead">Above</a> we had already seen such a structure in top codimension: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Sigma = *</annotation></semantics></math> is the point, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">[\Sigma, \mathbf{B}G_{conn}] \simeq [*, \mathbf{B}G_{conn}] \simeq \mathbf{B}G_{conn}</annotation></semantics></math>. So there should be a cricle 3-bundle with connection on this moduli stack. Taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">G = Spin</annotation></semantics></math>, for definiteness, then the differential first fractional Pontryagin class from <a href="#DifferentialWhitehead">above</a> is precisely of this form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spinn</mi> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tfrac{1}{2} \hat \mathbf{p}_1 : \mathbf{B} Spinn_{conn} \to \mathbf{B}^3 U(1) \,. </annotation></semantics></math></div> <p>And indeed, as stated there, this induces the Chern-Simons action functional itself. Indeed, it induces a whole tower of higher circle bundles, in each codimension:</p> <p>The operation of fiber integration of differential forms extends to an operation of <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a>, which in turn, as discussed there, extends to a morphism of smooth moduli stacks of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp(2 \pi i \int_{\Sigma_k}(-)) : [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k} U(1)_{conn} \,. </annotation></semantics></math></div> <p>If now</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} </annotation></semantics></math></div> <p>is any universal differential characteristic map, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> is compact closed of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, then the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow></msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \exp(2 \pi i \int_{\Sigma_k} \hat \mathbf{c}) : [\Sigma_k, \mathbf{B}G_{conn}] \stackrel{[\Sigma_k, \hat \mathbf{c}]}{\to} [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_k} (-))}{\to} \mathbf{B}^{n-k} U(1)_{conn} </annotation></semantics></math></div> <p>is a “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-extended action functional”.</p> <p>An important class of <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theories">∞-Chern-Simons theories</a> arising this way come from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> that are <a class="existingWikiWord" href="/nlab/show/cup+product">cup products</a> of two other differential classes (<a href="#FSSd">FSSd</a>). For instance in ordinary abelian <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> one starts with the tautological differential class</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>DD</mi></mstyle><mo stretchy="false">^</mo></mover><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \hat \mathbf{DD} : \mathbf{B}^{2k+1} U(1)_{conn} \to \mathbf{B}^{2k+1} U(1)_{conn} </annotation></semantics></math></div> <p>and then forms its <a class="existingWikiWord" href="/nlab/show/differential+cup+product">differential cup product</a> (<a href="#FSSd">FSSd</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo>≔</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mrow></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>×</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mrow><mover><mstyle mathvariant="bold"><mi>DD</mi></mstyle><mo stretchy="false">^</mo></mover><mo>∪</mo><mover><mstyle mathvariant="bold"><mi>DD</mi></mstyle><mo stretchy="false">^</mo></mover></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat \mathbf{c} \coloneqq \mathbf{B}^{2k+1} U(1)_{conn} \stackrel{}{\to} \mathbf{B}^{2k+1} U(1)_{conn} \times \mathbf{B}^{2k+1} U(1)_{conn} \stackrel{\hat \mathbf{DD} \cup \hat \mathbf{DD}}{\to} \mathbf{B}^{4k + 3} U(1)_{conn} \,. </annotation></semantics></math></div> <p>The action functional induced by this is that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(4k+3)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> which sends those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2k+1)</annotation></semantics></math>-form fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> whose underlying bundle happens to be trivial to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow></msub></mrow></msub><mi>C</mi><mo>∧</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(2 \pi i\int_{\Sigma_{4k+3}} C \wedge d_{dR} C)</annotation></semantics></math>.</p> <p>The anomaly line bundle which we now turn to arises in this kind of way, only for the slightly more general case that the <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> involved is not given by a differential square, but by a genuine differential cup of two different cocycles: the electric and the magnetic differential cocycles.</p> <h4 id="GaugeInteractionAndChargeAnomaly">Gauge interaction and the charge anomaly</h4> <p>We discuss now how the action functional of the (higher) gauge theory in the presence of <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> <a class="existingWikiWord" href="/nlab/show/current">current</a> and <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <a class="existingWikiWord" href="/nlab/show/current">current</a> has in general an <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">anomaly</a>, but this anomaly exhibits itself, in traditional language, as something living over <em>families</em> of gauge fields. But by the formula for the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> of sheaves/stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo>:</mo><mi>U</mi><mo>↦</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>U</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>G</mi> <mi>conn</mi> <mi>bg</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>background</mi><mspace width="thickmathspace"></mspace><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>fields</mi><mspace width="thickmathspace"></mspace><mi>on</mi><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>U</mi><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mi>U</mi><mo>−</mo><mi>parameterized</mi><mspace width="thickmathspace"></mspace><mi>famlily</mi><mspace width="thickmathspace"></mspace><mi>of</mi><mspace width="thickmathspace"></mspace><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>fields</mi><mspace width="thickmathspace"></mspace><mi>on</mi><mspace width="thickmathspace"></mspace><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> [X, \mathbf{Fields}] : U \mapsto \mathbf{H}(X \times U, \mathbf{B}G^{bg}_{conn}) = \{ background\;gauge\;fields\; on\; X \times U \} = \{ U-parameterized\;famlily\;of\;gauge\;fields\; on\; X \} </annotation></semantics></math></div> <p>this means effectively to work over the <em>smooth moduli stack of fields</em> itself. Notably, the anomaly is going to be (the non-triviality of) a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a> on “the space of all fields”, so we certainly need a smooth structure on that space. We indicate now how that line bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>anomaly</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \nabla_{anomaly} : [X, \mathbf{Fields}] \to \mathbf{B}U(1)_{conn} </annotation></semantics></math></div> <p>appears.</p> <p>First, the kinetic piece of the action functional is simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>kin</mi></msub><mo stretchy="false">(</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mi>F</mi><mo>∧</mo><mo>*</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(i S_{kin}(\hat F)) = \exp(i \int_X F \wedge \ast F)</annotation></semantics></math>.</p> <p>But suppose there is a charged particle with trajectory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : S^1 \to X</annotation></semantics></math>. Then there is an interaction term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msup><mi>γ</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(2\pi i \int_{S^1} \gamma^* A)</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>el</mi></msub></mrow><annotation encoding="application/x-tex">J_{el}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/Poincare+duality">Poincare dual</a> form. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mi>A</mi><mo>∧</mo><msub><mi>J</mi> <mi>el</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cdots = \exp(i \int_X A \wedge J_{el})</annotation></semantics></math>.</p> <p>This may be expressed using the <a class="existingWikiWord" href="/nlab/show/Beilinson-Deligne+cup+product">Beilinson-Deligne cup product</a> and the <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat F)</annotation></semantics></math>. (Here is where we need <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>el</mi></msub></mrow><annotation encoding="application/x-tex">J_{el}</annotation></semantics></math> to have <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a>.) This is a differential 2-cocycle on the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of field configurations: by the formula for the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> and for forms on a stack, we are evaluating for each test manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> on families of fields over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">X \times U</annotation></semantics></math> and then integrate out over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>But in the case where there is non-trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}_{mag}</annotation></semantics></math> this is no longer the case, there instead this is a trivialization of a twist</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow></mover><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>mag</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega \stackrel{\exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat F)}{\to} \exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat \mathbf{c}_{mag}) \,. </annotation></semantics></math></div> <p>Moreover, this twist matters in <a class="existingWikiWord" href="/nlab/show/compactly+supported+cohomology">compactly supported cohomology</a> (this is what <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a> sees), where it is in general not trivialized. So the action functional is not a function, but a <a class="existingWikiWord" href="/nlab/show/section">section</a> of a line bundle. Its <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> is the 2-class of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>higher</mi><mspace width="thickmathspace"></mspace><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub><mo>:</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>el</mi></msub><mo>∪</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>mag</mi></msub><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \nabla_{higher\;gauge\;anomaly} : [X, \mathbf{Fields}] \stackrel{ \exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat \mathbf{c}_{mag}) } { \to } \mathbf{B} U(1)_{conn} </annotation></semantics></math></div> <p>This is the <em>anomaly line bundle with connection on the moduli stack of fields</em>.</p> <p>For this to cancel, there needs to be a fermionic anomaly – the <a class="existingWikiWord" href="/nlab/show/Pfaffian+line+bundle">Pfaffian line bundle</a> of the <a class="existingWikiWord" href="/nlab/show/Dirac+operators">Dirac operators</a> on the fermions in the theory – of the same structure.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>fermion</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub><mo>:</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_{fermion\;anomaly} : [X, \mathbf{Fields}] \to \mathbf{B} U(1)_{conn} \,. </annotation></semantics></math></div> <p>The total action functional (higher gauge fields and fermions) is a section of the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of these two</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>total</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub><mo>=</mo><msub><mo>∇</mo> <mrow><mi>higher</mi><mspace width="thickmathspace"></mspace><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub><mo>⊗</mo><msub><mo>∇</mo> <mrow><mi>fermion</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub><mo>:</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_{total\;anomaly} = \nabla_{higher\;gauge\;anomaly} \otimes \nabla_{fermion\;anomaly} : [X, \mathbf{Fields}] \to \mathbf{B} U(1)_{conn} \,. </annotation></semantics></math></div> <p>The action functional needs to be a flat <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>total</mi><mspace width="thickmathspace"></mspace><mi>anomaly</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\nabla_{total\;anomaly}</annotation></semantics></math>. Hence the two line bundles need to be inverse to each other. This condition is the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>.</p> <h3 id="TwistedK"><strong>III)</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Fivebrane</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Fivebrane^c</annotation></semantics></math>-structure</h3> <p>In the <a href="#SpinStringFivebraneStructures">previous section</a> we have considered higher differential structures originating in the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>. In applications to <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> these structures receive twists originating in the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> (or representations through the unitary group of groups like <a class="existingWikiWord" href="/nlab/show/E8">E8</a>). (The orthogonal structures correspond to the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, while the unitary structures correspond to <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a>.)</p> <p>Accordingly, the <a href="#WhiteheadTower">above Whitehead tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}O</annotation></semantics></math> has stage-wise <em><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary</a> twistings</em>. In the first stage this is given by the familiar <a class="existingWikiWord" href="/nlab/show/spin%5Ec">spin^c</a>-group, then there is a <a class="existingWikiWord" href="/nlab/show/String%5Ec+2-group">String^c 2-group</a>, etc.</p> <p>After discussing some generalities of these higher unitary-twisted connected covers of the orthogonal group <a href="#HigherUnitaryTwistedCovers">below</a> we then turn to discussing a list of twisted structures and their appearance in string theory:</p> <table><thead><tr><th>unitary-twisted higher orthogonal structure</th><th>role in string theory</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted differential spin^c structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> for <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II strings</a> on <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">twisted differential String^c-structure</a></td><td style="text-align: left;">flux quantization in <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d sugra</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> with <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> in <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></td></tr> </tbody></table> <h4 id="HigherUnitaryTwistedCovers">Higher unitary-twisted covers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math></h4> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <a class="existingWikiWord" href="/nlab/show/spin%5Ec">spin^c</a> is traditionally defined by the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo>≔</mo><mi>Spin</mi><msub><mo>×</mo> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Spin^c \coloneqq Spin \times_{\mathbb{Z}_2} U(1) = (Spin \times U(1))/{\mathbb{Z}_2} \,, </annotation></semantics></math></div> <p>which denotes the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin \times U(1)</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <a class="existingWikiWord" href="/nlab/show/action">action</a> induced by the common canonical <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> <a class="existingWikiWord" href="/nlab/show/group+of+order+2">of order 2</a>.</p> <p>For our purposes it is useful to think of this as follows. We have the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">higher fiber bundle</a> classified by the smooth <a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>ℤ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} \mathbb{Z}_2 &\to& \mathbf{B} Spin \\ && \downarrow \\ && \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z} } </annotation></semantics></math></div> <p>and we also have the <a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">higher fiber bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} U(1) &\stackrel{\cdot 2}{\to}& \mathbf{B} U(1) \\ && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ && \mathbf{B}\mathbb{Z}_2 } </annotation></semantics></math></div> <p>which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated</a> to the <a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a> universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-bundle.</p> <p>One finds, using the presentation of these maps as discussed <a href="#SmoothWhitehead">above</a>, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c</annotation></semantics></math> is the corresponding <a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated bundle</a>, namely the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,. </annotation></semantics></math></div> <p>Equivalently we may write this as a <a class="existingWikiWord" href="/nlab/show/Mayer-Vietoris+sequence">Mayer-Vietoris sequence</a> and thus obtain the universal local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c</annotation></semantics></math>-coefficient bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 \mathbb{Z}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>SO</mi><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin^c &\to& \mathbf{B} (SO \times U(1)) \\ && \downarrow^{\mathrlap{\mathbf{w}_2 - \mathbf{c}_1 mod 2}} \\ && \mathbf{B}^2 \mathbb{Z}_2 } </annotation></semantics></math></div> <p>We may read this as saying:</p> <p><em>Where the moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin</annotation></semantics></math></em> is the homotopy fiber of the smooth second Stiefel-Whitney class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{w}_2</annotation></semantics></math>, the moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c</annotation></semantics></math> is that homotopy fiber universally twisted by the smooth <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbf{c}_1 mod 2</annotation></semantics></math>._</p> <p>In the following we consider higher analogs of this, where homotopy fibers of “orthogonal classes” are twisted by “unitary classes”.</p> <p>In particular, one step higher in the <a href="#WhiteheadTower">Whitehead tower of BO</a>, we can twist the smooth <a class="existingWikiWord" href="/nlab/show/first+fractional+Pontryagin+class">first fractional Pontryagin class</a> with the smooth <a class="existingWikiWord" href="/nlab/show/second+Chern+class">second Chern class</a> to obtain the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the smooth <a class="existingWikiWord" href="/nlab/show/String%5Ec+2-group">String^c 2-group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>String</mi> <mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><mo>×</mo><mi>SU</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} String^{\mathbf{c}_2} &\to& \mathbf{B}(Spin \times SU) \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - \mathbf{c}_2 }} \\ && \mathbf{B}^3 U(1) } \,. </annotation></semantics></math></div> <p>This controls the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>/<a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> as well as the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>, discussed below.</p> <p>Before we come to that we consider another variant, since that leads to the most familiar twisting, that of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>.</p> <p>One finds that there is also a universal local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c</annotation></semantics></math>-coefficient bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1)</annotation></semantics></math>, and this is given by the smooth third <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">integral Stiefel-Whitney class</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Spin^c &\to& \mathbf{B} SO \\ && \downarrow^{\mathrlap{\mathbf{W}_3}} \\ && \mathbf{B}^2 U(1) } \,. </annotation></semantics></math></div> <p>Since that now lands in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1)</annotation></semantics></math>, we can apply <em>one more</em> unitary twist by a corresponding class. The canonical such class is the universal <a class="existingWikiWord" href="/nlab/show/Dixmier-Douady+class">Dixmier-Douady class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>dd</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{dd}</annotation></semantics></math> of (stable) <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary</a> bundles</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msup><mi>Spin</mi> <mi>c</mi></msup><msup><mo stretchy="false">)</mo> <mstyle mathvariant="bold"><mi>dd</mi></mstyle></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>SO</mi><mo>×</mo><mi>PU</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub><mo>−</mo><mstyle mathvariant="bold"><mi>dd</mi></mstyle></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}(Spin^c)^{\mathbf{dd}} &\to& \mathbf{B} ( SO \times PU ) \\ && \downarrow^{\mathrlap{\mathbf{W}_3 - \mathbf{dd}}} \\ && \mathbf{B}^2 U(1) } \,. </annotation></semantics></math></div> <p>This universal local coefficient bundle controls the <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>. To which we now turn.</p> <h4 id="BFieldInK"><strong>a)</strong> Twisted differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structure: Freed-Witten mechanism</h4> <p>We discuss aspects of the twisted smooth cohomology involved over <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>: the <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> mechanism in terms of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>.</p> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> of Lie groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U(1) \to U(n) \to PU(n) </annotation></semantics></math></div> <p>that exhibits the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> as a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-extension of the <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a> induces the corresponding morphism of smooth <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \,. </annotation></semantics></math></div> <p>This is part of a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">long fiber sequence</a> which continues to the right by a <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{dd}_n</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Here the last morphism is presented in simplicial presheaves by the zig-zag of sheaves of <a class="existingWikiWord" href="/nlab/show/crossed+modules">crossed modules</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [U(1) \to U(n)] &\to& [U(1) \to 1] \\ {}^{\mathllap{\simeq}}\downarrow \\ PU(n) } \,. </annotation></semantics></math></div> <p>We have seen <a href="#SmoothWhitehead">above</a> that a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{B}^2 U(1)</annotation></semantics></math> classifies a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle</a> encoded by a Cech 2-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_{i j k} : U_i \cap U_j \cap U_k \to U(1))</annotation></semantics></math> . This means that the universal local coefficient bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} U(n) &\to& \mathbf{B} PU(n) \\ && \downarrow^{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) } </annotation></semantics></math></div> <p>induces a notion of unitary bundles that are <em>twisted</em> by a 2-bundle.</p> <p>Indeed, unwinding the definition one finds that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{dd}_n</annotation></semantics></math> cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{h}{\to}&& \mathbf{B} PU(n) \\ & {}_{\mathllap{\phi}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) } </annotation></semantics></math></div> <p>is, with respect to a resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math>, given by maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U_i\}) \to \mathbf{B}(U(1) \to U(n))</annotation></semantics></math> whose components read</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>h</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></munder></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>∈</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) } \right) \;\;\; \mapsto \;\;\; \left( \array{ && * \\ & {}^{\mathllap{h_{i j}(x)}}\nearrow &\Downarrow^{\phi_{i j k}(x)}& \searrow^{\mathrlap{h_{j k}}} \\ * &&\underset{h_{i k}}{\to}&& * } \right) \in \mathbf{B}(U(1) \to U(n)) \,. </annotation></semantics></math></div> <p>Hence these are collections of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math>-valued functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (h_{i j} : U_i \cap U_j \to U(n)) </annotation></semantics></math></div> <p>which satisfy on triple overlaps the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>h</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>h</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> h_{i j} h_{j k} = h_{i k} \phi_{i j k} \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phi_{i j k}</annotation></semantics></math> happens to be constant on the neutral element, then this is the condition for a cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>smooth</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1_{smooth}(X, U(n))</annotation></semantics></math>. So in general we say it is a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-twisted</em> such cocycle. And that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h_{i j})</annotation></semantics></math> classifies a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted unitary bundle</a></em>.