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elliptic cohomology in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 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 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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/9841/#Item_1" 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" tabindex="0"> <h3 id="context">Context</h3> <h4 id="elliptic_cohomology">Elliptic cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>, <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented</a><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a> 2</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersingular+elliptic+curve">supersingular elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+elliptic+curve">derived elliptic curve</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>, <a class="existingWikiWord" href="/nlab/show/Jacobi+form">Jacobi form</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a>, <a class="existingWikiWord" href="/nlab/show/j-invariant">j-invariant</a>, <a class="existingWikiWord" href="/nlab/show/Weierstrass+sigma-function">Weierstrass sigma-function</a>, <a class="existingWikiWord" href="/nlab/show/Dedekind+eta+function">Dedekind eta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+modular+form">topological modular form</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></li> </ul> </li> </ul></div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#genera_the_elliptic_genus_and_relation_to_string_theory'>Genera, the elliptic genus and relation to string theory</a></li> <li><a href='#equivariant_elliptic_cohomology_and_loop_group_representations'>Equivariant elliptic cohomology and loop group representations</a></li> <li><a href='#chromatic_filtration'>Chromatic filtration</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#elliptic_cohomology'>Elliptic cohomology</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#equivariant_elliptic_cohomology'>Equivariant elliptic cohomology</a></li> <li><a href='#via_derived_geometry'>Via derived <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-geometry</a></li> </ul> <li><a href='#elliptic_genera'>Elliptic genera</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#equivariant_elliptic_genera'>Equivariant elliptic genera</a></li> <li><a href='#twisted_elliptic_genera'>Twisted elliptic genera</a></li> </ul> <li><a href='#EllipticGeneraAsBranePartitionFunctionsReferences'>Elliptic genera as super <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>-brane partition functions</a></li> <ul> <li><a href='#formulations'>Formulations</a></li> <ul> <li><a href='#via_super_vertex_operator_algebra'>Via super vertex operator algebra</a></li> <li><a href='#via_diracramond_operators_on_free_loop_space'>Via Dirac-Ramond operators on free loop space</a></li> <li><a href='#via_conformal_nets'>Via conformal nets</a></li> </ul> <li><a href='#InterpretationInTMFCohomology'>Conjectural interpretation in tmf-cohomology</a></li> <li><a href='#occurrences_in_string_theory'>Occurrences in string theory</a></li> <ul> <li><a href='#ReferencesEllipticGeneraAsPartitionFunctionsOfHStrings'>H-string elliptic genus</a></li> <li><a href='#ReferencesEllipticGeneraAsPartitionFunctionsOfM5Branes'>M5-brane elliptic genus</a></li> <li><a href='#ReferencesEllipticGeneraAsPartitionFunctionsOfMStrings'>M-string elliptic genus</a></li> <li><a href='#ReferencesEllipticGeneraAsPartitionFunctionsOfEStrings'>E-string elliptic genus</a></li> </ul> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em>elliptic cohomology theory</em> is a type of <em><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></em> theory associated with the datum of an <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a>.</p> <p>Even (<a class="existingWikiWord" href="/nlab/show/weakly+periodic+cohomology+theory">weakly</a>) <a class="existingWikiWord" href="/nlab/show/periodic+cohomology+theory">periodic</a> <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theories <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> are characterized by the <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a> whose ring of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(\mathbb{C}P^\infty)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</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> evaluated on the <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^\infty</annotation></semantics></math> and whose group product is induced from the canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>×</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty</annotation></semantics></math> that describes the tensor product of complex <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a>s under the identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>≃</mo><mi>ℬ</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}P^\infty \simeq \mathcal{B} U(1)</annotation></semantics></math>.</p> <p>There are precisely three types of 1-dimensional such <a class="existingWikiWord" href="/nlab/show/formal+group+laws">formal group laws</a>:</p> <ul> <li> <p>the simple additive group structure – this corresponds to integral <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> given by the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a>;</p> </li> <li> <p>the multiplicative group that corresponds to complex <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p>the formal group law on an <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a>.</p> </li> </ul> <p>An <strong>elliptic cohomology</strong> theory is an even <a class="existingWikiWord" href="/nlab/show/periodic+cohomology+theory">periodic</a> <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory whose corresponding formal group is an elliptic curve, hence which is <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by an <a class="existingWikiWord" href="/nlab/show/elliptic+spectrum">elliptic spectrum</a>.</p> <p>(e.g. <a href="#Lurie">Lurie, def. 1.2</a>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/Goerss-Hopkins-Miller-Lurie+theorem">Goerss-Hopkins-Miller-Lurie theorem</a> shows that the assignment of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theories to <a class="existingWikiWord" href="/nlab/show/elliptic+curves">elliptic curves</a> lifts to an assignment of representing <a class="existingWikiWord" href="/nlab/show/spectrum">spectra</a> in a structure-preserving way.</p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> of this assignment functor, i.e. the “gluing” of all spectra representing all elliptic cohomology theories is the <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> that represents the cohomology theory called <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>.</p> <h2 id="properties">Properties</h2> <h3 id="genera_the_elliptic_genus_and_relation_to_string_theory">Genera, the elliptic genus and relation to string theory</h3> <blockquote> <p>A properly developed theory of <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> is likely to shed some light on what <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> really means. (<a href="#Witten87">Witten 87, very last sentence</a>)</p> </blockquote> <p>The following is <strong>rough material</strong> originating from notes taken live (and long ago), to be polished. See also at <em><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a></em> and <em><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></em></p> <p>Some topological invariants of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> that are of interest:</p> <p>we restricted attention to <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> connected <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</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></p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">e(X) \in \mathbb{Z}</annotation></semantics></math></p> <ul> <li> <p>takes all values in <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></p> </li> <li> <p>is the obstruction to the existence of a nowhere vanishing <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</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> <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>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mo>∃</mo><mi>v</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>∀</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>:</mo><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (e(X)= 0) \Leftrightarrow (\exists v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0) </annotation></semantics></math></div></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/signature">signature</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sgn</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sgn(X)</annotation></semantics></math></p> <p>this is the obstruction to <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> being cobordant to a fiber bundle over the circle:</p> <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> is <a class="existingWikiWord" href="/nlab/show/bordant+manifold">bordant</a> to a <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sgn</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">sgn(X) = 0</annotation></semantics></math></p> </li> <li> <p>when <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> has a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></p> <p>the index of the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ind</mi><msub><mi>D</mi> <mi>X</mi></msub><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> ind D_X \in \mathbb{Z} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>∈</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mi>dim</mi><mi>X</mi><mo>=</mo><mn>0</mn><mi>mod</mi><mn>4</mn></mtd></mtr> <mtr><mtd><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mi>dim</mi><mi>X</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mi>mod</mi><mn>8</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> \alpha(D) \in \left\{ \array{ \mathbb{Z} & dim X = 0 mod 4 \\ \mathbb{Z}_2 & dim X = 1, 2 mod 8 \\ 0 & otherwise } \right. </annotation></semantics></math></div> <p><strong>theorem</strong> (Gromov-Lawson / <a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stolz</a>) let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">dim X \geq 5</annotation></semantics></math> and</p> <p>then <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> admits a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> of positive <a class="existingWikiWord" href="/nlab/show/scalar+curvature">scalar curvature</a> precisely when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha(X) = 0</annotation></semantics></math></p> </li> </ul> <p>These invariants share the following properties:</p> <ul> <li> <p>they are additive under <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>s</p> </li> <li> <p>they are multiplicative under <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> of manifolds</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>mod</mi><mn>2</mn><mo>,</mo><mi>sgn</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ind</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X) mod 2, sgn(X), ind(D_X)</annotation></semantics></math> all vanish when <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 boundary, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo><mi>W</mi><mo>:</mo><mi>X</mi><mo>=</mo><mo>∂</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">\exists W : X = \partial W</annotation></semantics></math>, which means 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 class="existingWikiWord" href="/nlab/show/cobordant+manifold">cobordant</a> to the empty manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math>.</p> <p>in other words, these invariants are <a class="existingWikiWord" href="/nlab/show/genus">genera</a>, namely <a class="existingWikiWord" href="/nlab/show/ring">ring</a> homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex"> \Omega \to R </annotation></semantics></math></div> <p>form the <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> <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> to some commutative <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <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> </li> <li> <p>good <a class="existingWikiWord" href="/nlab/show/genus">genera</a> are those which reflect geometric properties 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>now 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/topological+space">topological space</a> consider the <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> <em>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></em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>f</mi><mo>:</mo><mi>M</mi><mover><mrow><mo>→</mo><mi>X</mi></mrow><mi>cont</mi></mover><mo stretchy="false">}</mo><msub><mo stretchy="false">/</mo> <mi>bordism</mi></msub></mrow><annotation encoding="application/x-tex"> \Omega(X) := \{(M,f)| f : M \stackrel{cont}{\to X}\}/_{bordism} </annotation></semantics></math></div> <p>where addition and multiplication are again given by disjoint union and cartesian product.</p> <p>this assignment of rings to topological spaces is a <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> theory: <a class="existingWikiWord" href="/nlab/show/cobordism+homology+theory">cobordism homology theory</a></p> <p><strong>question</strong> given a <a class="existingWikiWord" href="/nlab/show/genus">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\Omega \to R</annotation></semantics></math>, can we find a <a class="existingWikiWord" href="/nlab/show/homology+theory">homology theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(-)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>pt</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R = R(pt)</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/homology+ring">homology ring</a> over the point and such that it all fits into a natural 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>Ω</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>R</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>ρ</mi></mover></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega &\to& R \\ \uparrow && \uparrow \\ \Omega(X) &\stackrel{\rho}{\to}& R(X) } </annotation></semantics></math></div> <p>This would be a <em>parameterized extension</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho = R(-)</annotation></semantics></math> of <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>Now 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 closed manifold.</p> <p>consider <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>X</mi><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u_X : X \to K(\pi_1(X),1)</annotation></semantics></math> (on the right an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a>) which is the classifying map for the <a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mo>*</mo></msub><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>canon</mi></msub></mrow></mover><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to} \pi_1(K(\pi_1(X), 1)) </annotation></semantics></math></div> <p>then consider</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>X</mi></msub><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msub><mi>u</mi> <mi>X</mi></msub><mo stretchy="false">]</mo><mo>∈</mo><mi>R</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \rho_X[X, u_X] \in R(K(\pi_1(X),1)) </annotation></semantics></math></div> <p><strong>theorem (Julia Weber)</strong></p> <p>take the <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> mod 2, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Eu</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Eu(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><msup><mi>Ω</mi> <mn>0</mn></msup></mtd> <mtd><mover><mo>→</mo><mrow><mi>Eu</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>t</mi> <mrow><mi>dim</mi><mi>M</mi></mrow></msup></mrow></mover></mtd> <mtd><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mi>t</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><msup><mi>Ω</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Eu</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega^0 &\stackrel{Eu(M)\cdot t^{dim M}}{\to}& \mathbb{Z}_2[t] \\ \uparrow && \uparrow \\ \Omega^0(X) &\to& Eu(X) & \simeq H_\bullet(X; \mathbb{Z}_2[t]) } </annotation></semantics></math></div> <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> smooth we have then:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Eu</mi> <mi>X</mi></msub><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>id</mi><mo stretchy="false">]</mo><mo>=</mo><mi>Poincare</mi><mspace width="thickmathspace"></mspace><mi>dual</mi><mspace width="thickmathspace"></mspace><mi>of</mi><mspace width="thickmathspace"></mspace><mi>total</mi><mspace width="thickmathspace"></mspace><mi>Stiefel</mi><mo>−</mo><mi>Whitney</mi><mspace width="thickmathspace"></mspace><mi>class</mi></mrow><annotation encoding="application/x-tex"> Eu_X[X, id] = Poincare\; dual\; of\; total\; Stiefel-Whitney\; class </annotation></semantics></math></div> <p><strong>theorem</strong> (Minalta)</p> <p>something analogous for <a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</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><msubsup><mi>Ω</mi> <mo>•</mo> <mi>SO</mi></msubsup></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Sig</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega_\bullet^{SO} &\to& Sig_\bullet(X) } </annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sign</mi> <mi>X</mi></msub><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>u</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>sig</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">sign_X[X,u] \in sig_\bullet(K) \otimes \mathbb{Q}</annotation></semantics></math></p> <p>this is the Novikov higher signature</p> <p>now the same for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-genus</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>Ω</mi> <mi>X</mi> <mi>Spin</mi></msubsup></mtd> <mtd><mover><mo>→</mo><mi>α</mi></mover></mtd> <mtd><msub><mi>KO</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>pt</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><msubsup><mi>Ω</mi> <mo>•</mo> <mi>Spin</mi></msubsup></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>KO</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega_{X}^{Spin} &\stackrel{\alpha}{\to}& KO_\bullet(pt) \\ \uparrow && \uparrow \\ \Omega_\bullet^{Spin} &\to& KO_\bullet(X) } </annotation></semantics></math></div></li> </ul> <p>now towards elliptic genera: recall the notion of <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a> of 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>: a lift of the structure map <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>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \mathcal{B}O(n)</annotation></semantics></math> through the 4th connected universal cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>ℬ</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">⟨</mo><mn>4</mn><mo stretchy="false">⟩</mo><mo>→</mo><mi>ℬ</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}String(n) := \mathcal{B}O(n)\langle 4\rangle \to \mathcal{B} O</annotation></semantics></math>:</p> <p>so consider <a class="existingWikiWord" href="/nlab/show/string+structure">String manifold</a>s and the bordism ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>String</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega_\bullet^{String}</annotation></semantics></math> of