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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="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>, <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</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> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</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/circle+n-bundle+with+connection">circle n-bundle with connection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a></p> </li> </ul> </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> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/differential+cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="chernweil_theory"><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>-Chern-Weil theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</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+introduction">∞-Chern-Weil theory introduction</a></p> </li> </ul> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+an+%28%E2%88%9E%2C1%29-topos">differential cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></li> </ul> </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/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>, <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></li> </ul> </li> </ul> <h2 id="connection">Connection</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+valued+differential+forms">∞-Lie algebroid valued differential forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connection+on+a+principal+%E2%88%9E-bundle">∞-connection on a principal ∞-bundle</a></p> </li> </ul> <h2 id="curvature">Curvature</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a></p> </li> </ul> </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/Chern-Simons+form">Chern-Simons form</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Gauss-Bonnet+theorem">Chern-Gauss-Bonnet theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-Chern-Weil+theory+-+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='#Definition'>Definition</a></li> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#GeneralProperties'>General</a></li> <ul> <li><a href='#observation'>Observation</a></li> <li><a href='#observation_2'>Observation</a></li> </ul> <li><a href='#ChernWeilTheory'>Construction in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-Cech cocycles</a></li> <ul> <li><a href='#PresentationOfClassByFibration'>Presentation of the differential class by a fibration</a></li> <ul> <li><a href='#observation_3'>Observation</a></li> </ul> <li><a href='#ExplicitCocycles'>Explicit Cech cocycles</a></li> </ul> <li><a href='#differential_string_structures_and_fermionic_string_quantum_anomalies'>Differential string structures and fermionic string quantum anomalies</a></li> <li><a href='#InheteroticSugra'>The Green-Schwarz mechanism in heterotic supergravity</a></li> <li><a href='#relation_to_string_2connections'>Relation to string 2-connections</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Where a <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a> is a trivialization of a class in <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a>, a <em>differential string structure</em> or <em>geometric string structure</em> is the trivialization of this class refined to <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>:</p> <p>the first fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>B</mi><mi>Spin</mi><mo>→</mo><msup><mi>B</mi> <mn>4</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \frac{1}{2} p_1 : B Spin \to B^4 \mathbb{Z} </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> has a refinement to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>– the <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#SmoothFirstFracPontryaginClass">smooth first fractional Pontryagin class</a>.</p> <p>The induced morphism on <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}\mathbf{p}_1 : \mathbf{H}(X,\mathbf{B} Spin) \stackrel{}{\to} \mathbf{H}(X,\mathbf{B}^3 U(1)) </annotation></semantics></math></div> <p>sends a <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> to its corresponding <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}\mathbf{p}_1(P)</annotation></semantics></math>.</p> <p>A choice of trivialization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1(P)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>. The <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of smooth string structures is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\mathbf{p}_1</annotation></semantics></math> over the trivial <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a>.</p> <p>By <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a> this morphism may be further refined to a <a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1</annotation></semantics></math> that lands in the <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))</annotation></semantics></math>, classifying <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundles with connection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}\hat \mathbf{p}_1 : \mathbf{H}_{conn}(X,\mathbf{B} Spin) \stackrel{}{\to} \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1)) </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of <strong>differential string structures</strong> is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of this refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">trivial circle 3-bundle with trivial connection</a> or more generally over the trivial circle 3-bundles with possibly non-trivial connection</p> <p>Such a differential string structure over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is characterized by a tuple consisting of</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a choice of trivial <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a> with connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>H</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0, H_3)</annotation></semantics></math>, hence a differential 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>3</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_3 \in \Omega^3(X)</annotation></semantics></math>;</p> </li> <li> <p>a choice of <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> with connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat\mathbf{p}_1(\nabla)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> with this chosen 3-bundle</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>H</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lambda : \frac{1}{2}\hat \mathbf{p}_1(\nabla) \stackrel{\simeq}{\to} (0,H_3) \,. </annotation></semantics></math></div> <p>More generally, one can consider the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1</annotation></semantics></math> over arbitrary circle 3-bundles with connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>𝒢</mi><mo stretchy="false">^</mo></mover> <mn>4</mn></msub><mo>∈</mo><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat \mathcal{G}_4 \in \mathbf{H}_{diff}^4(X, \mathbf{B}^3 U(1))</annotation></semantics></math> and hence replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>H</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,H_3)</annotation></semantics></math> in the above with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>𝒢</mi><mo stretchy="false">^</mo></mover> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\hat \mathcal{G}_4</annotation></semantics></math>. According to the general notion of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, these may be thought of as <strong>twisted differential string structures</strong>, where the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>𝒢</mi> <mn>4</mn></msub><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\mathcal{G}_4] \in H^4_{diff}(X)</annotation></semantics></math> is the twist.