</p> <p>In generalization of how unitary bundles constitute cocycles for <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>, these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-twisted unitary bundles constitute cocycles for <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>.</p> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> there is a canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} U(n) \to \mathbf{B} U(n+1)</annotation></semantics></math>, exhibiting the fact that a rank-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> complex vector bundle canonically induces a rank-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-bundle by added a trivial line bundle.</p> <p>To get rid of the dependence on the rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> – to <em>stabilize</em> the rank – we may form the <a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a> of smooth moduli stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo>≔</mo><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>n</mi></msub></mrow></munder><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n) </annotation></semantics></math></div> <p><strong>Proposition</strong> The smooth stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} U</annotation></semantics></math> is a smooth refinement of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <a class="existingWikiWord" href="/nlab/show/BU"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>U</mi> </mrow> <annotation encoding="application/x-tex">B U</annotation> </semantics> </math></a> of reduced <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>. Also, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-principal bundles and smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-gauge transformations on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are represented by ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math>-bundles for some finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>n</mi></msub></mrow></munder><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \mathbf{B}U) \simeq \underset{\rightarrow_n}{\lim} \mathbf{H}(X, \mathbf{B} U(n)) \,. </annotation></semantics></math></div> <p>Now we think of the manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> for the <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a>, hence assume it to be orientable and spin: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w_1(X) = 0 </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w_2(X) = 0</annotation></semantics></math>. Consider moreover a <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>:</mo><mi>Q</mi><mo>↪</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \iota : Q \hookrightarrow X \,, </annotation></semantics></math></div> <p>to be thought of as the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> of a <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>, which is also or orientable and spin.</p> <p>Assume first that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> also admits a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>, hence that also the third <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">integral Stiefel-Whitney class</a> vanishes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">W_3(Q) = 0</annotation></semantics></math>.</p> <p>We can consider then a cocycle in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>dd</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{dd}</annotation></semantics></math>, namely a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>ga</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>ga</mi></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub><msub><mo stretchy="false">|</mo> <mi>Q</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>dd</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q &\stackrel{\phi_{ga}}{\to}& \mathbf{B} P U(n) \\ {}^{\mathllap{i}}\downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\mathbf{dd}_n}} \\ X &\stackrel{\phi_B}{\to}& \mathbf{B}^2 U(1) } \;\;\;\; \leftrightarrow \;\;\; \array{ Q &&\stackrel{\phi_{ga}}{\to}&& \mathbf{B} P U(n) \\ & {}_{\mathllap{\phi_B|_Q}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{dd}}} \\ && \mathbf{B}^2 U(1) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>This is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_B : X \to \mathbf{B}^2 U(1)</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: the underlying bundle of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>;</p> </li> <li> <p>a projective unitary bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>ga</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{ga}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/Chan-Paton+bundle">Chan-Paton bundle</a> on the D-brane;</p> </li> </ol> <ul> <li>together with an identification of the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub><msub><mo stretchy="false">|</mo> <mi>Q</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{B}|_Q</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-brane with the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>dd</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{dd}(\phi_{ga})</annotation></semantics></math> to lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>ga</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{ga}</annotation></semantics></math> to a genuine unitary bundle.</li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> this says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>dd</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>ϕ</mi> <mi>B</mi></msub><msub><mo stretchy="false">|</mo> <mi>Q</mi></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\mathbf{dd}(\phi_{ga})] = [\phi_B|_Q] \;\; \in H^3(Q) \,. </annotation></semantics></math></div> <p>This is the <em><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> condition</em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-branes with <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>.</p> <p>More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> does not necessarily have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structure, we consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a> with coefficients in the universal local coefficient bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>dd</mi></mstyle><mo>−</mo><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{dd} - \mathbf{W}_3</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo>,</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>SO</mi><mo>×</mo><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo>,</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>SO</mi><mo>×</mo><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub><msub><mo stretchy="false">|</mo> <mi>Q</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Q &\stackrel{(T X, \phi_{ga})}{\to}& \mathbf{B} ( SO \times P U(n)) \\ {}^{\mathllap{i}}\downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\mathbf{dd}_n - \mathbf{W}_3}} \\ X &\stackrel{\phi_B}{\to}& \mathbf{B}^2 U(1) } \;\;\;\; \leftrightarrow \;\;\; \array{ Q &&\stackrel{(T X, \phi_{ga})}{\to}&& \mathbf{B} (SO \times P U(n)) \\ & {}_{\mathllap{\phi_B|_Q}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{dd}_n-\mathbf{W}_3}} \\ && \mathbf{B}^2 U(1) } \,. </annotation></semantics></math></div> <p>This now is equivalently a twisted Chan-Paton bundle and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field such that in cohomology</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>dd</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>ϕ</mi> <mi>B</mi></msub><msub><mo stretchy="false">|</mo> <mi>Q</mi></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\mathbf{dd}(\phi_{ga})] + [W_3(Q)] = [\phi_B|_Q] \;\;\; \in H^3(X, \mathbb{Z}) \,. </annotation></semantics></math></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> condition for general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>.</p> <h4 id="SuGraFluxQuantization"><strong>b)</strong> Twisted differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">String^{\mathbf{c}_2}</annotation></semantics></math>-structure: M-theory flux quantization</h4> <p>We discuss the twisted smooth cohomology of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> in <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>. With the smooth and differential refinement of the Whitehead tower in hand, this proceeds essentially in higher analogy to the <a href="#BFieldInK">previous example</a>.</p> <p>From the <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective QFT</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> the bosonic massless field content consists locally of the <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> and a 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. We have the following information on how a model of this field content must behave globally</p> <ol> <li> <p>Due to the existence of <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, the graviton must be part of a <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_{gr} : X \to \mathbf{B}Spin_{conn} \,. </annotation></semantics></math></div></li> <li> <p>Due to the coupling to the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> the 3-form must lift to a well-dfined <a class="existingWikiWord" href="/nlab/show/higher+holonomy">3-holonomy</a> and hence must globally be a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>C</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_C : X \to \mathbf{B}^3 U(1)_{conn} \,. </annotation></semantics></math></div></li> <li> <p>Due to the coupling to the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>, there is an auxiliary <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex"> \phi_{ga} : X \to \mathbf{B} E_8 </annotation></semantics></math></div> <p>and these fields must satisfy what in string theory literature is called the <em>flux quantization condition</em>, and what in <a href="#HopkinsSinger">Hopkins-Singer 05</a> is called an <em><a class="existingWikiWord" href="/nlab/show/differential+integral+Wu+structure">differential integral Wu structure</a></em>, meaning that in <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ϕ</mi> <mi>C</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\phi_C] = [\frac{1}{2}p_1(\phi_{gr})] + [2\mathbf{a}(\phi_{ga})] \;\; \in H^4(X, \mathbb{Z}) \,. </annotation></semantics></math></div></li> </ol> <p>(Depending on convention one may write “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>ϕ</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">2 \phi_C</annotation></semantics></math>” for “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\phi_C</annotation></semantics></math>” here, regarding the physical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field as being “one half” of the differential cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^3 U(1)_conn</annotation></semantics></math> above, see the remark <a href="arxiv.org/pdf/hep-th/9609122v2.pdf#page=2">below (1.2)</a> in <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Witten</a>‘s <a href="http://arxiv.org/abs/hep-th/9609122">arXiv:hep-th/9609122</a>).</p> <p>A <a class="existingWikiWord" href="/nlab/show/discrete+infinity-groupoid">discrete</a> 1-groupoid model satisfying these points has been by <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Freed</a>-<a class="existingWikiWord" href="/nlab/show/Greg+Moore">Moore</a> and others (see the references <a href="supergravity+C-field#References">here</a>). Using <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a> and following (<a href="#FSSc">FSSc</a>) we now obtain naturally a genuine <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">smooth moduli 3-stack</a> of such field configurations: the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">interpretation</a> of the evident expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>CField</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo>,</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo>,</mo><msub><mi>ϕ</mi> <mi>C</mi></msub><mo stretchy="false">|</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>ϕ</mi> <mi>C</mi></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{CField}(X) \coloneqq \left\{ \phi_{gr}, \phi_{ga}, \phi_{C} | \frac{1}{2}\mathbf{p}_1(\phi_{gr}) + 2\mathbf{a}(\phi_{ga}) \simeq \phi_C \right\} </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is (see <a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a> for how this works) the smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>CField</mi></mstyle><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{CField} \in \mathbf{H}</annotation></semantics></math> given as the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>CField</mi></mstyle></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mn>8</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>+</mo><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle></mrow></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{CField} &\to& \mathbf{B} Spin_{conn} \times \mathbf{B}E_8 \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\tfrac{1}{2}\mathbf{p}_1 + 2\mathbf{a}} \\ \mathbf{B}^3 U(1)_{conn} &\stackrel{}{\to}& \mathbf{B}^3 U(1) } \,. </annotation></semantics></math></div> <p>On the right this has the universal local coefficient bundle for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>String</mi></mstyle> <mrow><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{String}^{2\mathbf{a}}</annotation></semantics></math> from <a href="#HigherUnitaryTwistedCovers">above</a>, and hence this identifies a <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>-<a class="existingWikiWord" href="/nlab/show/supergravity+C-field">C-field</a> configuration as being (a partial differential refinement of) a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ϕ</mi> <mi>C</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\phi_C]</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mrow><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">String^{2\mathbf{a}}</annotation></semantics></math>-structure.</p> <h4 id="TwistedStringStructure"><strong>c)</strong> Twisted differential String-structure – Green-Schwarz mechanism</h4> <p>By <span class="newWikiWord">Ho?ava-Witten theory<a href="/nlab/new/Ho%3Fava-Witten+theory">?</a></span>, the 10-dimensional <a class="existingWikiWord" href="/nlab/show/target+space">target</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> of the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> may be understood as being a boundary (or rather <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> fixed points, see <a href="#HWCompactifications">below</a>) of the 11-dimensional spacetime of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d SuGra</a>. Over this boundary</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 4-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_4(\phi_C)</annotation></semantics></math> vanishes;</p> </li> <li> <p>and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> picks up a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a>.</p> </li> </ol> <p>This means that the <a href="#SuGraFluxQuantization">above</a> defining homotopy pullback for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>CField</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{CField}</annotation></semantics></math> goes over into the one that defines the differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>String</mi> <mrow><mi>conn</mi><mo>′</mo></mrow> <mstyle mathvariant="bold"><mi>a</mi></mstyle></msubsup></mrow><annotation encoding="application/x-tex">String^{\mathbf{a}}_{conn'}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mstyle mathvariant="bold"><mi>a</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">String^{\mathbf{a}}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msubsup><mi>String</mi> <mrow><mi>conn</mi><mo>′</mo></mrow> <mstyle mathvariant="bold"><mi>a</mi></mstyle></msubsup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub></mrow></mpadded></msup><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>+</mo><mover><mstyle mathvariant="bold"><mi>a</mi></mstyle><mo stretchy="false">^</mo></mover></mrow></msup></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mrow><msub><mi>ϕ</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}String^{\mathbf{a}}_{conn'} &\to& \mathbf{B} Spin_{conn} \times \mathbf{B}(E_8 \times E_8)_{conn} \\ \downarrow &{}^{\mathllap{\phi_B}}\swArrow_{\simeq}& \downarrow^{\tfrac{1}{2}\hat \mathbf{p}_1 + \hat \mathbf{a}} \\ \Omega^3_{cl}(-) & \stackrel{\phi_C}{\hookrightarrow} & \mathbf{B}^3 U(1)_{conn} } \,. </annotation></semantics></math></div> <p>On cohomology classes this means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\tfrac{1}{2}p_1(\phi_{gr})] = [a(\phi_{ga})] \;\; \in H^4(X, \mathbb{Z}) \,. </annotation></semantics></math></div> <p>This is the integral part of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> for the <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string</a>.</p> <p>Since this is now refined not just to cocycles, but to differential cocycles – to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi mathvariant="normal">String</mi> <mstyle mathvariant="bold"><mi>a</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">\mathrm{String}^{\mathbf{a}}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">2-connections</a>, there is, locally over a cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math>, also an equation of differential forms that exhibits this in de Rham cohomology.</p> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msubsup><mi>String</mi> <mi>conn</mi> <mrow><mn>2</mn><mstyle mathvariant="bold"><mi>a</mi></mstyle></mrow></msubsup></mrow><annotation encoding="application/x-tex">X \to String^{2\mathbf{a}}_{conn}</annotation></semantics></math> classifies field content that is expressed with respect to a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> in particular over single patches <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> by (<a href="#SSSa">SSSa</a>, <a href="#FSSa">FSSa</a>)</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">gauge connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo>⊕</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_i \in \Omega^1(U_i, \mathfrak{e}_8 \oplus \mathfrak{e}_8)</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>𝔰𝔬</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega_i \in \Omega^1(U_i, \mathfrak{so})</annotation></semantics></math> (the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> in <a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a>);</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_B \in \Omega^2(U_i)</annotation></semantics></math>;</p> </li> </ul> <p>which come with <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a>/<a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> forms</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><msub><mi>A</mi> <mi>i</mi></msub></mrow></msub><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>A</mi> <mi>i</mi></msub><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>∧</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F_{A_i} = d_{dR} A_i + \tfrac{1}{2}[A_i \wedge A_i]</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><msub><mi>ω</mi> <mi>i</mi></msub></mrow></msub><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>ω</mi> <mi>i</mi></msub><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo>∧</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F_{\omega_i} = d_{dR} \omega_i + \tfrac{1}{2}[\omega_i \wedge \omega_i]</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>i</mi></msub><mo>=</mo><mi>d</mi><msub><mi>B</mi> <mi>i</mi></msub><mo>+</mo><mi>CS</mi><mo stretchy="false">(</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_i = d B_i + CS(\omega_i) - CS(A_i)</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field strength shifted by the difference of the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+forms">Chern-Simons forms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\omega_i</annotation></semantics></math>);</p> </li> </ul> <p>satisfying the (twisted) <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> (<a href="SSSa">SSSa</a>))</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>F</mi> <mrow><msub><mi>A</mi> <mi>i</mi></msub></mrow></msub><mo>+</mo><mo stretchy="false">[</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>∧</mo><msub><mi>F</mi> <mrow><msub><mi>A</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{dR} F_{A_i} + [A_i \wedge F_{A_i}] = 0</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>F</mi> <mrow><msub><mi>ω</mi> <mi>i</mi></msub></mrow></msub><mo>+</mo><mo stretchy="false">[</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo>∧</mo><msub><mi>F</mi> <mrow><msub><mi>ω</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{dR} F_{\omega_i} + [\omega_i \wedge F_{\omega_i}] = 0</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>H</mi><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">d_{dR} H = \langle F_\omega \wedge F_\omega \rangle - \langle F_A \wedge F_A \rangle</annotation></semantics></math></p> </li> </ul> <p>(together with more local cocycle components on higher overlaps). Notably the twisted Bianchi identity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> exhibits the above cohomological identity in de Rham cocycles.</p> <p>These formulas characterize the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz anomaly cancellation conditions</a> on the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> content, that makes the heterotic string be well defined. Accordingly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>String</mi> <mi>conn</mi> <mstyle mathvariant="bold"><mi>a</mi></mstyle></msubsup></mrow><annotation encoding="application/x-tex">String^{\mathbf{a}}_{conn}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli 2-stack</a> of anomaly free heterotic background fields (in the massless bosonic sector).</p> <p>Notice that if the twist <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mover><mstyle mathvariant="bold"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mover><mstyle mathvariant="bold"><mi>a</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gau</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}\hat \mathbf{p_1}(\phi_{gr}) - \hat\mathbf{a}(\phi_{gau})</annotation></semantics></math> happen to vanish (say because both the field of gravity and the gauge field are trivial), then the above homotopy pullback reduces to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub></mrow></mpadded></msup><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mrow><msub><mi>ϕ</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^2 U(1)_{conn} &\to& * \\ \downarrow &{}^{\mathllap{\phi_B}}\swArrow_{\simeq}& \downarrow^{} \\ \Omega^3_{cl}(-) & \stackrel{\phi_C}{\hookrightarrow} & \mathbf{B}^3 U(1)_{conn} } </annotation></semantics></math></div> <p>and exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\phi_B</annotation></semantics></math> as a genuine <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a> (and its 3-form curvature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\phi_C</annotation></semantics></math>.). Conversely, this shows how in the general situation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\phi_B</annotation></semantics></math> is a <em>twisted</em> circle 2-bundle, with the twist given by the “magnetic fivebrane current” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>gr</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mover><mstyle mathvariant="bold"><mi>a</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>ga</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}\hat \mathbf{p}_1(\phi_{gr}) - \hat \mathbf{a}(\phi_{ga})</annotation></semantics></math>.</p> <h4 id="d_twisted_differential_fivebrane_structure__dual_greenschwarz_mechanism"><strong>d)</strong> Twisted differential Fivebrane structure – dual Green-Schwarz mechanism</h4> <p>(…)</p> <h3 id="HigherOrientifold"><strong>IV)</strong> Higher orientifold structure</h3> <h4 id="orientifolds">Orientifolds</h4> <p>An <em><a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a></em> <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> for the bosonic string is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> or more generally <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/double+cover">double cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_X : X \to \mathbf{B} \mathbb{Z}_2</annotation></semantics></math>;</p> </li> <li> <p>a twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-equivariant circle 2-bundle, given by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>B</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_B : X \to \mathbf{B}Aut(\mathbf{B}U(1))</annotation></semantics></math> whose underlying double cover is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>.</p> </li> </ul> <p>This means that this background is a cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>-twisted cohomology for the local coefficient bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mstyle mathvariant="bold"><mi>J</mi></mstyle></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}U(1) &\to& \mathbf{B}Aut(\mathbf{B}U(1)) \\ && \downarrow^{\mathbf{J}} \\ && \mathbf{B}\mathbb{Z}_2 } \,. </annotation></semantics></math></div> <p>Hence the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> is now a cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>-twisted cohomology</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>b</mi></msub><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>w</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>J</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \phi_b \in \mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}( w_X , \mathbf{J}) \,, </annotation></semantics></math></div> <p>or rather a differential refinement thereof. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> of a <a class="existingWikiWord" href="/nlab/show/string">string</a>, an orientifold string configuration is a cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>φ</mi><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>w</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\varphi, \nu) \in \mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}(\mathbf{w}_1(\Sigma), w_X) \,, </annotation></semantics></math></div> <p>given in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> by a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>φ</mi></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mrow></mrow> <mi>ν</mi></msup><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>w</mi> <mi>X</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma &&\stackrel{\varphi}{\to}&& X \\ & {}_{\mathllap{\mathbf{w}_1}}\searrow &{}^{\nu}\swArrow_{\simeq}& \swarrow_{\mathrlap{w_X}} \\ && \mathbf{B}\mathbb{Z}_2 } </annotation></semantics></math></div> <p>consisting of</p> <ul> <li> <p>a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\varphi : \Sigma \to X</annotation></semantics></math>;</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>:</mo><msup><mi>φ</mi> <mo>*</mo></msup><msub><mi>w</mi> <mi>X</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nu : \varphi^* w_X \to \mathbf{w}_1(\Sigma)</annotation></semantics></math>.</p> </li> </ul> <h4 id="HWCompactifications">Hořava-Witten compactifications</h4> <p>In <span class="newWikiWord">Ho?ava-Witten theory<a href="/nlab/new/Ho%3Fava-Witten+theory">?</a></span> there is similarly a twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-action on the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>, exhibited by a local coefficient bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mstyle mathvariant="bold"><mi>J</mi></mstyle></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^2U(1) &\to& \mathbf{B}Aut(\mathbf{B}^2U(1)) \\ && \downarrow^{\mathbf{J}} \\ && \mathbf{B}\mathbb{Z}_2 } \,. </annotation></semantics></math></div> <h3 id="further_twists">Further twists</h3> <p>There are various further twisted cohomological structures in string theory known or conjectured (for some of which possibly no smooth refinement has been constructed yet). We briefly list some of them.</p> <h4 id="TwistedSuperBundle">Twisted super bundle</h4> <p>In work like <em><a class="existingWikiWord" href="/nlab/show/Loop+Groups+and+Twisted+K-Theory">Loop Groups and Twisted K-Theory</a></em> the following structure plays a role:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group">group</a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>G</mi><mo>→</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \epsilon : G \to \mathbb{Z}_2 </annotation></semantics></math></div> <p>be a fixed group homomorphism. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><msub><mi>E</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E = E_0 \oplus E_1 \to X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a>, an “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>-twist” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> such that an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> acts by an even automorphism if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon(g)</annotation></semantics></math> is even, and by an odd automorphism if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon(g)</annotation></semantics></math> is odd (<a href="#FreedLecture">Freed ESI lecture, (1.13)</a>).</p> <p>This is a special case of the general notion of twist discussed here by considering the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mstyle mathvariant="bold"><mi>e</mi></mstyle></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Aut(E) \\ \downarrow^{\mathbf{e}} \\ \mathbf{B} \mathbb{Z}_2 } </annotation></semantics></math></div> <p>as the local coefficient bundle, and considering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> as the twist: then an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>-twisting as above is a cocycle in the twisted cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ϵ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>e</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}(\mathbf{B}\epsilon, \mathbf{e})</annotation></semantics></math> given by a commuting triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ϵ</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>e</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}G &&\to&& \mathbf{B}Aut(E) \\ & {}_{\mathllap{\mathbf{B}\epsilon}}\searrow && \swarrow_{\mathrlap{\mathbf{e}}} \\ && \mathbf{B}\mathbb{Z}_2 } \,. </annotation></semantics></math></div> <h4 id="RelativeFields">Relative fields</h4> <p>Meanwhile in (<a href="#FreedTeleman">Freed-Teleman 2012</a>) special cases of the general notion of <em>twisted fields</em> <a href="#TableOfTwists">above</a> are being called <em>relative fields</em>. We briefly spell out how the definitions considered in that article are examples of the general notion above.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi \in Grp(Set) \hookrightarrow Grp(Smooth\infty Grpd)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>¯</mo></mover><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo></mrow><annotation encoding="application/x-tex">\overline{X} \in \mathbf{H} \coloneqq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> any object (for instance a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>) a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mover><mi>X</mi><mo>¯</mo></mover></msub><mo lspace="verythinmathspace">:</mo><mover><mi>X</mi><mo>¯</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>π</mi></mrow><annotation encoding="application/x-tex">\phi_{\overline{X}} \colon \overline{X} \to \mathbf{B}\pi</annotation></semantics></math> modulates a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">X \to \overline{X}</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, hence a free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">X \to \overline{X}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map.</p> <p>Then the corresponding <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>π</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>ϕ</mi> <mover><mi>X</mi><mo>¯</mo></mover></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}\pi}(-, \phi_{\overline{X}})</annotation></semantics></math> has</p> <ol> <li> <p>domains are objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, modulated by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>π</mi></mrow><annotation encoding="application/x-tex">\phi_\Sigma \colon \Sigma \to \mathbf{B}\pi</annotation></semantics></math>;</p> </li> <li> <p>cocycles are morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub><mo>→</mo><msub><mi>ϕ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\phi_\Sigma \to \phi_X</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>π</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}\pi}</annotation></semantics></math>, hence <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo>¯</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>θ</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mi>X</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>π</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma &&\stackrel{f}{\to}&& \overline{X} \\ & {}_{\mathllap{\phi_\Sigma}}\searrow &\swArrow_{\theta}& \swarrow_{\mathrlap{\phi_X}} \\ && \mathbf{B}\pi } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, hence maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">f \colon \Sigma \to \overline{X}</annotation></semantics></math> equipped with equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>ϕ</mi> <mi>X</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>ϕ</mi> <mi>Σ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \theta \colon f^* \phi_X \stackrel{\simeq}{\to} \phi_\Sigma \,. </annotation></semantics></math></div></li> </ol> <p>This class of examples is what appears as def. 3.4 in (<a href="#FreedTeleman">Freed-Teleman</a>). It contains in particular the above examples of <em><a href="#ReductionOfTheStructureGroup">Reduction of structure group</a></em> and <a href="#SpinConnection">its differential refinement</a>.</p> <p>Next, consider a compact Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{G}</annotation></semantics></math> and a central <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>→</mo><mi>G</mi><mo>→</mo><mover><mi>G</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\pi \to G \to \overline{G}</annotation></semantics></math>. This is classified by a cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo>¯</mo></mover><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>π</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{c} \colon \mathbf{B}\overline{G} \to \mathbf{B}^2 \pi \,. </annotation></semantics></math></div> <p>Then the corresponding twisted cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>π</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}^2 \pi}(-, \mathbf{c})</annotation></semantics></math> has</p> <ol> <li> <p>domains are objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>π</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\pi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> modulated by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>π</mi></mrow><annotation encoding="application/x-tex">\phi_\Sigma \colon \Sigma \to \mathbf{B}^2 \pi</annotation></semantics></math>;</p> </li> <li> <p>cocycles are morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi_\Sigma \to \mathbf{c}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>π</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}^2\pi}</annotation></semantics></math>, hence <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo>¯</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>θ</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>π</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma &&\stackrel{f}{\to}&& \overline{X} \\ & {}_{\mathllap{\phi_\Sigma}}\searrow &\swArrow_{\theta}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}^2\pi } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, hence maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">f \colon \Sigma \to \overline{X}</annotation></semantics></math> equipped with equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>ϕ</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex"> \theta \colon \mathbf{c}(f) \stackrel{\simeq}{\to} \phi_\Sigma </annotation></semantics></math></div> <p>between the principal 2-bundle/bundle gerbe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{c}(f)</annotation></semantics></math> induced by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{G}</annotation></semantics></math>-principal bundle modulated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, and the one modulated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\phi_\Sigma</annotation></semantics></math>.</p> </li> </ol> <p>Alternatively one can use here the differential refinement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\mathbf{B}\overline{G}</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mover><mi>G</mi><mo>¯</mo></mover> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\overline{G}_{conn}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>.</p> <p>Examples and further details are discussed in <a href="#Schreiber">Schreiber, section 7.1</a>. In (<a href="#FreedTeleman">Freed-Teleman</a>) this example appears as def. 4.6.</p> <h4 id="twisted_tmf">Twisted tmf</h4> <ul> <li>twisted <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> and charges for M5-branes (<a href="#Ando-Sati">Ando-Sati</a>).</li> </ul> <h4 id="twisted_morava_ktheory">Twisted Morava K-theory</h4> <ul> <li> <p>twisted <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a> (<a href="#SatiWesterland">Sati-Westerland</a>).</p> </li> <li> <p>(…)</p> </li> </ul> <h2 id="LocalPrequantumFieldTheory"><strong>B)</strong> Local boundary prequantum field theory from twisted smooth cohomology</h2> <blockquote> <p>This section originates in some talk notes (<a href="#SchreiberTwists2013">Schreiber, Twists 2013</a>).</p> </blockquote> <p>We indicate now how the twisted smooth cohomology data as in the examples above induces and in fact corresponds to data for <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a> <em>localized</em> (in the sense of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>) to <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> (<a href="#lpqft">lpqft</a>).</p> <p>In order to formalize this accurately, we first talk a bit more about the relevant <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>, <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> and <a class="existingWikiWord" href="/nlab/show/tangent+cohesion">tangent cohesion</a>. The way to understand this is as follows:</p> <p>In full generality, <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> (see there for details) is what is given by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-spaces">(∞,1)-categorical hom-spaces</a> in some <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-topos">∞-topos</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X,A \in \mathbf{H}</annotation></semantics></math> any two <a class="existingWikiWord" href="/nlab/show/objects">objects</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H(X,A) \coloneqq \pi_0 \mathbf{H}(X,A) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Therefore it is natural to ask: <em>What is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> to be like whose intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> is <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">stable</a> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>?</em></p> <p>And the answer we find is: it is to be the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-topos">tangent (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>For more on this see at <em><a class="existingWikiWord" href="/nlab/show/tangent+cohesion">tangent cohesion</a></em>.</p> <h3 id="cohesive_contexts_for_equivariant_differential_twisted_cohomology">Cohesive contexts for equivariant differential twisted cohomology</h3> <p>After the discovery of the role of <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> in <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> phenomena in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, the observation that more generally <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> involves various other <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> theories such as <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> and <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a>, has notably been highlighted by <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, surveyed in (<a href="#Sati10">Sati 10</a>).</p> <p>The suggestion that the right context for formulating the smooth and differential refinement of these twisted cohomology theories is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> We</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mo>=</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Funct</mi><mo stretchy="false">(</mo><msup><mi>SmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>KanCplx</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mi>stalkwise</mi><mspace width="thickmathspace"></mspace><mi>homotopy</mi><mspace width="thickmathspace"></mspace><mi>equivalences</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{H} &= Sh_\infty(SmthMfd) \\ & \simeq Funct(SmoothMfd^{op}, KanCplx)[\{stalkwise\;homotopy\;equivalences\}^{-1}] \end{aligned} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (the result of universally turning <a class="existingWikiWord" href="/nlab/show/stalk">stalkwise</a> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">presheaves of</a> <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> into actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>) is due to (<a href="#Schreiber09">Schreiber 09</a>).</p> <p>A full formalization of the classification of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> in such <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, and their classification by <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> was later given in (<a href="#NSS">Nikolaus-Schreiber-Stevenson 12</a>). There it is shown that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> of twists, the corresponding <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> is the plain cohomology of the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Topos</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}_{/\mathbf{B}G} \;\; \in (\infty,1)Topos \,. </annotation></semantics></math></div> <p>Precisely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\rho \in \mathbf{H}_{\mathbf{B}G}</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> incarnated as its universal <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> (the <a class="existingWikiWord" href="/nlab/show/local+coefficient+%E2%88%9E-bundle">local coefficient ∞-bundle</a>) and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\chi \colon X \longrightarrow \mathbf{B}G</annotation></semantics></math> a twist, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> with <a class="existingWikiWord" href="/nlab/show/local+coefficients">local coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>χ</mi> <mi>X</mi></msub><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>ϕ</mi></mover></mtd> <mtd></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>χ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}_{\mathbf{B}G}(\chi_X,\rho) = \left\{ \array{ X && \stackrel{\phi}{\longrightarrow} && V//G \\ & {}_{\mathllap{\chi}}\searrow && \swarrow_{\mathrlap{\rho}} \\ && \mathbf{B}G } \right\} \,. </annotation></semantics></math></div> <p>The observation that for the <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>-refinement of this <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> geometric cohomology it is the <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functors">(∞,1)-functors</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a> (“<a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><mover><munder><mo>←</mo><mi>coDisc</mi></munder><mover><mo>⟶</mo><mi>Γ</mi></mover></mover><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>⟶</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd </annotation></semantics></math></div> <p>(with <a class="existingWikiWord" href="/nlab/show/discrete+object">Disc</a> and <a class="existingWikiWord" href="/nlab/show/codiscrete+object">coDisc</a> <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functors">full and faithful (∞,1)-functors</a> and the <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos">fundamental ∞-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+homotopy+types">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limit">finite product</a>-preserving) which governs all the theory was observed first in (<a href="SatiSchreiberStasheff09">Sati-Schreiber-Stasheff 09 (11)</a>)</p> <p>There it was shown that this serves to construct and characterize the <a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structures">twisted differential string structures</a> and <a class="existingWikiWord" href="/nlab/show/twisted+differential+Fivebrane+structures">twisted differential Fivebrane structures</a> in (<a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual</a>) <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>.</p> <p>A comprehensive theory of (<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a>) <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> formulated by just this <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a> of “<a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>” was then laid out in the thesis (<a href="#Schreiber">Schreiber 11</a>). Parts of this appear in various articles, such as (<a href="FiorenzaRogersSchreiber13">Fiorenza-Schreiber-Rogers 13</a>). See (Schreiber Synthetic 13eiberSynthetic)) for a fairly comprehensive survey.</p> <p>Notice that such <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> is a very special property of some <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a>, not a generic property. In particular the existence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a> and a <a class="existingWikiWord" href="/nlab/show/globally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">globally ∞-connected (∞,1)-topos</a>, and the existence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coDisc</mi></mrow><annotation encoding="application/x-tex">coDisc</annotation></semantics></math> means that it is a <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> <p>Examples of <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> established and discussed in (<a href="#Schreiber">Schreiber 11</a>) include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+topological+%E2%88%9E-groupoids">Euclidean topological ∞-groupoids</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>TopMfd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(TopMfd)</annotation></semantics></math> is cohesive</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmoothMfd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(SmoothMfd)</annotation></semantics></math> is cohesive</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoids">super ∞-groupoids</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SuperPoints</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(SuperPoints)</annotation></semantics></math> is cohesive;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoids">smooth super ∞-groupoids</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SuperMfd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(SuperMfd)</annotation></semantics></math> is cohesive over <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoids">super ∞-groupoids</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoids">synthetic differential ∞-groupoids</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>FormalMfds</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(FormalMfds)</annotation></semantics></math> is even “<a class="existingWikiWord" href="/nlab/show/differential+cohesion">differentially cohesive</a>”, which allows to <a class="existingWikiWord" href="/nlab/show/axiom">axiomatize</a> also the notions of <em><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></em>, <em><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></em> etc.;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoids">synthetic differential super ∞-groupoids</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>FormalSuperMfds</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(FormalSuperMfds)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differentially cohesive</a> over <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoids">super ∞-groupoids</a>.</p> </li> </ul> <p>Clearly these examples are all of a similar kind (modeled on variants of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>). There are also some <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> categories which are usefully cohesive, for instance</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+objects">simplicial objects</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{\Delta^{op}}</annotation></semantics></math> is cohesive over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of simplicial objects in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>).</li> </ul> <p>In October 2013 <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a> announced a new kind of cohesion, namely:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Orb</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(Orb)</annotation></semantics></math> (hence <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaves">(∞,1)-presheaves</a> on the global <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a>) is cohesive (here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> produces <a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> produces topological <a class="existingWikiWord" href="/nlab/show/quotients">quotients</a> of <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a> acting on <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>).</li> </ul> <p>But of course a central feature desireable for <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a> is <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of cohesion/cohesive <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a>.</p> <p>In (<a href="tangent+cohesion#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13 (14?)</a>) is considered the <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth cohesion</a>, hence the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(Sh_\infty(SmthMfd))</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> in <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>, which carries an analogous <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> over the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+spectra">(∞,1)-category of spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Spectra</mi></mrow><annotation encoding="application/x-tex">Stab(\infty Grpd) \simeq Spectra</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>coDisc</mi> <mi>stab</mi></msup></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>Γ</mi> <mi>stab</mi></msup></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>Disc</mi> <mi>stab</mi></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><msup><mi>Π</mi> <mi>stab</mi></msup></mrow></mover></mover><mi>Spectra</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Stab(Sh_\infty(SmthMfd)) \stackrel{\overset{\Pi^{stab}}{\longrightarrow}}{\stackrel{\overset{Disc^{stab}}{\leftarrow}}{\stackrel{\overset{\Gamma^{stab}}{\longrightarrow}}{\underset{coDisc^{stab}}{\leftarrow}}}} Spectra \,. </annotation></semantics></math></div> <p>But this stable aspect is unified with the unstable cohesion by the notion of “<a class="existingWikiWord" href="/nlab/show/tangent+cohesion">tangent cohesion</a>”. This we turn to now.