String manifold, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">M_\bullet</annotation></semantics></math> be the ring of integral <a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>s, then there is a <a class="existingWikiWord" href="/nlab/show/genus">genus</a> – the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a> <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> <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>Ω</mi> <mo>•</mo> <mi>String</mi></msubsup></mtd> <mtd><mover><mo>→</mo><mi>W</mi></mover></mtd> <mtd><msub><mi>M</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mo>•</mo> <mi>String</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>M</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>tmf</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega_\bullet^{String} &\stackrel{W}{\to}& M_\bullet \\ \uparrow && \uparrow \\ \Omega_\bullet^{String}(X) &\to& M_\bullet(X) \\ &\searrow& \\ && tmf_\bullet(X) } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>String</mi></msubsup><mo stretchy="false">(</mo><mi>pt</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>tmf</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>pt</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega_\bullet^{String}(pt) \to tmf_\bullet(pt)</annotation></semantics></math> is surjective</p> <p><strong>conjecture (<a class="existingWikiWord" href="/nlab/show/Stolz+conjecture">Stolz conjecture</a>)</strong></p> <p>If a String manifold <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> has a positive <a class="existingWikiWord" href="/nlab/show/Ricci+curvature">Ricci curvature</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">metric</a>, then the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a> vanishes.</p> <p>The attempted “Proof” of this is the motivation for the <a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stolz</a>-<a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Teichner</a>-program for <a class="existingWikiWord" href="/nlab/show/geometric+models+for+elliptic+cohomology">geometric models for elliptic cohomology</a>:</p> <p><strong>“Proof”</strong> If <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> is String, then the <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">L Y</annotation></semantics></math> is has <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, so if <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> has positive <a class="existingWikiWord" href="/nlab/show/Ricci+curvature">Ricci curvature</a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">L Y</annotation></semantics></math> has positive <a class="existingWikiWord" href="/nlab/show/scalar+curvature">scalar curvature</a> which implies by the above that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ind</mi> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msup><msub><mi>D</mi> <mrow><mi>L</mi><mi>Y</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ind^{S^1} D_{L Y} = 0</annotation></semantics></math> which by the index formula is the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>.</p> <h3 id="equivariant_elliptic_cohomology_and_loop_group_representations">Equivariant elliptic cohomology and loop group representations</h3> <p>The analog of the <a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a> with <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> replaced by <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a> yields (aspects of) the <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> of <a class="existingWikiWord" href="/nlab/show/loop+groups">loop groups</a>. (<a href="#Ganter12">Ganter 12</a>)</p> <h3 id="chromatic_filtration">Chromatic filtration</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic homotopy theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>/<a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></th><th><a class="existingWikiWord" href="/nlab/show/real+oriented+cohomology+theory">real oriented cohomology theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/HZR-theory">HZR-theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">0th <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(0)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory+spectrum">complex K-theory spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">first <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">first <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+spectrum">elliptic spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ell</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">Ell_E</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">second <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(2)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">second <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(2)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> of <a class="existingWikiWord" href="/nlab/show/KU">KU</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(KU)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3 …10</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+spectrum">K3 spectrum</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><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>th <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(n)</annotation></semantics></math></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><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BPR-theory">BPR-theory</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> applied to chrom. level <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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(E_n)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/red-shift+conjecture">red-shift conjecture</a>)</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MU">MU</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MR-theory">MR-theory</a></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+spectrum">elliptic spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+genus">equivariant elliptic genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/SO+orientation+of+elliptic+cohomology">SO orientation of elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/spin+orientation+of+elliptic+cohomology">spin orientation of elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+Chern+character">elliptic Chern character</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-orientation">sigma-orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles+with+connection">line n-bundles with connection</a> on <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>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</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></strong></p> <table><thead><tr><th><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></th><th><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau n-fold</a></th><th><a class="existingWikiWord" href="/nlab/show/line+n-bundle">line n-bundle</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/flat+infinity-connection">flat</a>/degree-0 n-bundles</th><th><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> of <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation moduli</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/modular+functor">modular functor</a>/<a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></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><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> in <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>/<a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>/<a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+3-fold">Calabi-Yau 3-fold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+3-bundle">line 3-bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+cohomology">CY3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <div> <h3 id="elliptic_cohomology">Elliptic cohomology</h3> <h4 id="general">General</h4> <p>The concept of <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> originates around:</p> <ul> <li id="Landweber88"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <em>Elliptic Cohomology and Modular Forms</em>, in: <em>Elliptic Curves and Modular Forms in Algebraic Topology</em>, Lecture Notes in Mathematics <strong>1326</strong> (1988) 55-68 &lbrack;<a href="https://doi.org/10.1007/BFb0078038">doi:10.1007/BFb0078038</a>&rbrack;</p> </li> <li id="LandweberRavenelStong93"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, <em>Periodic cohomology theories defined by elliptic curves</em>, in <a class="existingWikiWord" href="/nlab/show/Haynes+Miller">Haynes Miller</a> et. al. (eds.), <em>The Cech centennial: A conference on homotopy theory</em>, June 1993, AMS (1995) (<a href="http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf">pdf</a>)</p> </li> </ul> <p>and in the universal guise of <a class="existingWikiWord" href="/nlab/show/topological+modular+forms">topological modular forms</a> in:</p> <ul> <li id="Hopkins02"><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>Algebraic topology and modular forms</em> in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (<a href="http://arxiv.org/abs/math/0212397">arXiv:math/0212397</a>)</li> </ul> <p>Surveys:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Greenberg">Matthew Greenberg</a>, <em>Constructing elliptic cohomology</em>, McGill University 2002 (<a href="https://www.bac-lac.gc.ca/eng/services/theses/Pages/item.aspx?idNumber=898194373">oclc:898194373</a>, <a class="existingWikiWord" href="/nlab/files/GreenbergEllipticCohomology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <em>Topological modular forms (after Hopkins, Miller, and Lurie)</em>, Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 (<a