</p> <h3 id="cohesive_contexts_for_stable_twisted_cohomology">Cohesive contexts for stable twisted cohomology</h3> <p>We first discuss generally how the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-category">tangent (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is itself an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> over the tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category of the original <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a> (<a href="#Joyal08">Joyal 08</a>). Then we observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is and is in fact an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> of the latter by its <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a>. a choice</p> <h4 id="tangent_topos">Tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos</h4> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>seq</mi></mrow><annotation encoding="application/x-tex">seq</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> category as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>seq</mi><mo>≔</mo><msub><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mi mathvariant="normal">id</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></munder></mtd> <mtd><msub><mi>x</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> seq \coloneqq \left\{ \array{ && \vdots && \vdots \\ && \downarrow && \\ \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast \\ &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}} \\ &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast \\ && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}} \\ && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots \\ && && && \downarrow \\ && && && \vdots } \right\}_{n \in \mathbb{Z}} \,. </annotation></semantics></math></div></div> <p>(<a href="#Joyal08">Joyal 08, section 35.5</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Given an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>seq</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> X_\bullet \;\colon\; seq \longrightarrow \mathbf{H} </annotation></semantics></math></div> <p>is equivalently</p> <ol> <li> <p>a choice of <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">B \in \mathbf{H}</annotation></semantics></math> (the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mi>in</mi><mi>seq</mi></mrow><annotation encoding="application/x-tex">\ast in seq</annotation></semantics></math>]);</p> </li> <li> <p>a sequence of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">}</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_n\} \in \mathbf{H}_{/B}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>;a choi</p> </li> <li> <p>a sequence of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>⟶</mo><msub><mi>Ω</mi> <mi>B</mi></msub><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_n \longrightarrow \Omega_B X_{n+1}</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math> in the slice.</p> </li> </ol> <p>This is a <a class="existingWikiWord" href="/nlab/show/prespectrum+object">prespectrum object</a> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/B}</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f \;\colon \;X_\bullet \to Y_\bullet</annotation></semantics></math> between two such functors with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mi>p</mi> <mi>n</mi> <mi>X</mi></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mi>p</mi> <mi>n</mi> <mi>Y</mi></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mi>b</mi></msub></mrow></mover></mtd> <mtd><msub><mi>B</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \array{ X_n &\stackrel{f_n}{\longrightarrow}& Y_n \\ \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}} \\ B_1 &\stackrel{f_b}{\longrightarrow}& B_2 } \right\} </annotation></semantics></math></div> <p>is equivalently a morphism of base objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>b</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>B</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>B</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_b \;\colon\; B_1 \longrightarrow B_2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> together with morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>⟶</mo><msubsup><mi>f</mi> <mi>b</mi> <mo>*</mo></msubsup><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n \longrightarrow f_b^\ast Y_n</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> of the components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Y_\bullet</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">f_b</annotation></semantics></math>.</p> <p>Therefore the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-presheaf (∞,1)-topos</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mo>≔</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>seq</mi><mo>,</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}^{seq} \coloneqq Func(seq, \mathbf{H}) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> with “fiberwise pre-stabilization”.</p> <p>A genuine <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> is a <a class="existingWikiWord" href="/nlab/show/prespectrum+object">prespectrum object</a> for which all the structure maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>Ω</mi> <mi>B</mi></msub><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalences</a>. The <a class="existingWikiWord" href="/nlab/show/full+sub-%28%E2%88%9E%2C1%29-category">full sub-(∞,1)-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup></mrow><annotation encoding="application/x-tex"> T \mathbf{H} \hookrightarrow \mathbf{H}^{seq} </annotation></semantics></math></div> <p>on the genuine <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> is therefore the “fiberwise <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a>” of the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a>, hence the tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category.</p> </div> <div class="num_lemma" id="SpectrificationLemma"> <h6 id="lemma">Lemma</h6> <p><strong>(spectrification is left exact reflective)</strong></p> <p>The inclusion of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+exact+%28infinity%2C1%29-functor">left</a> <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28infinity%2C1%29-category">reflective</a>, hence it has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> which preserves <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mo>↪</mo><mover><mo>←</mo><mrow><msub><mi>L</mi> <mi>lex</mi></msub></mrow></mover></mover><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T \mathbf{H} \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow} \mathbf{H}^{seq} \,. </annotation></semantics></math></div></div> <p>(<a href="#Joyal08">Joyal 08, section 35.1</a>)</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Forming degreewise <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> constitutes an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo lspace="verythinmathspace">:</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup></mrow><annotation encoding="application/x-tex">\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq}</annotation></semantics></math> and by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>seq</mi></mrow><annotation encoding="application/x-tex">seq</annotation></semantics></math> this comes with a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> out of the identity</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>id</mi><mo>⟶</mo><mi>Ω</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \theta \;\colon\; id \longrightarrow \Omega \,. </annotation></semantics></math></div> <p>This in turn is compatible with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>∘</mo><mi>Ω</mi><mo>≃</mo><mi>Ω</mi><mo>∘</mo><mi>θ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ω</mi><mo>⟶</mo><mi>Ω</mi><mo>∘</mo><mi>Ω</mi><mo>=</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \theta \circ \Omega \simeq \Omega \circ \theta \;\colon\; \Omega \longrightarrow \Omega \circ \Omega = \Omega^2 \,. </annotation></semantics></math></div> <p>Consider then a sufficiently deep <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mi>tf</mi></msup></mrow><annotation encoding="application/x-tex">\rho^{tf}</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> available in the <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> this stabilizes, and hence provides a <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28infinity%2C1%29-category">reflection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mo>⟶</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">L \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}</annotation></semantics></math>.</p> <p>Since <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> is a <a class="existingWikiWord" href="/nlab/show/filtered+%28%E2%88%9E%2C1%29-colimit">filtered (∞,1)-colimit</a> and since in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> these commute with <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a>, it follows that spectrum objects are an <a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">left exact</a> <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> over the <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>, its <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-category">tangent (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> over the base <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">T \infty Grpd</annotation></semantics></math> (and hence in particular also over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> itself).</p> </div> <p>(<a href="#Joyal08">Joyal 08, section 35.5</a>)</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the the spectrification lemma <a class="maruku-ref" href="#SpectrificationLemma"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> into the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-presheaf (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{seq}</annotation></semantics></math>, and this implies that it is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> (by the discussion there).</p> <p>Moreover, since both <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> in the <a class="existingWikiWord" href="/nlab/show/global+section+geometric+morphism">global section geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Δ</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd</annotation></semantics></math> preserve <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a> they preserve <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> and hence their immediate <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a> prolongation immediately restricts to the inclusion of spectrum objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><mi>T</mi><mi>Γ</mi></mrow></munder><mover><mo>←</mo><mrow><mi>T</mi><mi>Δ</mi></mrow></mover></mover></mtd> <mtd><mi>T</mi><mn>∞</mn><mi>Grpd</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Δ</mi></mover></mover></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ T \mathbf{H} &\stackrel{\overset{T \Delta}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}}& T \infty Grpd \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} & \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} & \infty Grpd } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>We may think of the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-topos">tangent (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> as being an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> by its <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">Stab(\mathbf{H}) \simeq T_\ast \mathbf{H}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow></munder><mover><mo>←</mo><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow></mover></mover></mtd> <mtd><mi>Spectra</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><mi>T</mi><mi>Γ</mi></mrow></munder><mover><mo>←</mo><mrow><mi>T</mi><mi>Δ</mi></mrow></mover></mover></mtd> <mtd><mi>T</mi><mn>∞</mn><mi>Grpd</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>base</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>base</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><munder><mo>⟶</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Δ</mi></mover></mover></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Stab(\mathbf{H}) &\stackrel{\overset{Stab(\Delta)}{\leftarrow}}{\underset{Stab(\Gamma)}{\longrightarrow}}& Spectra \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ T\mathbf{H} &\stackrel{\overset{T\Delta}{\leftarrow}}{\underset{T\Gamma}{\longrightarrow}}& T \infty Grpd \\ \downarrow^{\mathrlap{base}} && \downarrow^{\mathrlap{base}} \\ \mathbf{H} &\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}& \infty Grpd } \,. </annotation></semantics></math></div> <p>Crucial for the internal interpretation in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is that the <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable homotopy types</a>.</p> </div> <h4 id="tangent_cohesive_homotopy_theory">Tangent cohesive homotopy theory</h4> <p>Now consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, in that there is an <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><mover><munder><mo>←</mo><mi>coDisc</mi></munder><mover><mo>⟶</mo><mi>Γ</mi></mover></mover><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>⟶</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mo>,</mo><mi>coDisc</mi></mrow><annotation encoding="application/x-tex">Disc, coDisc</annotation></semantics></math> being <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functors">full and faithful (∞,1)-functors</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> preserving finite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-products">(∞,1)-products</a>.</p> <p>Since <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limits">(∞,1)-limits</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimits">(∞,1)-colimits</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-presheaf (∞,1)-topos</a> are computed objectwise, this <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> immediately prolongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{seq}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>coDisc</mi> <mi>seq</mi></msup></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>Γ</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>Disc</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><msup><mi>Π</mi> <mi>seq</mi></msup></mrow></mover></mover><mn>∞</mn><msup><mi>Grpd</mi> <mi>seq</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}^{seq} \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} \infty Grpd^{seq} \,. </annotation></semantics></math></div> <p>Moreover, all three <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a> preserves the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullbacks">(∞,1)-pullbacks</a> involved in the characterization of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> and hence restrict to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>coDisc</mi> <mi>seq</mi></msup></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>Γ</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>Disc</mi> <mi>seq</mi></msup></mrow></mover></mover><mrow></mrow></mover><mi>T</mi><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T\mathbf{H} \stackrel{}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} T\infty Grpd \,. </annotation></semantics></math></div> <p>But then we have a further <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> given as the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mover><munder><mo>←</mo><mrow><msup><mi>Disc</mi> <mi>seq</mi></msup></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>Π</mi> <mi>seq</mi></msup></mrow></mover></mover><mn>∞</mn><msup><mi>Grpd</mi> <mi>seq</mi></msup><mover><munder><mo>←</mo><mrow></mrow></munder><mover><mo>⟶</mo><mi>L</mi></mover></mover><mi>T</mi><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T\mathbf{H} \hookrightarrow \mathbf{H}^{seq} \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\underset{Disc^{seq}}{\leftarrow}} \infty Grpd^{seq} \stackrel{\overset{L}{\longrightarrow}}{\underset{}{\leftarrow}} T \infty Grpd \,. </annotation></semantics></math></div> <p>Again since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+exact+%28%E2%88%9E%2C1%29-functor">left exact (∞,1)-functor</a> this composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>Π</mi></mrow><annotation encoding="application/x-tex">L \Pi</annotation></semantics></math> preserves finite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-products">(∞,1)-products</a>.</p> <p>So it follows in conclusion that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> then its tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> is itself a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> over the tangent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">T \infty Grpd</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a>, which is an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> of the cohesion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> by the cohesion of the stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(\mathbf{H})</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Spec</mi></mrow><annotation encoding="application/x-tex">Stab(\infty Grpd) \simeq Spec</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>coDisc</mi> <mi>seq</mi></msup></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>Γ</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>Disc</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><mi>L</mi><msup><mi>Π</mi> <mi>seq</mi></msup></mrow></mover></mover></mtd> <mtd><mi>Stab</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Spectra</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>≃</mo></mtd> <mtd></mtd> <mtd><mo>≃</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd></mtd> <mtd><msub><mi>T</mi> <mo>*</mo></msub><mn>∞</mn><mi>Grpd</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><munder><mo>←</mo><mrow><msup><mi>Ω</mi> <mn>∞</mn></msup><mo>∘</mo><mi>tot</mi></mrow></munder><mover><mo>⟶</mo><mi>d</mi></mover></mover></mtd> <mtd><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><mover><mover><munder><mo>←</mo><mrow><msup><mi>coDisc</mi> <mi>seq</mi></msup></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>Γ</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>←</mo><mrow><msup><mi>Disc</mi> <mi>seq</mi></msup></mrow></mover></mover><mover><mo>⟶</mo><mrow><mi>L</mi><msup><mi>Π</mi> <mi>seq</mi></msup></mrow></mover></mover></mtd> <mtd><mi>T</mi><mn>∞</mn><mi>Grpd</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>base</mi></mpadded></msup><mo stretchy="false">↓</mo><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>base</mi></mpadded></msup><mo stretchy="false">↓</mo><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd> <mtd><mover><mover><mover><munder><mo>←</mo><mi>coDisc</mi></munder><mover><mo>⟶</mo><mi>Γ</mi></mover></mover><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>⟶</mo><mi>Π</mi></mover></mover></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \simeq Spectra \\ && \simeq && \simeq \\ && T_\ast \mathbf{H} && T_\ast \infty Grpd \\ && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}& T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} \\ && \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd } \,. </annotation></semantics></math></div> <p>Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>∞</mn></msup><mo>∘</mo><mi>tot</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>⟶</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\Omega^\infty \circ tot \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}</annotation></semantics></math> assigns the <em>total space</em> of a spectrum bundle;</p> <p>its <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> is the <a href="tangent+%28infinity%2C1%29-category#CotangentComplex">tangent complex functor</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>base</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>⟶</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">base \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}</annotation></semantics></math> assigns the <em>base space</em> of a spectrum bundle;</p> <p>its <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> produces the 0-bundle.</p> </li> </ul> <p>Where the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a> in a general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> constitute a notion of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, those of a <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-topos">tangent (∞,1)-topos</a> specifically constitute <a class="existingWikiWord" href="/nlab/show/twisted+generalized+cohomology">twisted generalized cohomology</a>, in fact <a class="existingWikiWord" href="/nlab/show/twisted+bivariant+cohomology">twisted bivariant cohomology</a>.</p> <p>For consider a <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">E \in T_\ast \mathbf{H}</annotation></semantics></math> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(E) \in Grp(\mathbf{H})</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+units">∞-group of units</a>. Then the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of this on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is (by the discussion there) exhibited by an object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>E</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ E//GL_1(E) \\ \downarrow \\ \mathbf{B}GL_1(E) } \right) \;\;\; \in \;\;\; T_{\mathbf{B}GL_1(E)}\mathbf{H} \hookrightarrow T\mathbf{H} \,. </annotation></semantics></math></div> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">Pic(E) \in \mathbf{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-groupoid">Picard ∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> there is the universal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-line+bundle">(∞,1)-line bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>→</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\widehat{Pic(E)} \to Pic(E)) \in T \mathbf{H} \,. </annotation></semantics></math></div> <p>Now for any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>E</mi><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>E</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mi>X</mi></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \times E \simeq \left( \array{ E \times X \\ \downarrow \\ X } \right) \;\;\; \in \;\;\; T_{X}\mathbf{H} \hookrightarrow T\mathbf{H} \,, </annotation></semantics></math></div> <p>then morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> from the latter to the former</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>E</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⟶</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>E</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ E \times X \\ \downarrow \\ X } \right) \longrightarrow \left( \array{ E//GL_1(E) \\ \downarrow \\ \mathbf{B}GL_1(E) } \right) </annotation></semantics></math></div> <p>are equivalently</p> <ol> <li> <p>a choice of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twist of E-cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi \;\colon \; X \longrightarrow \mathbf{B}GL_1(E)</annotation></semantics></math>;</p> </li> <li> <p>an element in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in E^{\bullet}(X,E)</annotation></semantics></math>.</p> </li> </ol> <p>If we consider the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> then we can use just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">X \times E</annotation></semantics></math>:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mo>↪</mo><mn>0</mn></mover><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric homotopy type</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">E \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a>, then the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>/<a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow></msub><mo>∈</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> [X,E]_{T \mathbf{H}} \in T \mathbf{H} </annotation></semantics></math></div> <p>(with respect to the Cartesian <a class="existingWikiWord" href="/nlab/show/closed+monoidal+%28%E2%88%9E%2C1%29-category">closed monoidal (∞,1)-category</a> structure on the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is equivalently the <a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mrow><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow></msub><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [\Sigma^\infty X, E]_{Stab(\mathbf{H})} \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H} \,, </annotation></semantics></math></div> <p>in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow></msub><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mrow><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X,E]_{T \mathbf{H}} \simeq [\Sigma^\infty X,E]_{Stab(\mathbf{H})} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Notice that as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup></mrow><annotation encoding="application/x-tex">T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}</annotation></semantics></math>, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the constant <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>seq</mi></mrow><annotation encoding="application/x-tex">seq</annotation></semantics></math>. By the formula for the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mo>•</mo></msub><mo>≃</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>seq</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mo>•</mo><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X,E]_\bullet \simeq \mathbf{H}^{seq}(X \times \bullet, E) \,. </annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is constant the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo>•</mo></mrow><annotation encoding="application/x-tex">X \times \bullet</annotation></semantics></math> is for each object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>seq</mi></mrow><annotation encoding="application/x-tex">seq</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/representable+functor">presheaf represented</a> by that object. Therefore by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mo>•</mo></msub><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X,E]_\bullet \simeq [X,E_\bullet] \,. </annotation></semantics></math></div> <p>This is manifestly the same formula as for the <a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma^\infty X</annotation></semantics></math>.