href="https://arxiv.org/abs/0910.5130">arXiv:0910.5130</a>, <a href="http://www.numdam.org/item/AST_2010__332__221_0">numdam:AST_2010__332__221_0</a>)</p> </li> <li id="Lurie"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology">A Survey of Elliptic Cohomology</a></em>, in: <em>Algebraic Topology</em>, Abel Symposia Volume 4, 2009, pp 219-277 (<a href="http://www.math.harvard.edu/~lurie/papers/survey.pdf">pdf</a>, <a href="https://doi.org/10.1007/978-3-642-01200-6_9">doi:10.1007/978-3-642-01200-6_9</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Elliptic cohomology and elliptic curves</em>, Felix Klein Lectures, Bonn (2015) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://www.hcm.uni-bonn.de/fkl-rezk">web</a>, <a href="https://rezk.web.illinois.edu/felix-klein-lectures-notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/RezkElliptic2015.pdf" title="pdf">pdf</a>, video rec: <a href="https://youtu.be/eFzB_2fkOxY">part 1</a>, <a href="https://youtu.be/4e9e3AtK7yg">2</a>, <a href="https://youtu.be/ueL5dMH9JoI">3</a>, <a href="https://youtu.be/r_7SsIoU9No">4</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="Thomas02"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Thomas">Charles Thomas</a>, <em>Elliptic cohomology</em>, Kluwer Academic, 2002 (<a href="https://link.springer.com/book/10.1007/b115001">doi:10.1007/b115001</a>, <a href="https://link.springer.com/content/pdf/10.1007%2Fb115001.pdf">pdf</a>)</p> </li> <li id="DFHH14"> <p><a class="existingWikiWord" href="/nlab/show/Christopher+Douglas">Christopher Douglas</a>, <a class="existingWikiWord" href="/nlab/show/John+Francis">John Francis</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a> (eds.), <em>Topological Modular Forms</em>, Mathematical Surveys and Monographs Volume 201, AMS 2014 (<a href="https://bookstore.ams.org/surv-201">ISBN:978-1-4704-1884-7</a>)</p> </li> </ul> <h4 id="equivariant_elliptic_cohomology">Equivariant elliptic cohomology</h4> <p>On <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a> and <a class="existingWikiWord" href="/nlab/show/positive+energy+representations">positive energy representations</a> of <a class="existingWikiWord" href="/nlab/show/loop+groups">loop groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eduard+Looijenga">Eduard Looijenga</a>, <em>Root systems and elliptic curves</em>, Invent. Math. <strong>38</strong> 1 (1976/77) 17-32 &lbrack;<a href="https://doi.org/10.1007/BF01390167">doi:10.1007/BF01390167</a>&rbrack;</p> </li> <li id="Grojnowski94"> <p><a class="existingWikiWord" href="/nlab/show/Ian+Grojnowski">Ian Grojnowski</a>, <em>Delocalised equivariant elliptic cohomology</em> (1994), in: <em>Elliptic cohomology</em>, Volume 342 of London Math. Soc. Lecture Note Ser., pages 114–121. Cambridge Univ. Press, Cambridge, 2007 (<a href="http://hopf.math.purdue.edu/Grojnowski/deloc.pdf">pdf</a>, <a href="https://doi.org/10.1017/CBO9780511721489.007">doi:10.1017/CBO9780511721489.007</a>)</p> </li> <li id="GinzburgKapranovVasserot95"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Ginzburg">Victor Ginzburg</a>, <a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, Eric Vasserot, <em>Elliptic Algebras and Equivariant Elliptic Cohomology</em> (<a href="http://arxiv.org/abs/q-alg/9505012">arXiv:q-alg/9505012</a>)</p> </li> <li id="Ando00"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <em>Power operations in elliptic cohomology and representations of loop groups</em>, Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (<a href="http://www.jstor.org/stable/221905">jstor:221905</a>, <a href="http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf">pdf</a>)</p> </li> <li id="Gepner05"> <p><a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em><a class="existingWikiWord" href="/nlab/show/Homotopy+topoi+and+equivariant+elliptic+cohomology">Homotopy topoi and equivariant elliptic cohomology</a></em>, University of Illinois at Urbana-Champaign, 2005 (<a class="existingWikiWord" href="/nlab/files/GepnerElliptic05.pdf" title="pdf">pdf</a>)</p> </li> <li id="GepnerMeier20"> <p><a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Lennart+Meier">Lennart Meier</a>, <em>On equivariant topological modular forms</em>, (<a href="https://arxiv.org/abs/2004.10254">arXiv:2004.10254</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mich%C3%A8le+Vergne">Michèle Vergne</a>, <em>Bouquets revisited and equivariant elliptic cohomology</em>, International Journal of Mathematics 2021 (<a href="https://arxiv.org/abs/2005.00312">arXiv:2005.00312</a>, <a href="https://doi.org/10.1142/S0129167X21400127">doi:10.1142/S0129167X21400127</a>)</p> </li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/Kac-Weyl+characters">Kac-Weyl characters</a> of <a class="existingWikiWord" href="/nlab/show/loop+group+representations">loop group representations</a></p> <ul> <li id="Brylinski90"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, <em>Representations of loop groups, Dirac operators on loop space, and modular forms</em>, Topology, 29(4):461–480, 1990 (<a href="https://doi.org/10.1016/0040-9383(90)90016-D">doi:10.1016/0040-9383(90)90016-D</a>)</p> </li> <li id="Ganter12"> <p><a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, <em>The elliptic Weyl character formula</em>, Compositio Mathematica, Vol 150, Issue 7 (2014), pp 1196-1234 (<a href="http://arxiv.org/abs/1206.0528">arXiv:1206.0528</a>)</p> </li> </ul> <p>The case of <a class="existingWikiWord" href="/nlab/show/twisted+ad-equivariant+Tate+K-theory">twisted ad-equivariant Tate K-theory</a>:</p> <ul> <li id="Ganter07"> <p><a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, Section 3.1 in: <em>Stringy power operations in Tate K-theory</em> (<a href="https://arxiv.org/abs/math/0701565">arXiv:math/0701565</a>)</p> </li> <li id="Ganter13"> <p><a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, <em>Power operations in orbifold Tate K-theory</em>, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. (<a href="https://arxiv.org/abs/1301.2754">arXiv:1301.2754</a>, <a href="https://projecteuclid.org/euclid.hha/1383943680">euclid:hha/1383943680</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zhen+Huan">Zhen Huan</a>, <em>Quasi-elliptic cohomology</em>, 2017 (<a href="http://hdl.handle.net/2142/97268">hdl</a>)</p> </li> <li id="Huan18"> <p><a class="existingWikiWord" href="/nlab/show/Zhen+Huan">Zhen Huan</a>, <em>Quasi-Elliptic Cohomology I</em>, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (<a href="https://arxiv.org/abs/1805.06305">arXiv:1805.06305</a>, <a href="https://doi.org/10.1016/j.aim.2018.08.007">doi:10.1016/j.aim.2018.08.007</a>)</p> </li> <li id="Huan18Quasi"> <p><a class="existingWikiWord" href="/nlab/show/Zhen+Huan">Zhen Huan</a>, <em>Quasi-theories</em> (<a href="https://arxiv.org/abs/1809.06651">arXiv:1809.06651</a>)</p> </li> <li id="Luecke19"> <p><a class="existingWikiWord" href="/nlab/show/Kiran+Luecke">Kiran Luecke</a>, <em>Completed K-theory and Equivariant Elliptic Cohomology</em>, Advances in Mathematics <strong>410</strong> B (2022) 108754 &lbrack;<a href="https://arxiv.org/abs/1904.00085">arXiv:1904.00085</a>, <a href="https://doi.org/10.1016/j.aim.2022.108754">doi:10.1016/j.aim.2022.108754</a>&rbrack;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Dove">Thomas Dove</a>, <em>Twisted Equivariant Tate K-Theory</em> (<a href="https://arxiv.org/abs/1912.02374">arXiv:1912.02374</a>)</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <a class="existingWikiWord" href="/nlab/show/Arnav+Tripathy">Arnav Tripathy</a>, <em>A geometric model for complex analytic equivariant elliptic cohomology</em>, (<a href="https://arxiv.org/abs/1805.04146">arXiv:1805.04146</a>)</p> </li> <li> <p>Nicolò Sibilla, Paolo Tomasini: <em>Equivariant Elliptic Cohomology and Mapping Stacks I</em> [<a href="https://arxiv.org/abs/2303.10146">arXiv:2303.10146</a>]</p> </li> </ul> <h4 id="via_derived_geometry">Via derived <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-geometry</h4> <p>Formulation of (<a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant</a>) <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> in <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a>/<a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+geometry">E-∞ geometry</a> (<a class="existingWikiWord" href="/nlab/show/derived+elliptic+curves">derived elliptic curves</a>):</p> <ul> <li id="GoerssHopkinsA"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>Moduli spaces of commutative ring spectra</em>, in <em>Structured ring spectra</em>, London</p> <p>Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. (<a href="http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf">pdf</a>, <a href="https://doi.org/10.1017/CBO9780511529955.009">doi:10.1017/CBO9780511529955.009</a>)</p> </li> <li id="GoerssHopkinsB"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>Moduli problems for structured ring spectra</em> (<a href="http://www.math.northwestern.edu/~pgoerss/spectra/obstruct.pdf">pdf</a>)</p> <p>(<a class="existingWikiWord" href="/nlab/show/Goerss-Hopkins-Miller+theorem">Goerss-Hopkins-Miller theorem</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Elliptic+Cohomology+I">Elliptic Cohomology I</a>: Spectral abelian varieties</em>, 2018. 141pp (<a href="http://math.harvard.edu/~lurie/papers/Elliptic-I.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Elliptic Cohomology II: Orientations</em>, 2018. 288pp (<a href="http://www.math.harvard.edu/~lurie/papers/Elliptic-II.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Elliptic Cohomology III: Tempered Cohomology</em>, 2019. 286pp (<a href="http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Elliptic Cohomology IV: Equivariant elliptic cohomology</em>, to appear.