</p> </div> <p>By the same kind of argument we have the following more general statement.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mo>↪</mo><mn>0</mn></mover><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric homotopy type</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msub><mi>E</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \in E_\infty(\mathbf{H})</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>→</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">(\widehat{Pic(E)} \to Pic(E)) \hookrightarrow T \mathbf{H}</annotation></semantics></math> its universal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-line+bundle">(∞,1)-line bundle</a> over its <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-groupoid">Picard ∞-groupoid</a>, then the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>/<a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mover><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><msub><mo stretchy="false">]</mo> <mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow></msub><mo>∈</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> [X,\widehat{Pic(E)}]_{T \mathbf{H}} \in T \mathbf{H} </annotation></semantics></math></div> <p>is the object whose</p> <ul> <li> <p>base homotopy type is the <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, Pic(E)]</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twist</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/spectrum+bundle">spectrum bundle</a> is the collection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted E-cohomology spectra</a> for all twists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>.</p> </li> </ul> </div> <h4 id="CohesiveAndDifferentialRefinement">Cohesive and differential refinement in tangent cohesion</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T\mathbf{H}</annotation></semantics></math> be a tangent cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T_\ast \mathbf{H}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> inside it.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in T_\ast \mathbf{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{}{\longrightarrow}& A/\flat A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) } </annotation></semantics></math></div> <p>(of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> applied to the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> of the <a class="existingWikiWord" href="/nlab/show/counit+of+a+comonad">counit</a> of the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a>) is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> square.</p> </div> <p>This was observed in (<a href="tangent+cohesion#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13</a>). It is an incarnation of a <a class="existingWikiWord" href="/nlab/show/fracture+theorem">fracture theorem</a>.</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos">cohesion</a> and <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stability</a> we have the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>♭</mo><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Π</mi><mo stretchy="false">(</mo><mo>♭</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \flat A &\longrightarrow & A &\stackrel{}{\longrightarrow}& A/\flat A \\ \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow \\ \Pi(\flat A) &\longrightarrow& \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) } </annotation></semantics></math></div> <p>where both rows are <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a>. By <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos">cohesion</a> the left vertical map is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a>. The claim now follows with the <a href="homotopy+pullback#HomotopyFiberCharacterization">homotopy fiber characterization</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>This means that in stable cohesion every cohesive stable homotopy type is in controled sense a cohesive extension/refinement of its <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+infinity-groupoids">geometric realization</a> <a class="existingWikiWord" href="/nlab/show/discrete+infinity-groupoid">geometrically discrete</a> (“bare”) stable <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> by the non-<a class="existingWikiWord" href="/nlab/show/discrete+object">discrete</a> part of its cohesive structure;</p> <p>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A/\flat A</annotation></semantics></math> may be identified with differential cycle data. Indeed, by stability and cohesion it is the <a href="cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology">flat de Rham coefficient object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi><mo>=</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex"> A/\flat A = \flat_{dR}\Sigma A </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. So</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) } </annotation></semantics></math></div> <p>exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>-coefficient of the <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(A)</annotation></semantics></math> (<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13</a>).</p> <p>It follows by the discussion at <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a> that the further differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{A}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> should be given by a further <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mi>A</mi><mo>^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mo>♭</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \widehat{A} &\longrightarrow& \Omega^1(-,Lie(A)) \\ \downarrow &{}^{(pb)}& \downarrow \\ A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) } \,. </annotation></semantics></math></div></div> <h3 id="local_prequantum_field_theory">Local prequantum field theory</h3> <p>We describe the formulation of <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (<a href="#lpqft">lpqft</a>).</p> <p>A <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>/<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> is traditionally defined by an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>: given a <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{Fields}_{traj}</annotation></semantics></math> “of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a>” of a given <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a>, then the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{Fields}_{traj} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ U(1) } </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>. The idea of producing a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> from this is to</p> <ol> <li> <p>choose a linearization in the form of the group <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1) \longrightarrow GL_1(\mathbb{C})</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> of the complex numbers,</p> </li> <li> <p>choose a <a class="existingWikiWord" href="/nlab/show/measure">measure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>μ</mi></mrow><annotation encoding="application/x-tex">d\mu</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{Fields}_{traj}</annotation></semantics></math></p> </li> </ol> <p>and then declare that the <a class="existingWikiWord" href="/nlab/show/integral">integral</a> (“<a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∫</mo><mrow><mi>ϕ</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mrow></munder><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>μ</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \underset{\phi \in \mathbf{Fields}_{traj}}{\int} \exp(i S(\phi))\, d\mu \in \mathbb{C} </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of the theory a kind of <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> with <a class="existingWikiWord" href="/nlab/show/probabilities">probabilities</a> replaced by <a class="existingWikiWord" href="/nlab/show/probability+amplitudes">probability amplitudes</a>.</p> <p>In order to make sense of this (for a full discussion of “<a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a>” in this sense see (<a href="#Nuiten13">Nuiten 13</a>), here we concentrate on the <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum aspects</a>), it is useful to allow some more conceptual wiggling room by passing to <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>. Notice that if we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/smooth+groupoid">smooth</a> universal <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>, then an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> as above is equivalently a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mn>0</mn></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mn>0</mn></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mn>0</mn></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mn>0</mn></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{Fields}_{traj} \\ & \swarrow && \searrow \\ \ast && \swArrow && \ast \\ & {}_{\mathllap{0}}\searrow && \swarrow_{\mathrlap{0}} \\ && \mathbf{B}U(1) } \;\;\;\; \simeq \;\;\;\; \array{ && \mathbf{Fields}_{traj} \\ & \swarrow &\downarrow^{\mathrlap{\exp(i S)}}& \searrow \\ \ast &\leftarrow& U(1) &\rightarrow& \ast \\ & {}_{\mathllap{0}}\searrow &{}^{(pb)}& \swarrow_{\mathrlap{0}} \\ && \mathbf{B}U(1) } \,, </annotation></semantics></math></div> <p>where on the right we used the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> which exhibits the smooth <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math>.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (“<a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>”) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \;\colon\; X \longrightarrow \mathbf{B}U(1)_{conn}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> (“<a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>”) then for <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of shape the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>, the canonical <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> (“<a class="existingWikiWord" href="/nlab/show/Lorentz+force">Lorentz force</a> gauge <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a>”) is the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> <a class="existingWikiWord" href="/nlab/show/nonlinear+functional">functional</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>Lor</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mo>∇</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>⟶</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp(i S_{Lor}) \coloneqq \exp(i \int_{S^1} [S^1, \nabla]) \;\colon\; [S^1, X] \stackrel{[S^1, X]}{\longrightarrow} [S^1, \mathbf{B}U(1)_{conn}] \stackrel{\exp(i \int_{S^1}(-))}{\longrightarrow} U(1) \,. </annotation></semantics></math></div> <p>But more generally, if the <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> have a <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>, hence if they are of the shape of an <a class="existingWikiWord" href="/nlab/show/interval">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>≔</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I \coloneqq [0,1]</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> functional on <a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^1, X]</annotation></semantics></math> generalizes to the <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> on the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,X]</annotation></semantics></math> and there it is no longer a function, but exists only as a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mn>0</mn></msub></mrow></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>I</mi></msub><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>χ</mi> <mo>∇</mo></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>χ</mi> <mo>∇</mo></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && [I,X] \\ & {}^{(-)|_0}\swarrow && \searrow^{\mathrlap{(-)|_1}} \\ X && \swArrow_{\exp(i \int_{I}[I,\nabla])} && X \\ & {}_{\mathllap{\chi_\nabla}}\searrow && \swarrow_{\mathrlap{\chi_\nabla}} \\ && \mathbf{B}U(1) } \,. </annotation></semantics></math></div> <p>Notice that this is a “local” description of the action functional: the data that determines it is the boundary</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∇</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \\ \downarrow^{\mathrlap{\nabla}} \\ \mathbf{B}U(1)_{conn} } </annotation></semantics></math></div> <p>and from this the rest is induced by <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a>.</p> <p>A related class of examples are <a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondences">prequantized Lagrangian correspondences</a>: Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ω</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \\ \downarrow^{\mathrlap{\omega}} \\ \mathbf{\Omega}^2 } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>. Then a <a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X \longrightarrow X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>graph</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ω</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>ω</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && graph(f) \\ & \swarrow && \searrow \\ X && && X \\ & {}_{\mathllap{\omega}}\searrow && \swarrow_{\mathrlap{\omega}} \\ && \mathbf{\Omega}^2 } \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> is a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mo>∇</mo></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}U(1)_{conn} \\ & \searrow & \downarrow^{\mathrlap{F_{(-)}}} \\ && \mathbf{\Omega}^2 } </annotation></semantics></math></div> <p>and so a <a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondence">prequantized Lagrangian correspondence</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>graph</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo></mo><mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>∇</mo></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && graph(f) \\ & \swarrow && \searrow \\ X && \swArrow && X \\ & _{\mathllap{\nabla}}\searrow && \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}U(1)_{conn} } \,. </annotation></semantics></math></div> <p>To conceptualize all this, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo><mi>SmoothGrpd</mi><mo>≔</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>SmoothMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>Grpd</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mi>stalkwise</mi><mspace width="thickmathspace"></mspace><mi>equivalences</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \coloneqq SmoothGrpd \coloneqq Func(SmoothMfd^{op}, Grpd)[\{stalkwise\;equivalences\}^{-1}] </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> obtained from the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>-valued <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on the <a class="existingWikiWord" href="/nlab/show/category">category</a> of all <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> by universally turning <a class="existingWikiWord" href="/nlab/show/stalk">stalkwise</a> <a class="existingWikiWord" href="/nlab/show/equivalences+of+groupoids">equivalences of groupoids</a> into genuine <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> (“<a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a>”).</p> <p>This is the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a>/smooth (<a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli</a>) <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> Corr_1(\mathbf{H}) \in (2,1)Cat </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}U(1)_{conn}}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-topos">slice</a> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> over the smooth <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle bundles with connection</a>. Then the abovve diagrams are morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>∈</mo><msub><mi>Corr</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla \in Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>, hence the <a class="existingWikiWord" href="/nlab/show/smooth+group">smooth group</a> which is the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete</a> smooth 1-parameter subgroup</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℝ</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>↪</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathbf{B}\mathbb{R} \longrightarrow \mathbf{B}\mathbf{Aut}_{/\mathbf{B}U(1)_{conn}}(\nabla) \hookrightarrow \mathbf{H}_{/\mathbf{B}U(1)_{conn}} </annotation></semantics></math></div> <p>is equivalently a choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in C^\infty(X)</annotation></semantics></math> of a smooth function and sends</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">{</mo><mi>H</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>t</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>∇</mo></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> t \;\; \mapsto \;\; \left( \array{ X &&\stackrel{\exp(t \{H,-\})}{\longrightarrow}&& X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow_{\mathrlap{\exp(i S_t)}}& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}U(1)_{conn} } \right) \,, </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">{</mo><mi>H</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(t \{H,-\})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+flow">Hamiltonian flow</a> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>t</mi></msub><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mi>t</mi></msubsup><mi>L</mi></mrow><annotation encoding="application/x-tex">S_t = \int_0^t L</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hamilton-Jacobi+theory">Hamilton-Jacobi</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, the <a class="existingWikiWord" href="/nlab/show/integral">integral</a> of the <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, hence of its <a class="existingWikiWord" href="/nlab/show/Legendre+transform">Legendre transform</a>.</p> </li> </ol> </div> <p>(see <a href="#SchreiberClassical">Schreiber 13</a>).</p> <p>It is now clear how to pass from this to <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> of higher dimension.</p> <p>Let now more generally</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo>≔</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>SuperMfd</mi> <mi>op</mi></msup><mo>,</mo><mi>KanCplx</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">{</mo><mi>stalkwise</mi><mspace width="thickmathspace"></mspace><mi>homotopy</mi><mspace width="thickmathspace"></mspace><mi>equivalences</mi><msup><mo stretchy="false">}</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \coloneqq Smooth\infty Grpd \coloneqq Func(SuperMfd^{op}, KanCplx)[\{stalkwise\;homotopy\;equivalences\}^{-1}] </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> obtained from the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-valued <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on the category of all <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a> by universally turning <a class="existingWikiWord" href="/nlab/show/stalk">stalkwise</a> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> into actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>.</p> <p>We say that this is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <em><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoids">smooth super ∞-groupoids</a></em>/_<a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometric</a> <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a><em>.</em></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> Corr_n(\mathbf{H}) \in (\infty,n)Cat </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. This is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-category">symmetric monoidal (∞,n)-category</a> under the objectwise <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p><a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> has the special property that it is <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a> in that it is equipped with an <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> of <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functors">adjoint (∞,1)-functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><mover><munder><mo>←</mo><mi>coDisc</mi></munder><mover><mo>⟶</mo><mi>Γ</mi></mover></mover><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>⟶</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd </annotation></semantics></math></div> <p>which induce an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monads">(∞,1)-monads</a>/comonads</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>Π</mi><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>⟶</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \left( \Pi \dashv \flat \dashv \sharp \right) \;\colon\; \mathbf{H} \longrightarrow \mathbf{H} </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cartesian+product">product</a>-preserving, called</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <em><a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <em><a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a></em> .</li> </ul> <p>Here the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> sends a <a class="existingWikiWord" href="/nlab/show/simplicial+manifold">simplicial manifold</a> to the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">fat geometric realization</a> of the underlying <a class="existingWikiWord" href="/nlab/show/simplicial+topological+space">simplicial topological space</a>, hence in particular sends a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> to its <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Bord_n</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of framed n-dimensional cobordisms</a>.</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2Cn%29-functor">monoidal (∞,n)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Bord</mi> <mi>n</mi></msub><mo>⟶</mo><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; Bord_n \longrightarrow Corr_n(\mathbf{H}) </annotation></semantics></math></div> <p>is equivalently a choice of object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \in \mathbf{H}</annotation></semantics></math>. It sends a <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> of its <a class="existingWikiWord" href="/nlab/show/shape">shape</a> into the <a class="existingWikiWord" href="/nlab/show/higher+moduli+stack">higher moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><msub><mi>Σ</mi> <mi>in</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>Σ</mi> <mi>out</mi></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mi>Σ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msub><mi>Σ</mi> <mi>in</mi></msub></mrow></msub></mrow></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msub><mi>Σ</mi> <mi>out</mi></msub></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mi>in</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mi>out</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ && \Sigma \\ & \nearrow && \nwarrow \\ \Sigma_{in} && && \Sigma_{out} } \right) \;\; \mapsto \;\; \left( \array{ && [\Pi\Sigma, \mathbf{Fields}] \\ & {}^{(-)|_{\Sigma_{in}}}\swarrow && \searrow^{(-)|_{\Sigma_{out}}} \\ [\Pi(\Sigma_{in}), \mathbf{Fields}] && && [\Pi(\Sigma_{out}), \mathbf{Fields}] } \right) \,. </annotation></semantics></math></div></div> <p>(<a href="#lpqft">lpqft</a>)</p> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>Under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mo lspace="verythinmathspace">:</mo><mi>ChainComplexes</mi><mover><mo>⟶</mo><mo>≃</mo></mover><mi>SimplicialAbelianGroups</mi><mover><mo>⟶</mo><mi>forget</mi></mover><mi>KanComplexes</mi></mrow><annotation encoding="application/x-tex"> DK \colon ChainComplexes \stackrel{\simeq}{\longrightarrow} SimplicialAbelianGroups \stackrel{forget}{\longrightarrow} KanComplexes </annotation></semantics></math></div> <p>we have for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>DK</mi><mrow><mo>(</mo><munder><mi>U</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \flat \mathbf{B}^{n+1}U(1) \simeq DK \left( \underline{U}(1) \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^2 \stackrel{\mathbf{d}}{\longrightarrow} \cdots \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{n+1}_{cl} \right) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> </div> <div class="num_example" id="tYM"> <h6 id="example">Example</h6> <p>Consider the induced canonical inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^{n+1} \longrightarrow \flat \mathbf{B}^{n+1}U(1) \,. </annotation></semantics></math></div> <p>By the above we may regard this as an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> with <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^{n+1}_{cl}</annotation></semantics></math>. As such we denote it</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>tYM</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{\Omega}^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \flat \mathbf{B}^{n+1}U(1) } \,, </annotation></semantics></math></div> <p>where the subscript is supposed to refer to “universal higher <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a>”.