</p> </li> </ul> <h3 id="elliptic_genera">Elliptic genera</h3> <h4 id="general_2">General</h4> <p>The general concept of <a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a> originates with:</p> <ul> <li id="Ochanine87"><a class="existingWikiWord" href="/nlab/show/Serge+Ochanine">Serge Ochanine</a>, <em>Sur les genres multiplicatifs definis par des integrales elliptiques</em>, Topology, Vol. 26, No. 2, 1987 (<a href="http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Mult-genre-Ochanine.pdf">pdf</a>, <a href="https://doi.org/10.1016/0040-9383(87)90055-3">doi:10.1016/0040-9383(87)90055-3</a>)</li> </ul> <p>Early development:</p> <ul> <li id="Zagier86"> <p><a class="existingWikiWord" href="/nlab/show/Don+Zagier">Don Zagier</a>, <em>Note on the Landweber-Stong elliptic genus</em> 1986 (<a href="http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0078047/fulltext.pdf">pdf</a>, <a href="http://edoc.mpg.de/744944">edoc:744944</a>)</p> </li> <li id="HirzebruchGergerJung92"> <p><a class="existingWikiWord" href="/nlab/show/Friedrich+Hirzebruch">Friedrich Hirzebruch</a>, Thomas Berger, Rainer Jung, chapter 2 of: <em>Manifolds and Modular Forms</em>, Aspects of Mathematics <strong>20</strong>, Viehweg (1992), Springer (1994) &lbrack;<a href="https://doi.org/10.1007/978-3-663-10726-2">doi:10.1007/978-3-663-10726-2</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf">pdf</a>&rbrack;</p> </li> <li id="ChudnovskyChudnovsky88"> <p>D.V. Chudnovsky, G.V. Chudnovsky, <em>Elliptic modular functions and elliptic genera</em>, Topology, Volume 27, Issue 2, 1988, Pages 163–170 (<a href="https://doi.org/10.1016/0040-9383(88)90035-3">doi:10.1016/0040-9383(88)90035-3</a>)</p> </li> <li id="Hovey91"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Spin Bordism and Elliptic Homology</em>, Math Z 219, 163–170 (1995) (<a href="https://doi.org/10.1007/BF02572356">doi:10.1007/BF02572356</a>)</p> </li> <li id="KreckStolz93"> <p><a class="existingWikiWord" href="/nlab/show/Matthias+Kreck">Matthias Kreck</a>, <a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>HP</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">HP^2</annotation></semantics></math>-bundles and elliptic homology</em>, Acta Math, 171 (1993) 231-261 (<a class="existingWikiWord" href="/nlab/files/KreckStolzElliptic.pdf" title="pdf">pdf</a>, <a href="https://projecteuclid.org/euclid.acta/1485890737">euclid:acta/1485890737</a>)</p> </li> </ul> <p>Review:</p> <ul> <li id="Landweber88"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <em>Elliptic genera: An introductory overview</em> In: P. Landweber (eds.) <em>Elliptic Curves and Modular Forms in Algebraic Topology</em>, Lecture Notes in Mathematics, vol 1326. Springer (1988) (<a href="https://doi.org/10.1007/BFb0078036">doi:10.1007/BFb0078036</a>)</p> </li> <li id="KL96"> <p><a class="existingWikiWord" href="/nlab/show/Kefeng+Liu">Kefeng Liu</a>, <em>Modular forms and topology</em>, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (<a href="https://www.math.ucla.edu/~liu/Research/vam.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/LiuModularFormsTopology.pdf" title="pdf">pdf</a>, <a href="http://dx.doi.org/10.1090/conm/193">doi:10.1090/conm/193</a>)</p> </li> <li id="Ochanine09"> <p><a class="existingWikiWord" href="/nlab/show/Serge+Ochanine">Serge Ochanine</a>, <em>What is… an elliptic genus?</em>, Notices of the AMS, volume 56, number 6 (2009) (<a href="http://www.ams.org/notices/200906/rtx090600720p.pdf">pdf</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Stolz+conjecture">Stolz conjecture</a> on the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>:</p> <ul> <li id="Stolz96"> <p><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <em>A conjecture concerning positive Ricci curvature and the Witten genus</em>, Mathematische Annalen Volume 304, Number 1 (1996) (<a href="https://doi.org/10.1007/BF01446319">doi:10.1007/BF01446319</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anand+Dessai">Anand Dessai</a>, <em>Some geometric properties of the Witten genus</em>, in: <a class="existingWikiWord" href="/nlab/show/Christian+Ausoni">Christian Ausoni</a>, <a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, <a class="existingWikiWord" href="/nlab/show/J%C3%A9r%C3%B4me+Scherer">Jérôme Scherer</a> (eds.) <em>Alpine Perspectives on Algebraic Topology</em>, Contemporary Mathematics 504 (2009) (<a href="http://homeweb2.unifr.ch/dessaia/pub/papers/Arolla/DessaiArollaFinalRevised30June09.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DessaiEllipticGenus.pdf" title="pdf">pdf</a>,</p> <p><a href="http://dx.doi.org/10.1090/conm/504">doi:10.1090/conm/504</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Jacobi+form">Jacobi form</a>-property of the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>:</p> <ul> <li id="AndoFrenchGanter08"><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, Christopher French, <a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, <em>The Jacobi orientation and the two-variable elliptic genus</em>, Algebraic and Geometric Topology 8 (2008) p. 493-539 (<a href="http://www.msp.warwick.ac.uk/agt/2008/08-01/agt-2008-08-016s.pdf">pdf</a>)</li> </ul> <p>The identification of <a class="existingWikiWord" href="/nlab/show/elliptic+genera">elliptic genera</a>, via <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>/<a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+collapse">Pontrjagin-Thom collapse</a>, as <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex orientations</a> of <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> (<a class="existingWikiWord" href="/nlab/show/sigma-orientation">sigma-orientation</a>/<a class="existingWikiWord" href="/nlab/show/string-orientation+of+tmf">string-orientation of tmf</a>/<a class="existingWikiWord" href="/nlab/show/spin-orientation+of+Tate+K-theory">spin-orientation of Tate K-theory</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>Topological modular forms, the Witten genus, and the theorem of the cube</em>, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 (<a href="http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf">pdf</a>)</p> </li> <li id="AndoHopkinsStrickland01"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Elliptic spectra, the Witten genus and the theorem of the cube</em>, Invent. Math. 146 (2001) 595–687 MR1869850 (<a href="https://doi.org/10.1007/s002220100175">doi:10.1007/s002220100175</a>, <a href="http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf">pdf</a>)</p> </li> <li id="AndoHopkinsStrickland02"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>The sigma orientation is an H-infinity map</em>, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (<a href="http://arxiv.org/abs/math/0204053">arXiv:math/0204053</a>, <a href="https://doi.org/10.1353/ajm.2004.0008">doi:10.1353/ajm.2004.0008</a>)</p> </li> <li id="AndoHopkinsRezk"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Multiplicative orientations of KO-theory and the spectrum of topological modular forms</em>, 2010 (<a href="http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/AndoHopkinsRezkStringOrientation.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>For the <a class="existingWikiWord" href="/nlab/show/Ochanine+genus">Ochanine genus</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dylan+Wilson">Dylan Wilson</a>, <em>Orientations and Topological Modular Forms with Level Structure</em> (<a href="http://arxiv.org/abs/1507.05116">arXiv:1507.05116</a>)</li> </ul> <h4 id="equivariant_elliptic_genera">Equivariant elliptic genera</h4> <p>Genera in <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a> and the <a class="existingWikiWord" href="/nlab/show/rigidity+theorem+for+equivariant+elliptic+genera">rigidity theorem for equivariant elliptic genera</a>:</p> <p>The statement, with a <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>-motivated plausibility argument, is due to <a href="#Witten87">Witten 87</a>.</p> <p>The first <a class="existingWikiWord" href="/nlab/show/proof">proof</a> was given in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+Taubes">Clifford Taubes</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-actions and elliptic genera</em>, Comm. Math. Phys. Volume 122, Number 3 (1989), 455-526 (<a href="https://projecteuclid.org/euclid.cmp/1104178471">euclid:cmp/1104178471</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Clifford+Taubes">Clifford Taubes</a>, <em>On the Rigidity Theorems of Witten</em>, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 (<a href="https://doi.org/10.2307/1990915">doi:10.2307/1990915</a>)</p> </li> </ul> <p>Reviewed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <em>On the Fixed Point Formula and the Rigidity Theorems of Witten</em>, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) <em>Nonperturbative Quantum Field Theory</em>. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) (<a href="https://doi.org/10.1007/978-1-4613-0729-7_2">doi:10.1007/978-1-4613-0729-7_2</a>)</li> </ul> <p>Further proofs and constructions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Friedrich+Hirzebruch">Friedrich Hirzebruch</a>, <em>Elliptic Genera of Level <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> for Complex Manifolds</em>, In: Bleuler K., Werner M. (eds) <em>Differential Geometrical Methods in Theoretical Physics</em> NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 250. Springer (1988) (<a href="https://doi.org/10.1007/978-94-015-7809-7_3">doi:10.1007/978-94-015-7809-7_3</a>)</p> </li> <li> <p>I. M. Krichever, <em>Generalized elliptic genera and Baker-Akhiezer functions</em>, Mathematical Notes of the Academy of Sciences of the USSR 47, 132–142 (1990) (<a href="https://doi.org/10.1007/BF01156822">doi:10.1007/BF01156822</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kefeng+Liu">Kefeng Liu</a>, <em>On modular invariance and rigidity theorems</em>, J. Differential Geom. Volume 41, Number 2 (1995), 343-396 (<a href="https://projecteuclid.org/euclid.jdg/1214456221">euclid:jdg/1214456221</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kefeng+Liu">Kefeng Liu</a>, <em>On elliptic genera and theta-functions</em>, Topology Volume 35, Issue 3, July 1996, Pages 617-640 (<a href="https://doi.org/10.1016/0040-9383(95)00042-9">doi:10.1016/0040-9383(95)00042-9</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anand+Dessai">Anand Dessai</a>, Rainer Jung, <em>On the Rigidity Theorem for Elliptic Genera</em>, Transactions of the American Mathematical Society Vol. 350, No. 10 (Oct., 1998), pp. 4195-4220 (26 pages) (<a href="https://www.jstor.org/stable/117694">jstor:117694</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioanid+Rosu">Ioanid Rosu</a>, <em>Equivariant Elliptic Cohomology and Rigidity</em>, American Journal of Mathematics 123 (2001), 647-677 (<a href="https://arxiv.org/abs/math/9912089">arXiv:math/9912089</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <em>Circle-equivariant classifying spaces and the rational equivariant sigma genus</em>, Math. Z. 269, 1021–1104 (2011) (<a href="https://arxiv.org/abs/0705.2687">arXiv:0705.2687</a>, <a href="https://doi.org/10.1007/s00209-010-0773-7">doi:10.1007/s00209-010-0773-7</a>)</p> </li> </ul> <p>On manifolds with <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>-<a class="existingWikiWord" href="/nlab/show/action">action</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anand+Dessai">Anand Dessai</a>, <em>The Witten genus and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math>-actions on manifolds</em>, 1994 (<a href="https://homeweb.unifr.ch/dessaia/pub/papers/MZpreprint6_Witten_S3.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DessaiWittenGenusS3.pdf" title="pdf">pdf</a>)</li> </ul> <h4 id="twisted_elliptic_genera">Twisted elliptic genera</h4> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/elliptic+genera">elliptic genera</a> <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> by a gauge bundle, i.e. for <a class="existingWikiWord" href="/nlab/show/string%5Ec+structure">string^c structure</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Maria+Basterra">Maria Basterra</a>, <em>The Witten genus and equivariant elliptic cohomology</em>, Math Z 240, 787–822 (2002) (<a href="https://arxiv.org/abs/math/0008192">arXiv:math/0008192</a>, <a href="https://doi.org/10.1007/s002090100399">doi:10.1007/s002090100399</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <em>The sigma orientation for analytic circle equivariant elliptic cohomology</em>, Geom. Topol., 7:91–153, 2003 (<a href="http://arxiv.org/abs/math/0201092">arXiv:math/0201092</a>, <a href="https://projecteuclid.org/euclid.gt/1513883094">euclid:gt/1513883094</a>)</p> </li> <li id="ABG"> <p><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>, Section 11 of <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>, American Mathematical Society, 2010 (<a href="http://arxiv.org/abs/1002.3004">arXiv:1002.3004</a>)</p> </li> <li id="ChenHanZhang10"> <p><a class="existingWikiWord" href="/nlab/show/Qingtao+Chen">Qingtao Chen</a>, <a class="existingWikiWord" href="/nlab/show/Fei+Han">Fei Han</a>, <a class="existingWikiWord" href="/nlab/show/Weiping+Zhang">Weiping Zhang</a>, <em>Generalized Witten Genus and Vanishing Theorems</em>, Journal of Differential Geometry 88.1 (2011): 1-39. (<a href="http://arxiv.org/abs/1003.2325">arXiv:1003.2325</a>)</p> </li> <li> <p>Jianqing Yu, Bo Liu, <em>On the Witten Rigidity Theorem for <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> Manifolds</em>, Pacific Journal of Mathematics 266.2 (2013): 477-508. (<a href="http://arxiv.org/abs/1206.5955">arXiv:1206.5955</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fei+Han">Fei Han</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <em>Projective elliptic genera and elliptic pseudodifferential genera</em>, Adv. Math. 358 (2019) 106860 (<a href="https://arxiv.org/abs/1903.07035">arXiv:1903.07035</a>)</p> </li> <li> <p>Haibao Duan, <a class="existingWikiWord" href="/nlab/show/Fei+Han">Fei Han</a>, Ruizhi Huang, <em><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 and Modular Invariants</em>, Trans. AMS 2020 (<a href="https://arxiv.org/abs/1905.02093">arXiv:1905.02093</a>)</p> </li> </ul> <div> <h3 id="EllipticGeneraAsBranePartitionFunctionsReferences">Elliptic genera as super <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>-brane partition functions</h3> <p>The interpretation of <a class="existingWikiWord" href="/nlab/show/elliptic+genera">elliptic genera</a> (especially the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>) as the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of a <a class="existingWikiWord" href="/nlab/show/2d+superconformal+field+theory">2d superconformal field theory</a> (or <a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a>) – and especially of the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> (“H-string”) or <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a> <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> theory has precursors in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A.+N.+Schellekens">A. N. Schellekens</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+P.+Warner">Nicholas P. Warner</a>, <em>Anomalies and modular invariance in string theory</em>, Physics Letters B 177 (3-4), 317-323, 1986 (<a href="https://doi.org/10.1016/0370-2693(86)90760-4">doi:10.1016/0370-2693(86)90760-4</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A.+N.+Schellekens">A. N. Schellekens</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+P.+Warner">Nicholas P. Warner</a>, <em>Anomalies, characters and strings</em>, Nuclear Physics B Volume 287, 1987, Pages 317-361 (<a href="https://doi.org/10.1016/0550-3213(87)90108-8">doi:10.1016/0550-3213(87)90108-8</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wolfgang+Lerche">Wolfgang Lerche</a>, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <a class="existingWikiWord" href="/nlab/show/A.+N.+Schellekens">A. N. Schellekens</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+P.+Warner">Nicholas P. Warner</a>, <em>Anomaly cancelling terms from the elliptic genus</em>, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (<a href="https://doi.org/10.1016/0550-3213(88)90468-3">doi:10.1016/0550-3213(88)90468-3</a>)</p> </li> </ul> <p>and then strictly originates with:</p> <ul> <li id="Witten87"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Elliptic genera and quantum field theory</em>, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536. (<a href="http://projecteuclid.org/euclid.cmp/1104117076">euclid:cmp/1104117076</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>On the Landau-Ginzburg Description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N=2</annotation></semantics></math> Minimal Models</em>, Int. J. Mod. Phys.A9:4783-4800,1994 (<a href="http://arxiv.org/abs/hep-th/9304026">arXiv:hep-th/9304026</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Toshiya+Kawai">Toshiya Kawai</a>, <a class="existingWikiWord" href="/nlab/show/Yasuhiko+Yamada">Yasuhiko Yamada</a>, Sung-Kil Yang, <em>Elliptic Genera and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N=2</annotation></semantics></math> Superconformal Field Theory</em>, Nucl. Phys. B414:191-212, 1994 (<a href="http://arxiv.org/abs/hep-th/9306096">arXiv:hep-th/9306096</a>, <a href="https://doi.org/10.1016/0550-3213(94)90428-6">doi:10.1016/0550-3213(94)90428-6</a>)</p> </li> <li> <p>Sujay K. Ashok, Jan Troost, <em>A Twisted Non-compact Elliptic Genus</em>, JHEP 1103:067, 2011 (<a href="http://arxiv.org/abs/1101.1059">arXiv:1101.1059</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, <em>Elliptic genera of Landau-Ginzburg models over nontrivial spaces</em>, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 (<a href="https://arxiv.org/abs/0905.1285">arXiv:0905.1285</a>)</p> </li> </ul> <p>Review in:</p> <ul> <li id="Cheng13"> <p><a class="existingWikiWord" href="/nlab/show/Miranda+Cheng">Miranda Cheng</a>, <em>(Mock) Modular Forms in String Theory and Moonshine</em>, lecture notes 2016 (<a class="existingWikiWord" href="/nlab/files/ChengModularForms16.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Katrin+Wendland">Katrin Wendland</a>, Section 2.4 in: <em>Snapshots of Conformal Field Theory</em>, in: <em>Mathematical Aspects of Quantum Field Theories</em> Mathematical Physics Studies. Springer 2015 (<a href="http://de.arxiv.org/abs/1404.3108">arXiv:1404.3108</a>, <a href="https://doi.org/10.1007/978-3-319-09949-1_4">doi:10.1007/978-3-319-09949-1_4</a>)</p> </li> </ul> <p>With emphasis on <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> CFTs:</p> <ul> <li>Sujay K. Ashok, <a class="existingWikiWord" href="/nlab/show/Jan+Troost">Jan Troost</a>, <em>Orbifolded Elliptic Genera of Non-Compact Models</em> [<a href="https://arxiv.org/abs/2405.08370">arXiv:2405.08370</a>]</li> </ul> <h4 id="formulations">Formulations</h4> <h5 id="via_super_vertex_operator_algebra">Via super vertex operator algebra</h5> <p>Formulation via <a class="existingWikiWord" href="/nlab/show/super+vertex+operator+algebras">super vertex operator algebras</a>:</p> <ul> <li> <p>Hirotaka Tamanoi, <em>Elliptic Genera and Vertex Operator Super-Algebras</em>, Springer 1999 (<a href="https://link.springer.com/book/10.1007/BFb0092541">doi:10.1007/BFb0092541</a>)</p> </li> <li id="DLM02"> <p>Chongying Dong, <a class="existingWikiWord" href="/nlab/show/Kefeng+Liu">Kefeng Liu</a>, Xiaonan Ma, <em>Elliptic genus and vertex operator algebras</em>, Algebr. Geom. Topol. 