</p> </div> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2Cn%29-functors">monoidal (∞,n)-functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Bord</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mstyle mathvariant="bold"><mi>Fields</mi></mstyle></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Bord_n &\stackrel{\exp(i S)}{\longrightarrow}& Corr_n(\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}) \\ & {}_{\mathbf{Fields}} \searrow & \downarrow \\ && Corr_n(\mathbf{H}) } </annotation></semantics></math></div> <p>are equivalent to objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>↪</mo><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \flat \mathbf{B}^{n+1}U(1) } \right) \;\;\; \in \;\;\; \mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)} \hookrightarrow Corr_n(\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}) \,. </annotation></semantics></math></div> <p>This sends the dual point to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(- i S)</annotation></semantics></math> and sends the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> to the <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(i S)</annotation></semantics></math> to the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>k</mi></msup><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^k , \mathbf{Fields}]</annotation></semantics></math>.</p> </div> <p>(<a href="#lpqft">lpqft</a>)</p> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Consider the induced canonical inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^{n+1} \longrightarrow \flat \mathbf{B}^{n+1}U(1) \,. </annotation></semantics></math></div> <p>By the above we may regard this as an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> with <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^{n+1}_{cl}</annotation></semantics></math>. As such we denote it</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>tYM</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{\Omega}^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \flat \mathbf{B}^{n+1}U(1) } \,, </annotation></semantics></math></div> <p>where the subscript is supposed to refer to “universal higher <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a>”.</p> </div> <p>Observe that by the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Bord_n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-category">symmetric monoidal (∞,n)-category</a> with <a class="existingWikiWord" href="/nlab/show/fully+dualizable+objects">fully dualizable objects</a> generated from a single object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub><mo>≃</mo><mi>FreeSMwD</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Bord_n \simeq FreeSMwD(\{\ast\}) \,. </annotation></semantics></math></div> <p>Let then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mo>∂</mo></msubsup><mo>≔</mo><mi>FreeSMwD</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>∅</mi><mo>⟶</mo><mo>*</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Bord_n^{\partial} \coloneqq FreeSMwD(\{\emptyset \longrightarrow \ast\}) </annotation></semantics></math></div> <p>the free <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-category">symmetric monoidal (∞,n)-category</a> with <a class="existingWikiWord" href="/nlab/show/fully+dualizable+objects">fully dualizable objects</a> generated from a single object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> and a single <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>⟶</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\emptyset \longrightarrow \ast</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> to the generating object. By the <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a>/<a class="existingWikiWord" href="/nlab/show/defect+field+theory">defect</a> version of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>, this is equivalently the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> with possibly a <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> component of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>.</p> <p>Hence a <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>Bord</mi> <mi>n</mi> <mo>∂</mo></msubsup></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><msup><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mo>∂</mo></msup></mrow></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Bord_n^\partial &\stackrel{\exp(i S)}{\longrightarrow}& Corr_n(\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}) \\ & {}_{\mathbf{Fields}^\partial} \searrow & \downarrow \\ && Corr_n(\mathbf{H}) } </annotation></semantics></math></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a> as above is equivalently a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bdr</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{Fields}_{bdr} \\ & \swarrow && \searrow \\ \ast && \swArrow && \mathbf{Fields} \\ & \searrow && \swarrow_{\mathrlap{\exp(i S)}} \\ && \flat \mathbf{B}^{n+1}U(1) } \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>The <em>universal</em> <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary condition</a> for the universal higher topological Yang-Mills theory of example <a class="maruku-ref" href="#tYM"></a> is the <a class="existingWikiWord" href="/nlab/show/higher+moduli+stack">higher moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>, hence a general boundary condition for this higher <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> is a <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a>].</p> </div> <p>The <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a> that we are after are boundaries of these boundary field theories, hence “corner field theories” (<a href="#Sati11">Sati 11</a>, <a href="#lpqft">lpqft</a>) of the higher universal topological Yang-Mills theory. This we turn to now.</p> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>The earliest and the only rigorously understood example of the <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> is the <a class="existingWikiWord" href="/nlab/show/AdS3-CFT2+and+CS-WZW+correspondence">AdS3-CFT2 and CS-WZW correspondence</a> between the <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> on a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and 3d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>.</p> <p>In (<a href="7d+Chern-Simons+theory#Witten98">Witten 98</a>) it is argued that all examples of the <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a> are governed by the <a class="existingWikiWord" href="/schreiber/show/infinity-Chern-Simons+theory">higher Chern-Simons theory</a> terms in the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> on one side of the correspondence, hence that the corresponding <span class="newWikiWord">conformal field theories] are higher dimensional analogs of the traditional [[WZW model<a href="/nlab/new/conformal+field+theories%5D+are+higher+dimensional+analogs+of+the+traditional+%5B%5BWZW+model">?</a></span>: that they are “<a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a>”-type models.</p> <p>In particular for <a class="existingWikiWord" href="/nlab/show/AdS7-CFT6">AdS7-CFT6</a> this means that the <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> should be a 6d-dimensional WZW model holographically related to the <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> which appears when <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> is <a class="existingWikiWord" href="/nlab/show/KK-reduction">KK-reduced</a> on a 4-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">3d <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></td><td style="text-align: left;"></td><td style="text-align: left;">2d <a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+model">Wess-Zumino-Witten model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> from <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> on <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> </tbody></table> <p>In (<a href="http://ncatlab.org/nlab/show/7d+Chern-Simons+theory#WittenI">Witten 96</a>) this is argued, by <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> after <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> to <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> 1, for the <em>bosonic and abelian</em> contribution in <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a>. (The subtle <a class="existingWikiWord" href="/nlab/show/theta+characteristic">theta characteristic</a> involved was later formalized in <a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Hopkins-Singer 02</a>.)</p> <p>In order to formalize this in generality, one needs a general formalization of <a class="existingWikiWord" href="/nlab/show/holography">holography</a> for <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> as these. How are <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a>-models higher holographic boundaries of <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a>? This we are dealing with at <em><a class="existingWikiWord" href="/nlab/show/Super+Gerbes">Super Gerbes</a></em>.</p> </div> <h3 id="motivic_quantization_of_twisted_fields">Motivic quantization of twisted fields</h3> <p>So far we have considered <a class="existingWikiWord" href="/nlab/show/configuration+spaces">configuration spaces</a> of fields, refined to <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a>. The next step is to consider aspects of the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of these fields, at least as an <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> (the full <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> being the corresponding <a class="existingWikiWord" href="/nlab/show/UV-completion">UV-completion</a>).</p> <p>By the <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> and specifically by <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a>, various of the twisted field configurations considered <a href="#Examples">above</a> participate either in <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> or in the corresponding <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a>.</p> <p>For instance the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>, after <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactification</a> to <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 7 in the context of <a href="AdS-CFT#AdS7CFT6">AdS7-CFT6</a>, has a <a class="existingWikiWord" href="/nlab/show/TQFT">topological</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> given by the <a href="intersection+pairing#SecondaryIntersectionPairing">secondary intersection pairing</a> <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">7d Chern-Simons theory</a> (or in fact, if quantum corrections are taken into account, a generalization of that to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math>-2-form fields <a href="#FSSb">FSSb</a>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of these higher CS theories yields canonical <a class="existingWikiWord" href="/nlab/show/states">states</a> in codimension 1, which by <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS-CFT</a> are interpreted as parts of the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+fields">self-dual higher gauge fields</a>.</p> <p>This is described at <em><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a></em> (<a href="#Nuiten2013">Nuiten 13</a>).</p> <h4 id="linearization_by_twisted_cohomology_spectra">Linearization by twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology spectra</h4> <p>Before actually quantizing a <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[ \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \mathbf{B}^n U(1)_{conn}}\right]</annotation></semantics></math> as above, we choose linear <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, given by</p> <ol> <li> <p>a choice of ground <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></p> <p>(playing the role of the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> in plain <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>);</p> </li> <li> <p>a choice of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \rho \;\colon\; \mathbf{B}^{n-1}U(1) \longrightarrow GL_1(E) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> of <a class="existingWikiWord" href="/nlab/show/phase+and+phase+space+in+physics">phases</a> to the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+units">∞-group of units</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, hence an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a> of the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></p> <p>(playing the role of the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">U(1) \hookrightarrow \mathbb{C}^\times</annotation></semantics></math> in plain <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>).</p> </li> </ol> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \longrightarrow \mathbf{B}^n U(1)</annotation></semantics></math> modulating a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ρ</mi></mover><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> X \longrightarrow \mathbf{B}^n U(1) \stackrel{\rho}{\longrightarrow} B GL_1(E) \longrightarrow E Mod </annotation></semantics></math></div> <p>modulates the <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>, which is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a>.</p> <p>Specifically, given the <a class="existingWikiWord" href="/nlab/show/higher+prequantum+bundle">higher prequantum bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\exp(i S) \;\colon\; \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn}</annotation></semantics></math> as above, the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mover><mo>⟶</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ρ</mi></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \chi \; \colon \; \mathbf{Fields} \stackrel{\exp(i S)}{\longrightarrow} \mathbf{B}^n U(1) \stackrel{\rho}{\longrightarrow} \mathbf{B} GL_1(E) \stackrel{}{\longrightarrow} Pic(E) \longrightarrow E Mod </annotation></semantics></math></div> <p>modulates the associated <em><a class="existingWikiWord" href="/nlab/show/higher+prequantum+line+bundle">higher prequantum E-line bundle</a></em>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/section">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> is a <em>higher <a class="existingWikiWord" href="/nlab/show/wavefunction">wavefunction</a></em>, hence a <em>higher <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></em>.</p> <p>(At this point this looks un-<a class="existingWikiWord" href="/nlab/show/polarization">polarized</a>, but in fact we will see in the next section that the notion of <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a> in <a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a> is automatic, but appears in a <a class="existingWikiWord" href="/nlab/show/holography">holographic</a>/<a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a> way in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math> instead here in codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.)</p> <p>Accordingly, the <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> is the higher <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> 0.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a> then the space of sections has a particularly nice description, on which we focus for a bit:</p> <p>The space of co-sections is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>→</mo></munder><mi>χ</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>E</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{\bullet + \chi}(X) \; \coloneqq\; \underset{\to}{\lim} \chi \; \in E Mod \,. </annotation></semantics></math></div> <p>This is also known as the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (<a href="#ABG">Ando-Blumberg-Gepner 10</a>).</p> <ul> <li> <p>a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \to E_{\bullet + \chi}(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E_{\bullet + \chi}(X) \to E</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <p>Hence we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>[</mo><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>,</mo><mspace width="thinmathspace"></mspace><mi>E</mi><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{\bullet + \chi}\left(X\right) \coloneqq \left[ E_{\bullet + \chi}\left(X\right), \, E \right] \,. </annotation></semantics></math></div></li> </ul> <p>Generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\chi_i \colon X_i \to E Mod</annotation></semantics></math> two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundles">(∞,1)-module bundles</a> over two spaces, a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>χ</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>χ</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_{\bullet + \chi_1}(X_1) \longrightarrow E_{\bullet + \chi_2}(X_2) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>χ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>χ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\chi_1, \chi_2)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/bivariant+cohomology">bivariant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>.</p> <p>Now given a <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a> on a space of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a>, hence a <a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a> as above, this induces an <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a> for linear maps between sections of <a class="existingWikiWord" href="/nlab/show/higher+prequantum+line+bundles">higher prequantum line bundles</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>in</mi></msub></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>out</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>χ</mi> <mi>in</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>χ</mi> <mi>out</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi><mi>Mod</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>in</mi></msub></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mi>ξ</mi></msub></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>out</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>in</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>out</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi><mi>Mod</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{Fields}_{traj} \\ & \swarrow && \searrow \\ \mathbf{Fields}_{in} && \swArrow_{\rho(\xi)} && \mathbf{Fields}_{out} \\ & {}_{\mathllap{\chi_{in}}}\searrow && \swarrow_{\mathrlap{\chi_{out}}} \\ && E Mod } \;\;\; \coloneqq \;\;\; \array{ && \mathbf{Fields}_{traj} \\ & \swarrow && \searrow \\ \mathbf{Fields}_{in} && \swArrow_\xi && \mathbf{Fields}_{out} \\ & {}_{\mathllap{\exp(i S_{in})}}\searrow && \swarrow_{\mathrlap{\exp(i S_{out})}} \\ && \mathbf{B}^n U(1) \\ && \downarrow \\ && B GL_1(E) \\ && \downarrow \\ && Pic(E) \\ && \downarrow \\ && E Mod } </annotation></semantics></math></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a> induced by the action functional, and acting on spaces of sections of the <a class="existingWikiWord" href="/nlab/show/higher+prequantum+line+bundle">higher prequantum line bundle</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> induced by these higher <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> is to be the <em><a class="existingWikiWord" href="/nlab/show/quantum+propagator">quantum propagator</a></em>. This we come to in the next section.</p> <p>Notice that forming co-sections constitutes an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></msub><mo>⟶</mo><mi>E</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{\bullet + (-)}(-) \;\colon\; \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \longrightarrow E Mod \,. </annotation></semantics></math></div> <p>Therefore forming co-sections sends an <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a> as above to a <a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-modules">(∞,1)-modules</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>i</mi> <mi>in</mi></msub></mrow></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>out</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>in</mi></msub></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>out</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>χ</mi> <mi>in</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>χ</mi> <mi>out</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi><mi>Mod</mi></mtd></mtr> <mtr><mtd><mspace width="thinmathspace"></mspace></mtd></mtr> <mtr><mtd><mspace width="thinmathspace"></mspace></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>χ</mi> <mi>in</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>in</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>in</mi></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow></mover></mtd> <mtd><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mi>i</mi> <mi>in</mi> <mo>*</mo></msubsup><msub><mi>χ</mi> <mi>in</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>≃</mo></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mi>i</mi> <mi>out</mi> <mo>*</mo></msubsup><msub><mi>χ</mi> <mi>out</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>traj</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>out</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>χ</mi> <mi>out</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>out</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{Fields}_{traj} \\ & {}^{i_{in}}\swarrow && \searrow^{\mathrlap{i_{out}}} \\ \mathbf{Fields}_{in} & & \swArrow_{\simeq} & & \mathbf{Fields}_{out} \\ & {}_{\mathllap{\chi_{in}}}\searrow && \swarrow_{\mathrlap{\chi_{out}}} \\ && E Mod \\ \, \\ \, \\ E_{\bullet + \chi_{in}}(\mathbf{Fields}_{in}) & \stackrel{(i_{in})_\ast}{\longleftarrow}& \array{ E_{\bullet + i_{in}^\ast\chi_{in}}(\mathbf{Fields}_{traj}) \\ \simeq \\ E_{\bullet + i_{out}^\ast\chi_{out}}(\mathbf{Fields}_{traj}) } & \stackrel{(i_{out})^\ast}{\longrightarrow}& E_{\bullet + \chi_{out}}(\mathbf{Fields}_{out}) } \,. </annotation></semantics></math></div> <p>The actual <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>/<a class="existingWikiWord" href="/nlab/show/path+integral+as+a+pull-push+transform">path integral as a pull-push transform</a> map now consists in forming <a class="existingWikiWord" href="/nlab/show/dual+morphism">dual morphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">E Mod</annotation></semantics></math> such as to turn one of the projections of such a correspondence arround a produce a <a class="existingWikiWord" href="/nlab/show/quantum+propagator">quantum propagator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>χ</mi> <mi>in</mi></msub></mrow></msup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>in</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>χ</mi> <mi>out</mi></msub></mrow></msup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>out</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^{\bullet + \chi_{in}}(\mathbf{Fields}_{in}) \longrightarrow E^{\bullet + \chi_{out}}(\mathbf{Fields}_{out}) </annotation></semantics></math></div> <p>that maps the incoming <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a>/<a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> to the outgoing ones.</p> <h4 id="ExpositionCohomologicalQuantization">Cohomological quantization by pull-push</h4> <p>What we need now for <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> is a <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> map that adds up the values of the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> over the space of trajectories, a functor of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Corr</mi> <mi>n</mi></msub><msup><mrow><mo>(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow> <mo>⊗</mo></msup><mo>→</mo><mo stretchy="false">(</mo><mi>E</mi><msup><mi>Mod</mi> <mrow><msup><mo>□</mo> <mi>n</mi></msup></mrow></msup><msup><mo stretchy="false">)</mo> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex"> \int (-) \;\; \colon \;\; Corr_n\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes \to (E Mod^{\Box^n})^\otimes </annotation></semantics></math></div> <p>As such this will in general only exist for <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Dijkgraaf-Witten+theory">∞-Dijkgraaf-Witten theory</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a> and hence has a “counting measure”. This case has been considered in (<a href="#FreedHopkinsLurieTeleman09">Freed-Hopkins-Lurie-Teleman 09</a>, <a href="#Morton10">Morton 10</a>).</p> <p>In the general case the path integral requires that we <em>choose</em> a suitable <a class="existingWikiWord" href="/nlab/show/measure">measure</a>/<a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> on the spaces of fields. We see below what this means, for the moment we just write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Corr</mi> <mi>n</mi> <mi>or</mi></msubsup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msup><mo stretchy="false">)</mo> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex"> Corr^{or}_n(\mathbf{H}_{/\mathbf{B}^n U(1)})^\otimes </annotation></semantics></math></div> <p>(i.e. with an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mi>or</mi></msup></mrow><annotation encoding="application/x-tex">{(-)}^{or}</annotation></semantics></math>-superscript) as a mnemonic for a suitable <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> of suitably oriented/measured spaces of fields with action functional. Then we may consider lifts of the action functional to <em>measure-valued</em> action functionals</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>μ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Corr</mi> <mi>n</mi> <mi>or</mi></msubsup><msup><mrow><mo>(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow> <mo>⊗</mo></msup><mo>→</mo><msup><mrow><mo>(</mo><mi>E</mi><msup><mi>Mod</mi> <mrow><msup><mo>□</mo> <mi>n</mi></msup></mrow></msup><mo>)</mo></mrow> <mo>⊗</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp(i S) \, d\mu \;\colon\; Corr_n^{or}\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes \to \left( E Mod^{\Box^n} \right)^\otimes \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> is then to be a monoidal functor of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Corr</mi> <mi>n</mi> <mi>or</mi></msubsup><msup><mrow><mo>(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow> <mo>⊗</mo></msup><mo>→</mo><msup><mrow><mo>(</mo><mi>E</mi><msup><mi>Mod</mi> <mrow><msup><mo>□</mo> <mi>n</mi></msup></mrow></msup><mo>)</mo></mrow> <mo>⊗</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int(-) \;\colon\; Corr_n^{or}\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes \to \left( E Mod^{\Box^n} \right)^\otimes \,. </annotation></semantics></math></div> <p>This we discuss now below. Once we have such a <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> functor, the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> process is its <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with the given <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>μ</mi></mrow><annotation encoding="application/x-tex">\exp(i S) \, d \mu</annotation></semantics></math> to obtain the genuine quantized <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>∫</mo><mrow><mi>ϕ</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow></munder><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>μ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msubsup><mi>Bord</mi> <mi>n</mi> <mo>⊗</mo></msubsup></mtd> <mtd><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>μ</mi></mrow></mover></mtd> <mtd><msubsup><mi>Corr</mi> <mi>n</mi> <mi>or</mi></msubsup><msup><mrow><mo>(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow> <mo>⊗</mo></msup></mtd> <mtd><mover><mo>→</mo><mrow><mo>∫</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mrow><mo>(</mo><mi>E</mi><msup><mi>Mod</mi> <mrow><msup><mo>□</mo> <mi>n</mi></msup></mrow></msup><mo>)</mo></mrow> <mo>⊗</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Fields</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>Corr</mi> <mi>n</mi></msub><msup><mrow><mo>(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>)</mo></mrow> <mo>⊗</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{\phi \in \mathbf{Fields}}{\int} \exp(i S(\phi)) \, d \mu(\phi) & \colon & Bord_n^\otimes &\stackrel{\exp(i S)\, d\mu}{\to}& Corr_n^{or}\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes &\stackrel{\int (-) }{\to}& \left( E Mod^{\Box^n} \right)^\otimes \\ && & {}_{\mathllap{Fields}}\searrow & \downarrow \\ && && Corr_n\left(\mathbf{H}\right)^\otimes } \,. </annotation></semantics></math></div> <p>We realize this now by <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">fiber integration in generalized cohomology</a>.</p> <p>While traditionally the definition of <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> is notoriously elusive, here we make use of general abstract but basic facts of <a class="existingWikiWord" href="/nlab/show/higher+linear+algebra">higher linear algebra</a> in a <em><a class="existingWikiWord" href="/nlab/show/tensor+%28%E2%88%9E%2C1%29-category">tensor (∞,1)-category</a></em> (a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable</a> and <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>): the simple basic idea is that</p> <p><em>Cohomological integration</em></p> <ol> <li> <p><em>Fiber integration of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-modules along a map is forming the <a class="existingWikiWord" href="/nlab/show/dual+morphisms">dual morphisms</a></em> of pulling back <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-modules.</p> </li> <li> <p>The choice of <em>measure</em> against which one integrates is the choice of identification of <a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a>.</p> </li> </ol> <p>More in detail, given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^\otimes</annotation></semantics></math> and given a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f \;\colon\; V_1 \to V_2 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, an <em><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>/<a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward</a>/<a class="existingWikiWord" href="/nlab/show/index">index map</a></em> is just</p> <ul> <li> <p>forming the <a class="existingWikiWord" href="/nlab/show/dual+morphism">dual morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>∨</mo></msup><mo lspace="verythinmathspace">:</mo><msubsup><mi>V</mi> <mn>2</mn> <mo>∨</mo></msubsup><mo>→</mo><msubsup><mi>V</mi> <mn>1</mn> <mo>∨</mo></msubsup></mrow><annotation encoding="application/x-tex">f^\vee \colon V_2^\vee \to V_1^\vee</annotation></semantics></math>;</p> </li> <li> <p>such that <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>V</mi> <mi>i</mi> <mo>∨</mo></msubsup><mo>≃</mo><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i^\vee \simeq V_i</annotation></semantics></math> exhbiting self-<a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a> exist (<a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a>) and have been chosen (<a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation</a>).</p> </li> </ul> <p>This allows in total to have a morphism between the same objects, but in the opposite direction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>!</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>V</mi> <mn>2</mn> <mo>∨</mo></msubsup><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mo>∨</mo></msup><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>V</mi> <mn>1</mn> <mo>∨</mo></msubsup><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>V</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^! \;\colon\; V_2 \stackrel{\;\simeq\;}{\to} V_2^\vee \stackrel{\; f^\vee \;}{\to} V_1^\vee \stackrel{\; \simeq \;}{\to} V_1 \,. </annotation></semantics></math></div> <p>That this is also the mechanism of <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">fiber integration in generalized cohomology</a> is almost explicit in the literature (<a class="existingWikiWord" href="/nlab/show/Alexander-Whitehead-Atiyah+duality">Alexander-Whitehead-Atiyah duality</a>), if maybe not fully clearly so. The statement is discussed explicitly in (<a href="#Nuiten13">Nuiten 13, section 4.1</a>).</p> <p>First, the basic example to keep in mind of is integration in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>H</mi><mi>R</mi><mo>=</mo><mi>H</mi><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">E = H R = H \mathbb{C}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> of the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a manifold, the <a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><msup><mi>R</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>H</mi><mi>R</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> H R^\bullet(X) \coloneqq [X,H R] </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, its dual the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>, with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> asserts that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msup><mi>R</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>H</mi><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">H R^\bullet(X) \in H R Mod</annotation></semantics></math> is essentially a self-<a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a>, except for a shift in degree: a choice of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><msup><mi>R</mi> <mo>•</mo></msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><munderover><mo>→</mo><mo>≃</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>PD</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munderover><msup><mrow><mo>(</mo><mi>H</mi><msup><mi>R</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow> <mo>∨</mo></msup><mo>≃</mo><mi>H</mi><msub><mi>R</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H R^\bullet\left(X\right) \underoverset{\simeq}{\;\; PD_X\;\;}{\to} \left( H R^{\bullet+ dim(X)}\left(X\right)\right)^\vee \simeq H R_{\bullet + dim(X)}\left(X\right) \,. </annotation></semantics></math></div> <p>Using this, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a map of <a class="existingWikiWord" href="/nlab/show/closed+manifolds">closed manifolds</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>, a compatible choice of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> induces from the canonical push-forward map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_\ast</annotation></semantics></math> on homology the <a class="existingWikiWord" href="/nlab/show/Umkehr+map">Umkehr map</a>/push-forward map on cohomology, by the composition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>!</mo></msup><mo>=</mo><msub><mo>∫</mo> <mi>f</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>H</mi><msup><mi>R</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>→</mo><mo>≃</mo><mrow><mspace width="thickmathspace"></mspace><msub><mi>PD</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace></mrow></munderover><mi>H</mi><msub><mi>R</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><msub><mi>f</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace></mrow></mover><mi>H</mi><msub><mi>R</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><munderover><mo>→</mo><mo>≃</mo><mrow><mspace width="thickmathspace"></mspace><msubsup><mi>PD</mi> <mi>Y</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace></mrow></munderover><mi>H</mi><msup><mi>R</mi> <mrow><mi>n</mi><mo>−</mo><mo stretchy="false">(</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^! = \int_f \;\colon\; H R^\bullet(X) \underoverset{\simeq}{\;PD_X\;}{\to} H R_{\bullet + dim(X)}(X) \stackrel{\; f_\ast \;}{\to} H R_{\bullet + dim(X)}(Y) \underoverset{\simeq}{\;PD_Y^{-1}\;}{\to} H R^{n-(dim(X)-dim(Y))}(Y) \,. </annotation></semantics></math></div> <p>This is ordinary <a class="existingWikiWord" href="/nlab/show/integration">integration</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msup><mi>ℝ</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \mathbb{R}^\bullet(X)</annotation></semantics></math> is modeled by <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PD</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">PD_X</annotation></semantics></math> is given by a choice of <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>!</mo></msup><mo>=</mo><msub><mo>∫</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">f^! = \int_{f}</annotation></semantics></math> is ordinary <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a>.</p> <p>The shift in degree here seems to somewhat break the simple pattern. In fact this is not so, if only we realize that since we are working over spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we should use a <em>relative/fiberwise</em> point of view and regard not duality in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">E Mod</annotation></semantics></math> itself, but in the functor categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(X, E Mod)</annotation></semantics></math>, which is fiberwise duality in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">E Mod</annotation></semantics></math>.</p> <p>Accordingly, given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \chi \;\colon \; X \to E Mod </annotation></semantics></math></div> <p>we form not just the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E^\bullet(X) = [X, E]</annotation></semantics></math> as above, but form the space of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of this bundle, which we write:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^{\bullet + \chi}(X) \coloneqq \Gamma_X(\chi) </annotation></semantics></math></div> <p>Here for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">[</mo><munder><mi>lim</mi><mo>→</mo></munder><mi>χ</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(\chi) \coloneqq [\underset{\to}{\lim} \chi, E] \,. </annotation></semantics></math></div> <p>Consider now a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>χ</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>χ</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi><mi>Mod</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X && \stackrel{f}{\to} && Y \\ & {}_{\mathllap{f^\ast \chi}}\searrow &\swArrow& \swarrow_{\mathrlap{\chi}} \\ && E Mod } </annotation></semantics></math></div> <p>along which we want to integrate, whith <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> invertible in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(Y, E Mod)</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><msup><mi>χ</mi> <mo>∨</mo></msup><mo>)</mo></mrow> <mo>∨</mo></msup><mo>≃</mo><mi>χ</mi></mrow><annotation encoding="application/x-tex">\left(\chi^\vee\right)^\vee \simeq \chi</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><msup><mi>χ</mi> <mo>∨</mo></msup><mo>)</mo></mrow> <mo>∨</mo></msup><mo>≃</mo><mi>χ</mi></mrow><annotation encoding="application/x-tex">\left(\chi^\vee\right)^\vee \simeq \chi</annotation></semantics></math>.</p> <p>Observe that we have the pair of <a class="existingWikiWord" href="/nlab/show/adjoint+triples">adjoint triples</a> of left/right <a class="existingWikiWord" href="/nlab/show/Kan+extensions">Kan extensions</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>/<a class="existingWikiWord" href="/nlab/show/limits">limits</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>Func</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mover><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow></mover></mover></mtd> <mtd><mi>Func</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mover><munder><mo>→</mo><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow></mover></mover></mtd> <mtd><mi>E</mi><mi>Mod</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\to}& Y &\stackrel{p}{\to}& \ast \\ \\ Func(X, E Mod) &\stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\to}}} & Func(Y, E Mod) &\stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^\ast}{\leftarrow}}{\underset{p_\ast}{\to}}} & E Mod } \,. </annotation></semantics></math></div> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^\ast</annotation></semantics></math> preserves duals, but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> may not.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><msup><mi>χ</mi> <mo>∨</mo></msup></mrow><annotation encoding="application/x-tex">f_! f^\ast \chi^\vee</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a>, say that a <em>choice of twisted orientation</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> is a choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\beta \colon X \to E Mod</annotation></semantics></math> together with a choice of a <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> (if such exists) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PD</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mrow><mo>(</mo><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><msup><mi>χ</mi> <mo>∨</mo></msup><mo>)</mo></mrow> <mo>∨</mo></msup><mo>≃</mo><msub><mi>f</mi> <mo>!</mo></msub><mrow><mo>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>χ</mi><mo>+</mo><mi>β</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> PD \;\colon \; \left( f_! f^\ast \chi^\vee \right)^\vee \simeq f_!\left( f^\ast \chi + \beta \right) \,, </annotation></semantics></math></div> <p>hence a choice of correction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> preserving the duality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>χ</mi></mrow><annotation encoding="application/x-tex">f^\ast \chi</annotation></semantics></math>.</p> <p>Then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_! \dashv f^\ast)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><msup><mi>χ</mi> <mo>∨</mo></msup><mo>→</mo><msup><mi>χ</mi> <mo>∨</mo></msup></mrow><annotation encoding="application/x-tex"> f_! f^\ast \chi^\vee \to \chi^\vee </annotation></semantics></math></div> <p>induces the <a class="existingWikiWord" href="/nlab/show/dual+morphism">dual morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>⟶</mo><msup><mrow><mo>(</mo><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><msup><mi>χ</mi> <mo>∨</mo></msup><mo>)</mo></mrow> <mo>∨</mo></msup><munderover><mo>⟶</mo><mo>≃</mo><mi>PD</mi></munderover><msub><mi>f</mi> <mo>!</mo></msub><mrow><mo>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>χ</mi><mo>+</mo><mi>β</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \chi \longrightarrow \left(f_! f^\ast \chi^\vee \right)^\vee \underoverset{\simeq}{PD}{\longrightarrow} f_!\left( f^\ast \chi + \beta \right) </annotation></semantics></math></div> <p>and under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><msub><mi>p</mi> <mo>!</mo></msub><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[ p_! \left( - \right), E \right]</annotation></semantics></math> this becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><msub><mi>p</mi> <mo>!</mo></msub><msub><mi>f</mi> <mo>!</mo></msub><mrow><mo>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>χ</mi><mo>+</mo><mi>β</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mo>]</mo></mrow><mo>⟶</mo><mrow><mo>[</mo><msub><mi>p</mi> <mo>!</mo></msub><mi>χ</mi><mo>,</mo><mi>E</mi><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> \left[p_! f_! \left(f^\ast \chi + \beta\right), E\right] \longrightarrow \left[p_! \chi , E\right] </annotation></semantics></math></div> <p>which is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>f</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>χ</mi><mo>+</mo><mi>β</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>χ</mi></mrow></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int_f \;\colon \; E^{\bullet + f^\ast \chi + \beta}(X) \longrightarrow E^{\bullet + \chi}(Y) \,. </annotation></semantics></math></div> <p>This we may call the the <em><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">fiber integration</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, or the <em>twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/index">index</a> map</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math><em>, induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>β</mi><mo>,</mo><mi>PD</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\beta, PD)</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\beta = 0</annotation></semantics></math> then we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PD</mi></mrow><annotation encoding="application/x-tex">PD</annotation></semantics></math> an</em><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation</a>_ of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted cohomology.</p> <p>Notice that</p> <ul> <li> <p>Under fiber integration in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, the twist may change.</p> </li> <li> <p>Grading in cohomology is just one incarnation of twist. Hence the fact that the twist changes under duality was already seen above in the ordinary case of <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>.</p> </li> <li> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+duality">Atiyah duality</a> identifies the dual cohomology spectrum with the <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> cohomology spectrum. Then a choice of orientation amounts to a choice of <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a>, as traditionally considered.</p> </li> </ul> <h2 id="GeneralTheory">General theory</h2> <p>We survey here some key aspects of a general theory of geometric twisted differential cohomology, following (<a href="#Schreiber">DCCT</a>), in which the above examples find a formal home. This is meant as a reference for readers of the <em><a href="#Examples">Examples</a></em>-section who wish to see pointers to formal details.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>We base the formulation of <a class="existingWikiWord" href="/nlab/show/physics">physics</a>/<a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> on the <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> of <em><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></em>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">interpreted</a> in <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a></em>. This provides a nicely natural and expressive language for the purpose of twisted smooth cohomology in string theory.</p> <p>The following table indicates the hierarchy of <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> that we invoke, the fragments of <a class="existingWikiWord" href="/nlab/show/theory">theory</a> that can be <a class="existingWikiWord" href="/nlab/show/categorical+semantics">interpreted</a> with these and the <a class="existingWikiWord" href="/nlab/show/models">models</a> that we need. Essentially all of the above discussion works in the model <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. A more encompassing treatment uses <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> and works in the model <a class="existingWikiWord" href="/nlab/show/SmoothSuper%E2%88%9EGrpd">SmoothSuper∞Grpd</a>.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/axioms">axioms</a></th><th><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></th><th><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></th><th>typical <a class="existingWikiWord" href="/nlab/show/models">models</a></th><th>expressiveness</th></tr></thead><tbody><tr><td style="text-align: left;">bare minimum</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, <a class="existingWikiWord" href="/nlab/show/Super%E2%88%9EGrpd">Super∞Grpd</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, <a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal bundles</a>, <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, <a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated</a> and <a class="existingWikiWord" href="/nlab/show/twisted+bundles">twisted bundles</a></td></tr> <tr><td style="text-align: left;">+<a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">locality</a>+<a class="existingWikiWord" href="/nlab/show/locally+infinity-connected+%28infinity%2C1%29-topos">∞-connectedness</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, <a class="existingWikiWord" href="/nlab/show/SmoothSuper%E2%88%9EGrpd">SmoothSuper∞Grpd</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/infinity-Chern-Weil+theory">Chern-Weil theory</a> <a class="existingWikiWord" href="/schreiber/show/infinity-Chern-Simons+theory">Chern-Simons theory</a>, <a class="existingWikiWord" href="/schreiber/show/infinity-geometric+prequantization">geometric quantization</a></td></tr> <tr><td style="text-align: left;">+<a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion">infinitesimals</a></td><td style="text-align: left;">differential homotopy type theory</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion">differential (∞,1)-topos</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a>, <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+infinity-groupoid">étale groupoid</a>, <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></td></tr> </tbody></table> <h3 id="homotopy_type_theory">Homotopy type theory</h3> <p>Traditionally a <em><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></em> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> regarded up to <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, hence equivalently an <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></em>. More generally, we think of parameterized homotopy types – of <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a></em> or <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a></em> – as <em>geometric homotopy types</em>. The collection of such forms an <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. One regards <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a> as part of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, and, more specifically, the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy-type theory</a></em>.</p> <p>We discuss now some basic structures that are expressible in such bare homotopy-type theory. (The fundamentals are due to <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Rezk</a> and <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Lurie</a>, see <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em>. We point out the perspective of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> in slices and add some aspects about higher bundle theory from (<a href="#NSS">NSS</a>)).</p> <p>Where useful, we indicate some of the discussion in formal <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a>, see <em><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></em> for more along such lines.</p> <h4 id="Cohomology">Cohomology</h4> <p>A <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28infinity%2C1%29-category">group object</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is a groupal <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞</a>-homotopy type: an <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></em>.