1 (2001) 743-762 (<a href="http://arxiv.org/abs/math/0201135">arXiv:math/0201135</a>, <a href="http://dx.doi.org/10.2140/agt.2001.1.743">doi:10.2140/agt.2001.1.743</a>)</p> </li> </ul> <p>and for the <a class="existingWikiWord" href="/nlab/show/topological+twist">topologically twisted</a> <a class="existingWikiWord" href="/nlab/show/2d+%282%2C0%29-superconformal+QFT">2d (2,0)-superconformal QFT</a> (the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> with enhanced supersymmetry) via <a class="existingWikiWord" href="/nlab/show/sheaves+of+vertex+operator+algebras">sheaves of vertex operator algebras</a> in</p> <ul> <li id="Cheung10"><a class="existingWikiWord" href="/nlab/show/Pokman+Cheung">Pokman Cheung</a>, <em>The Witten genus and vertex algebras</em> (<a href="http://arxiv.org/abs/0811.1418">arXiv:0811.1418</a>)</li> </ul> <p>based on <a class="existingWikiWord" href="/nlab/show/chiral+differential+operators">chiral differential operators</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vassily+Gorbounov">Vassily Gorbounov</a>, <a class="existingWikiWord" href="/nlab/show/Fyodor+Malikov">Fyodor Malikov</a>, <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <em>Gerbes of chiral differential operators</em>, Math. Res. Lett. 7(1), 55–66 (2000) (<a href="http://arxiv.org/abs/math/9906117">arXiv:math/9906117</a>, <a href="http://arxiv.org/abs/math/0003170">arXiv:math/0003170</a>, <a href="http://arxiv.org/abs/math/0005201">arXiv:math/0005201</a>)</li> </ul> <p>In relation to error-correcting codes:</p> <ul> <li>Kohki Kawabata, Shinichiro Yahagi, <em>Elliptic genera from classical error-correcting codes</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2308.12592">arXiv:2308.12592</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <h5 id="via_diracramond_operators_on_free_loop_space">Via Dirac-Ramond operators on free loop space</h5> <p>Tentative interpretation as <a class="existingWikiWord" href="/nlab/show/indices">indices</a> of <a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operators">Dirac-Ramond operators</a> as would-be <a class="existingWikiWord" href="/nlab/show/Dirac+operators+on+smooth+loop+space">Dirac operators on smooth loop space</a>:</p> <ul> <li id="Witten87b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>The Index Of The Dirac Operator In Loop Space</em>, in: <em>Elliptic Curves and Modular Forms in Algebraic Topology</em>, Lecture Notes in Mathematics <strong>1326</strong>, Springer (1988) 161-181 &lbrack;<a href="https://doi.org/10.1007/BFb0078045">doi:10.1007/BFb0078045</a>, <a href="http://inspirehep.net/record/245523">spire</a>&rbrack;</p> <p>originating from:</p> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, p. 92-94 in: <em>Global anomalies in string theory</em>, in: W. Bardeen and A. White (eds.) <em><a href="https://inspirehep.net/conferences/965785?ui-citation-summary=true">Symposium on Anomalies, Geometry, Topology</a></em>, World Scientific (1985) 61-99 &lbrack;<a class="existingWikiWord" href="/nlab/files/WittenGlobalAnomaliesInStringTheory.pdf" title="pdf">pdf</a>, <a href="https://inspirehep.net/literature/214913">spire:214913</a>&rbrack;</p> </li> <li id="AlvarezKillingbackManganoWindey87"> <p><a class="existingWikiWord" href="/nlab/show/Orlando+Alvarez">Orlando Alvarez</a>, T. P. Killingback, Michelangelo Mangano, <a class="existingWikiWord" href="/nlab/show/Paul+Windey">Paul Windey</a>, <em>The Dirac-Ramond operator in string theory and loop space index theorems</em>, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: <em>Nonperturbative methods in field theory</em>, 1987 (<a href="https://doi.org/10.1016/0920-5632(87)90110-1">doi"10.1016/0920-5632(87)90110-1</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Orlando+Alvarez">Orlando Alvarez</a>, T. P. Killingback, Michelangelo Mangano, <a class="existingWikiWord" href="/nlab/show/Paul+Windey">Paul Windey</a>, <em>String theory and loop space index theorems</em>, Comm. Math. Phys., 111(1):1–10, 1987 (<a href="https://projecteuclid.org/euclid.cmp/1104159462">euclid:cmp/1104159462</a>)</p> </li> <li id="Landweber99"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Landweber">Gregory Landweber</a>, <em>Dirac operators on loop space</em>, PhD thesis (Harvard 1999) (<a href="http://math.bard.edu/greg/LoopDirac.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Orlando+Alvarez">Orlando Alvarez</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Windey">Paul Windey</a>, <em>Analytic index for a family of Dirac-Ramond operators</em>, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 (<a href="https://arxiv.org/abs/0904.4748">arXiv:0904.4748</a>)</p> </li> </ul> <h5 id="via_conformal_nets">Via conformal nets</h5> <p>Tentative formulation via <a class="existingWikiWord" href="/nlab/show/conformal+nets">conformal nets</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <em>Topological modular forms and conformal nets</em>, in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Proceedings of Symposia in Pure Mathematics, AMS (2011) (<a href="https://arxiv.org/abs/1103.4187">arXiv:1103.4187</a>, <a href="https://doi.org/10.1090/pspum/083">doi:10.1090/pspum/083</a>)</li> </ul> <h4 id="InterpretationInTMFCohomology">Conjectural interpretation in tmf-cohomology</h4> <p>The resulting suggestion that, roughly, deformation-classes (<a class="existingWikiWord" href="/nlab/show/concordance">concordance</a> classes) of <a class="existingWikiWord" href="/nlab/show/2d+SCFTs">2d SCFTs</a> with <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> are the <a class="existingWikiWord" href="/nlab/show/Whitehead-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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> of <a class="existingWikiWord" href="/nlab/show/topological+modular+forms">topological modular forms</a> (<a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>):</p> <ul> <li id="StolzTeichner11"><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>Supersymmetric field theories and generalized cohomology</em>, in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a href="http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionStolzTeichner">Mathematical foundations of Quantum field theory and String theory</a></em>, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS 2011 (<a href="https://arxiv.org/abs/1108.0189">arXiv:1108.0189</a>)</li> </ul> <p>and the more explicit suggestion that, under this identification, the <a class="existingWikiWord" href="/nlab/show/Chern-Dold+character">Chern-Dold character</a> from <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> to <a class="existingWikiWord" href="/nlab/show/modular+forms">modular forms</a>, sends a <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> to its <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a>/<a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a>/supersymmetric index:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, Section 5 of: <em>Holomorphic SCFTs with small index</em>, Canadian Journal of Mathematics (<a href="https://arxiv.org/abs/1811.00589">arXiv:1811.00589</a>, <a href="https://doi.org/10.4153/S0008414X2100002X">doi:10.4153/S0008414X2100002X</a>)</li> </ul> <p>This perspective is also picked up in <a href="#GukovPeiPutrovVafa18">Gukov, Pei, Putrov & Vafa 18</a>.</p> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/2d+SCFTs">2d SCFTs</a> (namely <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>-<a class="existingWikiWord" href="/nlab/show/WZW-models">WZW-models</a>) conjecturally corresponding, under this conjectural identification, to the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/24</annotation></semantics></math> <a href="tmf#BoardmanHomomorphismInTmfIs6Connected"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>tmf</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>3</mn></mrow></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>tmf</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> tmf^{-3}(\ast) = \pi_3(tmf) </annotation></semantics></math> <a href="tmf#BoardmanHomomorphismInTmfIs6Connected"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_3(\mathbb{S})</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/third+stable+homotopy+group+of+spheres">third stable homotopy group of spheres</a>):</p> <ul> <li id="GaiottoJohnsonFreydWitten19"> <p><a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, p. 17 of: <em>A Note On Some Minimally Supersymmetric Models In Two Dimensions</em>, (<a href="https://arxiv.org/abs/1902.10249">arXiv:1902.10249</a>) in S. Novikov et al. <em>Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry</em>, Proc. Symposia Pure Math., 103(2), 2021 (<a href="https://bookstore.ams.org/pspum-103-2">ISBN: 978-1-4704-5592-7</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>Mock modularity and a secondary elliptic genus</em> (<a href="https://arxiv.org/abs/1904.05788">arXiv:1904.05788</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>Topological Mathieu Moonshine</em> (<a href="https://arxiv.org/abs/2006.02922">arXiv:2006.02922</a>)</p> </li> </ul> <p>Discussion properly via <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <em>How do field theories detect the torsion in topological modular forms?</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2303.09138">arXiv:2303.09138</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <em>How do field theories detect the torsion in topological modular forms?</em>, talk at <em><a href="https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html">QFT and Cobordism</a></em>, <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Mar 2023) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="Center+for+Quantum+and+Topological+Systems#BerwickEvansMar23">web</a>, video:<a href="https://www.youtube.com/watch?v=fw7yxFsDmjs">YT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>See also:</p> <ul> <li> <p>Ying-Hsuan Lin, <a class="existingWikiWord" href="/nlab/show/Du+Pei">Du Pei</a>, <em>Holomorphic CFTs and topological modular forms</em> &lbrack;<a href="https://arxiv.org/abs/2112.10724">arXiv:2112.10724</a>&rbrack;</p> </li> <li> <p>Jan Albert, Justin Kaidi, Ying-Hsuan Lin, <em>Topological modularity of Supermoonshine</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2210.14923">arXiv:2210.14923</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <a class="existingWikiWord" href="/nlab/show/Mayuko+Yamashita">Mayuko Yamashita</a>, <a class="existingWikiWord" href="/nlab/show/Kazuya+Yonekura">Kazuya Yonekura</a>, <em>Remarks on mod-2 elliptic genus</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2302.07548">arXiv:2302.07548</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, Hao Y. Zhang, <em>On a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</annotation></semantics></math>-valued discrete topological term in 10d heterotic string theories</em> [<a href="https://arxiv.org/abs/2403.08861">arXiv:2403.08861</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <a class="existingWikiWord" href="/nlab/show/Mayuko+Yamashita">Mayuko Yamashita</a>, <em>On the 576-fold periodicity of the spectrum SQFT: The proof of the lower bound via the Anderson duality pairing</em> [<a href="https://arxiv.org/abs/2404.06333">arXiv:2404.06333</a>]</p> </li> <li> <p>Vivek Saxena, <em>A T-Duality of Non-Supersymmetric Heterotic Strings and an implication for Topological Modular Forms</em> [<a href="https://arxiv.org/abs/2405.19409">arXiv:2405.19409</a>]</p> </li> </ul> <h4 id="occurrences_in_string_theory">Occurrences in string theory</h4> <h5 id="ReferencesEllipticGeneraAsPartitionFunctionsOfHStrings">H-string elliptic genus</h5> <p>Further on the elliptic genus of the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> being the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>:</p> <p>The interpretation of <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+genera">equivariant elliptic genera</a> as <a class="existingWikiWord" href="/nlab/show/partition+functions">partition functions</a> of <a class="existingWikiWord" href="/nlab/show/parametrized+WZW+models">parametrized WZW models</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>:</p> <ul> <li id="DistlerSharpe07"> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, section 8.5 of <em>Heterotic compactifications with principal bundles for general groups and general levels</em>, Adv. Theor. Math. Phys. 14:335-398, 2010 (<a href="http://arxiv.org/abs/hep-th/0701244">arXiv:hep-th/0701244</a>)</p> </li> <li id="Ando07"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <em>Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe</em>, <a href="http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php">talk 2007</a> (<a href="http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf">lecture notes pdf</a>)</p> </li> </ul> <p>Proposals on physics aspects of lifting the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a> to <a class="existingWikiWord" href="/nlab/show/topological+modular+forms">topological modular forms</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <em>Topological modular forms and the absence of a heterotic global anomaly</em>, Progress of Theoretical and Experimental Physics, <strong>2022</strong> 4 (2022) 04A107 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2103.12211">arXiv:2103.12211</a>, <a href="https://doi.org/10.1093/ptep/ptab060">doi:10.1093/ptep/ptab060</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <a class="existingWikiWord" href="/nlab/show/Mayuko+Yamashita">Mayuko Yamashita</a>, <em>Topological modular forms and the absence of all heterotic global anomalies</em>, Comm. Math. Phys. <strong>402</strong> (2023) 1585-1620 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2108.13542">arXiv:2108.13542</a>, <a href="https://doi.org/10.1007/s00220-023-04761-2">doi:10.1007/s00220-023-04761-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <a class="existingWikiWord" href="/nlab/show/Mayuko+Yamashita">Mayuko Yamashita</a>, <em>Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2305.06196">arXiv:2305.06196</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <h5 id="ReferencesEllipticGeneraAsPartitionFunctionsOfM5Branes">M5-brane elliptic genus</h5> <p>On the <a class="existingWikiWord" href="/nlab/show/M5-brane+elliptic+genus">M5-brane elliptic genus</a>:</p> <p>A <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> argued to describe the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> on a <a class="existingWikiWord" href="/nlab/show/4-manifold">4-manifold</a> (specifically: a <a class="existingWikiWord" href="/nlab/show/complex+surface">complex surface</a>) originates with</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Black Hole Entropy in M-Theory</em>, JHEP 9712:002, 1997 (<a href="https://arxiv.org/abs/hep-th/9711053">arXiv:hep-th/9711053</a>)</li> </ul> <p>Discussion of the resulting <a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a> (<a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a>) originates with:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Xi+Yin">Xi Yin</a>, <em>The M5-Brane Elliptic Genus: Modularity and BPS States</em>, JHEP 0708:070, 2007 (<a href="https://arxiv.org/abs/hep-th/0607010">hep-th/0607010</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Xi+Yin">Xi Yin</a>, <em>Examples of M5-Brane Elliptic Genera</em>, JHEP 0711:004, 2007 (<a href="https://arxiv.org/abs/hep-th/0702012">arXiv:hep-th/0702012</a>)</p> </li> </ul> <p>Further discussion in:</p> <ul> <li> <p>Murad Alim, <a class="existingWikiWord" href="/nlab/show/Babak+Haghighat">Babak Haghighat</a>, Michael Hecht, <a class="existingWikiWord" href="/nlab/show/Albrecht+Klemm">Albrecht Klemm</a>, Marco Rauch, Thomas Wotschke, <em>Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes</em>, Comm. Math. Phys. <strong>339</strong> (2015) 773–814 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1012.1608">arXiv:1012.1608</a>, <a href="https://doi.org/10.1007/s00220-015-2436-3">doi:10.1007/s00220-015-2436-3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="GukovPeiPutrovVafa18"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Du+Pei">Du Pei</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Putrov">Pavel Putrov</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>4-manifolds and topological modular forms</em>, J. High Energ. Phys. <strong>2021</strong> 84 (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1811.07884">arXiv:1811.07884</a>, <a href="https://doi.org/10.1007/JHEP05(2021)084">doi:10.1007/JHEP05(2021)084</a>, <a href="https://inspirehep.net/literature/1704312">spire:1704312</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <h5 id="ReferencesEllipticGeneraAsPartitionFunctionsOfMStrings">M-string elliptic genus</h5> <p>On the <a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a> of <a class="existingWikiWord" href="/nlab/show/M-strings">M-strings</a> inside <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Hohenegger">Stefan Hohenegger</a>, <a class="existingWikiWord" href="/nlab/show/Amer+Iqbal">Amer Iqbal</a>, <em>M-strings, Elliptic Genera and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\mathcal{N}=4</annotation></semantics></math> String Amplitudes</em>, Fortschritte der PhysikVolume 62, Issue 3 (<a href="http://arxiv.org/abs/1310.1325">arXiv:1310.1325</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Hohenegger">Stefan Hohenegger</a>, <a class="existingWikiWord" href="/nlab/show/Amer+Iqbal">Amer Iqbal</a>, Soo-Jong Rey, <em>M String, Monopole String and Modular Forms</em>, Phys. Rev. D 92, 066005 (2015) (<a href="https://arxiv.org/abs/1503.06983">arXiv:1503.06983</a>)</p> </li> <li> <p>M. Nouman Muteeb, <em>Domain walls and M2-branes partition functions: M-theory and ABJM Theory</em> (<a href="https://arxiv.org/abs/2010.04233">arXiv:2010.04233</a>)</p> </li> <li> <p>Kimyeong Lee, Kaiwen Sun, Xin Wang, <em>Twisted Elliptic Genera</em> &lbrack;<a href="https://arxiv.org/abs/2212.07341">arXiv:2212.07341</a>&rbrack;</p> </li> </ul> <h5 id="ReferencesEllipticGeneraAsPartitionFunctionsOfEStrings">E-string elliptic genus</h5> <p>On the <a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a> of <a class="existingWikiWord" href="/nlab/show/E-strings">E-strings</a> as <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>:</p> <ul> <li> <p>J. A. Minahan, D. Nemeschansky, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, N. P. Warner, <em>E-Strings and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">N=4</annotation></semantics></math> Topological Yang-Mills Theories</em>, Nucl. Phys. B527 (1998) 581-623 (<a href="https://arxiv.org/abs/hep-th/9802168">arXiv:hep-th/9802168</a>)</p> </li> <li> <p>Wenhe Cai, Min-xin Huang, Kaiwen Sun, <em>On the Elliptic Genus of Three E-strings and Heterotic Strings</em>, J. High Energ. Phys. 2015, 79 (2015). (<a href="https://arxiv.org/abs/1411.2801">arXiv:1411.2801</a>, <a href="https://doi.org/10.1007/JHEP01(2015)079">doi:10.1007/JHEP01(2015)079</a>)</p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a> of <a class="existingWikiWord" href="/nlab/show/E-strings">E-strings</a> as <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> <a class="existingWikiWord" href="/nlab/show/brane+intersection">ending on</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>:</p> <ul> <li id="KKLPV14">Joonho Kim, Seok Kim, Kimyeong Lee, <a class="existingWikiWord" href="/nlab/show/Jaemo+Park">Jaemo Park</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>Elliptic Genus of E-strings</em>, JHEP 1709 (2017) 098 (<a href="https://arxiv.org/abs/1411.2324">arXiv:1411.2324</a>)</li> </ul> </div></div></body></html> </div> <div class="revisedby"> <p> Last revised on September 15, 2021 at 07:32:48. 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