</p> <p>By <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a> there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mover><munderover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>≃</mo></munderover><mover><mo>←</mo><mi>Ω</mi></mover></mover><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}} \mathbf{H}_{\geq 1}^{*/} </annotation></semantics></math></div> <p>between group objects and <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/n-connected+object+of+an+%28%E2%88%9E%2C1%29-topos">connected</a> homotopy types.</p> <div class="num_remark" id="DeloopingInTypeTheory"> <h6 id="remark_6">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">pt : \mathbf{B}G</annotation></semantics></math> is the essentially uniqe point of the <a class="existingWikiWord" href="/nlab/show/n-connected+object+of+an+%28%E2%88%9E%2C1%29-topos">connected</a> type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>, then the group type itself is simply</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>≔</mo><mo>⊢</mo><mo stretchy="false">(</mo><mi>pt</mi><mo>≃</mo><mi>pt</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Type</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G \coloneqq \vdash (pt \simeq pt) : Type \,, </annotation></semantics></math></div> <p>the type of auto-equivalences of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a group object which admits an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> any object, we write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \mathbf{B}^n A)</annotation></semantics></math> for the space of degree-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">c : X \to A</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X, \mathbf{B}^n A)</annotation></semantics></math> for the degree-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> </ul> <h4 id="PrincipalBundles">Principal bundles</h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-cocycles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> have an equivalent geometric interpretation as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>: these are types</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \\ \downarrow \\ X } </annotation></semantics></math></div> <p>over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho : P \times G \to P</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">∞-quotient</a> of the action is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p><strong>Theorem</strong> There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-cohomology over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mover><munderover><mo>→</mo><mi>fib</mi><mo>≃</mo></munderover><mover><mo>←</mo><munder><mi>lim</mi><mo>→</mo></munder></mover></mover><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \mathbf{B}G) \stackrel{\overset{\underset{\to}{\lim}}{\leftarrow}}{\underoverset{fib}{\simeq}{\to}} G Bund(X) </annotation></semantics></math></div> <p>From left to righr the equivalence is established by sending a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X\to \mathbf{B}G</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>.</p> <p>(<a href="#NSS">NSS</a>)</p> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> the type as in remark <a class="maruku-ref" href="#DeloopingInTypeTheory"></a>, the principal bundle corresponding to a coycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{B}G</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>:</mo><mi>X</mi><mo>⊢</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>pt</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex"> P \; \coloneqq \; x : X \vdash ( \phi(x) \simeq pt ) : Type </annotation></semantics></math></div> <p>And the action is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>ρ</mi><mo>:</mo><mi>P</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>X</mi><mo>;</mo><mi>p</mi><mo>:</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>pt</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo>:</mo><mo stretchy="false">(</mo><mi>pt</mi><mo>≃</mo><mi>pt</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>x</mi><mo>:</mo><mi>X</mi><mo>;</mo><mi>g</mi><mo>∘</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \rho : P \times G \to G \\ & \;\;\coloneqq (( x : X; p : (\phi(x) \simeq pt)), g : (pt \simeq pt) ) \mapsto (x : X; g \circ \phi) \end{aligned} </annotation></semantics></math></div></div> <h4 id="TwistedCohomology">Twisted cohomology</h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in \mathbf{H}</annotation></semantics></math>, the collection of morphisms into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/ \mathbf{B}G}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math> is naturally interpreted as follows</p> <ol> <li> <p>a local coefficient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>:</mo><mi>E</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{c} : E \to \mathbf{B}G</annotation></semantics></math> is a <em>universal bundle of local coefficients</em>;</p> </li> <li> <p>a domain object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{B}G</annotation></semantics></math> is a <em>twisting bundle</em>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>σ</mi></mover></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\sigma}{\to}&& E \\ & {}_{\mathllap{\phi}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>is equivalently a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi^* E \to X</annotation></semantics></math>;</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>E</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \phi^* E &\to & E \\ &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow^{\mathbf{c}} \\ X &\stackrel{id}{\to}& X &\stackrel{\phi}{\to}& \mathbf{B}G } </annotation></semantics></math></div></li> </ol> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><msub><mi>Struc</mi> <mi>ϕ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{c}Struc_{\phi}(X) \coloneqq \mathbf{c}(\phi) \coloneqq \mathbf{H}_{/\mathbf{B}G}(\phi, \mathbf{c}) </annotation></semantics></math></div> <p>is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (with local coefficients in the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>).</p> </li> </ul> <div class="num_remark"> <h6 id="remark_8">Remark</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> of the <a class="existingWikiWord" href="/nlab/show/theory">theory</a> the <a class="existingWikiWord" href="/nlab/show/type">type</a> of twisted cocycles is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ϕ</mi><mo>,</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">]</mo><mo>≔</mo><mo>⊢</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>b</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Type</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>X</mi></mrow></munder><mi>E</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex"> [\phi, \mathbf{c}] \coloneqq \vdash \prod_{b \in \mathbf{B}G} (X(b) \to E(b)) : Type \;\; = \;\; \vdash \prod_{x : X} E(\phi(x)) : Type </annotation></semantics></math></div> <p>(see <a href="http://www.ncatlab.org/nlab/show/locally%20cartesian%20closed%20category#RelationCartesianClosureBaseChangeInTypeTheory">here</a>).</p> <p>While on the right this expresses the collection of sections of the pullback bundle, the left hand side expresses explicitly a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>-parameterized collection of cocycles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(b) \to E(b)</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_9">Remark</h6> <p>Cocycles in twisted cohomology relative to a local coefficient bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>:</mo><mi>E</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{c} : E \to \mathbf{B}G</annotation></semantics></math> do not pull back along morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (unless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is trivial), but do pull back along morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> that are lifted to morphisms in the slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/ \mathbf{B}G}</annotation></semantics></math>.</p> </div> <h4 id="AssociatedAndTwistedBundles">Associated and twisted bundles</h4> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}A &\to& \mathbf{B}\hat G \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>a universal local coefficient bundle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{B}G</annotation></semantics></math> a twist and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\sigma : X \to \mathbf{B}\hat G</annotation></semantics></math> a section, hence a cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, the corresponding geometric object is the <a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \sigma</annotation></semantics></math> on the total space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mi>P</mi><mo stretchy="false">˜</mo></mover></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mover><mi>σ</mi><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>σ</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde P &\to& * \\ \downarrow && \downarrow \\ P &\stackrel{\tilde \sigma}{\to}& \mathbf{B}A &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{\sigma}{\to}& \mathbf{B}\hat G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>(<a href="#NSS">NSS</a>)</p> <p>(…)</p> <h3 id="cohesive_homotopy_type_theory">Cohesive homotopy type theory</h3> <p>In general the <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> encoded by an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> can be exotic. Two extra axioms ensure that it is modeled locally on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connected geometrical archetypes, such as for instance on <em><a class="existingWikiWord" href="/nlab/show/open+disks">open disks</a></em> for <a class="existingWikiWord" href="/nlab/show/Euclidean-topological+%E2%88%9E-groupoid">Euclidean-topological geometry</a> and <em>smooth open disks</em> for <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth geometry</a>. Following <a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Lawvere</a>, we call this refinement <em><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></em> <a class="existingWikiWord" href="/nlab/show/categorical+semantics">interpreted</a> in <em><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-toposes">cohesive (∞,1)-toposes</a></em>.</p> <h4 id="geometric_realization">Geometric realization</h4> <p>A <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> is in particular equipped with a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">derived adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>Disc</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>↩</mo><mi>Disc</mi></munder><mover><mo>→</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi \dashv Disc) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\underset{Disc}{\hookleftarrow}} \infty Grpd \,. </annotation></semantics></math></div> <p>We may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> as being the functor that sends a <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos">fundamental path ∞-groupoid</a></p> <p>Under the identification (see <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis theorem</a></em>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> \infty Grpd \stackrel{{\vert-\vert}}{\to} Top </annotation></semantics></math></div> <p>this identifies with <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of geometric homotopy types / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>.</p> <p>We say that a lift of a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is a <strong>geometric refinement</strong> of the diagram.</p> <h4 id="differential_cohomology">Differential cohomology</h4> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo>≔</mo><mi>Disc</mi><mo>∘</mo><mi>Π</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Pi} \coloneqq Disc \circ \Pi</annotation></semantics></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo>≔</mo><mi>Disc</mi><mo>∘</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\flat \coloneqq Disc \circ \Gamma</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≔</mo><mo>*</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\flat_{dR} \mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G</annotation></semantics></math>: the <em>de Rham coefficient object</em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> <p>There is a canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>G</mi><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\theta : G \to \flat_{dR} \mathbf{B}G</annotation></semantics></math>: this identifies as the canonical <em><a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a></em> on the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = \mathbf{B}^n U(1)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle (n+1)-group</a> we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><msub><mo>≔</mo> <mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">curv \coloneqq_{\mathbf{B}^n U(1)} : \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1} U(1)</annotation></semantics></math> the <em>universal <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> class</em> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>.</p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi></mrow><annotation encoding="application/x-tex">curv</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> identifies with <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n U(1)_{conn} &\to& \Omega^{n+1}_{cl}(-) \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{curv}{\to}& \mathbf{B}^{n+1} U(1)_{conn} } </annotation></semantics></math></div> <h4 id="chernweil_theory">Chern-Weil theory</h4> <p>(…)</p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory+introduction">∞-Chern-Weil theory introduction</a></p> <h3 id="differential_homotopytype_theory">Differential homotopy-type theory</h3> <p>(…)</p> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion">infinitesimals</a></p> <h4 id="de_rham_stack">de Rham stack</h4> <p>(…)</p> <h4 id="tale_stacks__orbifolds">Étale stacks / orbifolds</h4> <p>(…)</p> <h2 id="RelatedEntries">Related entries</h2> <p>An entry with discussion of and list of examples of twisted differential structures is</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></em>.</li> </ul> <p>Related expositions include the following:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory+FAQ">string theory FAQ</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motives+in+physics">motives in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert%27s+sixth+problem">Hilbert's sixth problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+theory+and+physics">model theory and physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-infinity+algebras+in+physics">L-infinity algebras in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+higher+differential+geometry">motivation for higher differential geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesion">motivation for cohesion</a></p> </li> </ul> <h2 id="References">References</h2> <p>The string-theoretic aspects of the above discussion owe a lot to <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, who has pointed out the appearance of various twisted structures in string theory, notably in the series of articles</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em><a class="existingWikiWord" href="/nlab/show/Geometric+and+topological+structures+related+to+M-branes">Geometric and topological structures related to M-branes</a></em> , part I (<a href="http://arXiv.org/abs/1001.5020">arXiv:1001.5020</a>), part <em>II: Twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math> structures</em> (<a href="http://arxiv/1007.5419">arXiv:1007.5419</a>); part <em>III: Twisted higher structures</em> (<a href="http://arxiv.org/abs/1008.1755">arXiv:1008.1755</a>)</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> by sections of twisting coeffcient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles used here is similar to that in</p> <ul id="ABG"> <li><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Twists of K-theory and TMF</em>, in Robert S. Doran, Greg Friedman, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <em>Superstrings, Geometry, Topology, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras</em>, Proceedings of Symposia in Pure Mathematics <a href="http://www.ams.org/bookstore-getitem/item=PSPUM-81">vol 81</a> (<a href="http://arxiv.org/abs/1002.3004">arXiv:1002.3004</a>)</li> </ul> <p>but considered in the non-stable context of <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> and refined from bare homotopy types to <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">geometric homotopy types</a>.</p> <p>The fundamental observation that background gauge fields in string theory are modeled by (twisted) <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> goes back to</p> <ul id="Freed"> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization+and+generalized+differential+cohomology">Dirac charge quantization and generalized differential cohomology</a></em></li> </ul> <p>and literature referenced there. For this classical literature, notably on the example of <em><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a></em> and <em><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a> <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></em>, as well as on <em><a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a></em>, see the lists of references provided at these entries, notably</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>K-Theory in Quantum Field Theory</em>, Current developments in Mathematics (2001) International Press Somerville (<a href="http://arxiv.org/abs/math-ph/0206031">arXiv:math-ph/0206031</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">7d Chern-Simons theory</a> that the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> participates in, the relation of the flux quantization to the corresponding <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic</a> description of the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+field">self-dual</a> field on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> has been discussed in</p> <ul id="Witten96"> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Five-Brane Effective Action In M-Theory</em> (<a href="http://arxiv.org/abs/hep-th/9610234">arXiv:hep-th/9610234</a>)</li> </ul> <ul id="Witten96b"> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>On Flux Quantization In M-Theory And The Effective Action</em>, J.Geom.Phys.22:1-13 (1997) (<a href="http://arxiv.org/abs/hep-th/9609122">arXiv:hep-th/9609122</a>)</li> </ul> <p>A precise mathematical formulation of the proposal made there is given in</p> <ul id="HopkinsSinger"> <li><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em><a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Quadratic Functions in Geometry, Topology, and M-Theory</a></em></li> </ul> <p>in terms of <a class="existingWikiWord" href="/nlab/show/quadratic+refinement">quadratic refinement</a> of <a class="existingWikiWord" href="/nlab/show/secondary+intersection+pairing">secondary intersection pairing</a> via <a class="existingWikiWord" href="/nlab/show/differential+integral+Wu+structures">differential integral Wu structures</a>. This also lays the mathematical foundation of much of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>.</p> <p>The suggestion that it is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack+%28%E2%88%9E%2C1%29-topos">∞-stack (∞,1)-topos</a> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> which is the right context for studying the twisted differential smooth cohomology in string theory was made in</p> <ul id="Schreiber09"> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Background+fields+in+twisted+differential+nonabelian+cohomology">Background fields in twisted differential nonabelian cohomology</a></em>, talk at <em><a class="existingWikiWord" href="/nlab/show/Oberwolfach+Workshop%2C+June+2009+--+Strings%2C+Fields%2C+Topology">Oberwolfach Workshop, June 2009 – Strings, Fields, Topology</a></em></li> </ul> <p>The smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack refinements of these structures, as discussed above, have been developed in articles including the following</p> <ul id="SSSa"> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/L-%E2%88%9E+algebra+connections">L-∞ algebra connections</a></em></li> </ul> <ul id="SSSb"> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>,_<a class="existingWikiWord" href="/schreiber/show/Fivebrane+structures">Fivebrane structures</a>_</li> </ul> <ul id="SatiSchreiberStasheff09"> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em>, Communications in Mathematical Physics, October 2012, Volume 315, Issue 1, pp 169-213 (<a href="http://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>)</li> </ul> <ul id="FSSa"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Cech+Cocycles+for+Differential+characteristic+Classes">Cech Cocycles for Differential characteristic Classes</a></em></li> </ul> <ul id="FSSb"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory</a></em></li> </ul> <ul id="FSSc"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+moduli+3-stack+of+the+C-field">The moduli 3-stack of the C-field</a></em></li> </ul> <ul id="FSSd"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Extended+higher+cup-product+Chern-Simons+theories">Extended higher cup-product Chern-Simons theories</a></em></li> </ul> <ul id="NSS"> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">Principal ∞-bundles – theory, presentations and applications</a></em> (<a href="http://arxiv.org/abs/1207.0248">arXiv:1207.0248</a>, <a href="#http://arxiv.org/abs/1207.0249">arXiv:1207.0249</a>)</li> </ul> <ul> <li> <p id="FiorenzaRogersSchreiber13"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Higher+geometric+prequantum+theory">Higher geometric prequantum theory</a></em>, 2013 (<a href="http://arxiv.org/abs/1304.0236">arXiv:1304.0236</a>)</p> </li> <li> <p id="lpqft"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> et. al., <em><a class="existingWikiWord" href="/schreiber/show/Local+prequantum+field+theory">Local prequantum field theory</a></em></p> </li> <li> <p id="SchreiberClassical"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Classical+field+theory+via+Cohesive+homotopy+types">Classical field theory via Cohesive homotopy types</a></em></p> </li> <li> <p id="Nuiten2013"><a class="existingWikiWord" href="/nlab/show/Joost+Nuiten">Joost Nuiten</a>, <em><a class="existingWikiWord" href="/schreiber/show/master+thesis+Nuiten">Cohomological quantization of local boundary prequantum field theory</a></em>, Master thesis Utrecht 2013</p> </li> <li> <p id="SchreiberSynthetic"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Synthetic+Quantum+Field+Theory">Synthetic Quantum Field Theory</a></em></p> </li> </ul> <p>A general theory of such smooth homotopy-types is laid out in</p> <ul id="Schreiber"> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em>, Habilitation thesis, Hamburg 2011</li> </ul> <p>The observation of the <a class="existingWikiWord" href="/nlab/show/tangent+%28infinity%2C1%29-topos">tangent (infinity,1)-topos</a> is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Notes on Logoi</em>, 2008 (<a href="http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf">pdf</a>)</li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>Section A) above originates in notes for an introductory lecture:</p> <ul id="SchreiberLect"> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="http://maths-old.anu.edu.au/esi/2012/program.html#Schreiber">Twisted differential structures in string theory</a></em> at <em><a href="http://maths-old.anu.edu.au/esi/2012/">ESI Program on K-Theory and Quantum Fields</a></em> (2012)</li> </ul> <p>Closely related lectures at the same program included</p> <ul id="FreedLecture"> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>Lectures on twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-theory and orientifolds</em> at <em><a href="http://maths-old.anu.edu.au/esi/2012/">ESI Program on K-Theory and Quantum Fields</a></em> (2012) (<a href="http://www.ma.utexas.edu/users/dafr/ESI.pdf">pdf</a>)</li> </ul> <p>Later, some special cases of the general notion of twisted fields considered <a href="#TableOfTwists">above</a> are being called <em>relative fields</em> in</p> <ul id="FreedTeleman"> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Relative quantum field theory</em> (<a href="http://arxiv.org/abs/1212.1692">arXiv:1212.1692</a>)</li> </ul> <p>as discussed above in the section <em><a href="#RelativeFields">Relative fields</a></em> .</p> <p>Section B) originates in notes for a talk</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Local+prequantum+field+theory">Local prequantum boundary field theory</a></em>, talk at <em><a href="http://wwwmath.uni-muenster.de/sfb/sfb878/2013/twists.html">Twists, generalised cohomology and applications</a></em>, 2013</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 5, 2024 at 00:31:19. See the <a href="/nlab/history/twisted+smooth+cohomology+in+string+theory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/twisted+smooth+cohomology+in+string+theory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3865/#Item_18">Discuss</a><span class="backintime"><a href="/nlab/revision/twisted+smooth+cohomology+in+string+theory/77" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/twisted+smooth+cohomology+in+string+theory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/twisted+smooth+cohomology+in+string+theory" accesskey="S" class="navlink" id="history" rel="nofollow">History (77 revisions)</a> <a href="/nlab/show/twisted+smooth+cohomology+in+string+theory/cite" style="color: black">Cite</a> <a href="/nlab/print/twisted+smooth+cohomology+in+string+theory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/twisted+smooth+cohomology+in+string+theory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>