</p> <h2 id="Definition">Definition</h2> <p>We will assume that the reader is familiar with basics of the discussion at <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. We often write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>:</mo><mo>=</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} := Smooth \infty Grpd</annotation></semantics></math> for short.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">Spin(n) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> be the <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a>, for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> and thus canonically as an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> object in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. We shall notationally suppress the <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> in the following. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. (See the discussion <a href="http://nlab.mathforge.org/nlab/show/smooth+infinity-groupoid#LieGroups">here</a>). Let moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1) \in Smooth \infty Grpd</annotation></semantics></math> be the <a href="http://nlab.mathforge.org/nlab/show/smooth+infinity-groupoid#CircleLienGroup">circle Lie 3-group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^3 U(1)</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>.</p> <p>At <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a> the following statement is proven (<a href="#FSS">FSS</a>).</p> <div class="num_prop" id="LieIntegrationToPOne"> <h6 id="proposition">Proposition</h6> <p>The image under <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the canonical <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra 3-cocycle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>:</mo><mi>𝔰𝔬</mi><mo>→</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \mu = \langle -,[-,-]\rangle : \mathfrak{so} \to b^2 \mathbb{R} </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{so}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> – the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> – is a morphism in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2} \mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>whose image under the <a href="http://nlab.mathforge.org/nlab/show/Euclidean-topological%20infinity-groupoid#GeometricHomotopy">the fundamental ∞-groupoid (∞,1)-functor/ geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\Pi : Smooth \infty Grpd \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is the ordinary fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>B</mi><mi>Spin</mi><mo>→</mo><msup><mi>B</mi> <mn>4</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \frac{1}{2}p_1 : B Spin \to B^4 \mathbb{Z} </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. Moreover, the corresponding <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#ChernWeilTheory">refined differential characteristic class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}\hat \mathbf{p}_1 : \mathbf{H}_{conn}(-,\mathbf{B}Spin) \to \mathbf{H}_{diff}(-, \mathbf{B}^3 U(1)) </annotation></semantics></math></div> <p>is in <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> the corresponding refined <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>:</mo><msubsup><mi>H</mi> <mi>Smooth</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Spin</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [\frac{1}{2}\hat \mathbf{p}_1] : H^1_{Smooth}(X,Spin) \to H_{diff}^4(X) </annotation></semantics></math></div> <p>with values in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> that corresponds to the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle - , - \rangle </annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{so}</annotation></semantics></math>.</p> </div> <div class="num_defn" id="DifferentialStringStructure"> <h6 id="definition_2">Definition</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of <strong>differential string-structures</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{diff}(X)</annotation></semantics></math> – is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1(X)</annotation></semantics></math> over the trivial differential cocycle.</p> <p>More generally (see <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>) the 2-groupoid of <strong>twisted differential string structures</strong> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{diff,tw}(X)</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>String</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ String_{diff,tw}(X) &\to& H_{diff}^4(X) \\ \downarrow && \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}Spin) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1)) } \,, </annotation></semantics></math></div> <p>where the right vertical morphism is a choice of (any) one point in each <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> (differential cohomology class) of the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> is independent of this choice).</p> <p>More specifically, a <strong>geometric string structure</strong> is a twisted differential string structure whose differential twist has underlying trivial class.</p> </div> <div class="num_note"> <h6 id="note">Note</h6> <p>In terms of local <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+valued+differential+forms">∞-Lie algebra valued differential forms</a> data this has been considered in (<a href="#SSSIII">SSSIII</a>), as we shall discuss <a href="#ChernWeilTheory">below</a>.</p> <p>For the case where the the underlying integral class of the twist is trivial – geometric string structures – something close to this definition, explicitly modeled on <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a>s, has been given in (<a href="#Waldorf">Waldorf</a>). See the discussion <a href="#Properties">below</a>.</p> </div> <h2 id="Properties">Properties</h2> <h3 id="GeneralProperties">General</h3> <div class="num_prop" id="GroupoidOfStringStructures"> <h6 id="observation">Observation</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mrow><mi>tw</mi><mo>,</mo><mi>diff</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{tw,diff}(X)</annotation></semantics></math> of twisted differential string structures is <a class="existingWikiWord" href="/nlab/show/2-truncated">2-truncated</a>, hence is a <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This follows from the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> associated to the defining <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>, using that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \mathbf{B}Spin)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4_{diff}(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/0-groupoid">0-groupoid</a>.</p> </li> </ul> </div> <p>See also (<a href="#Waldorf">Waldorf, cor. 1.1.5</a>).</p> <div class="num_prop" id="PropertiesOfTrivialIntegralTwistClass"> <h6 id="observation_2">Observation</h6> <p>If the underlying <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> class of the twist is trivial, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>tw</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c(tw) = 0 \in H^3(X, \mathbb{Z})</annotation></semantics></math>, then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tw</mi></mrow><annotation encoding="application/x-tex">tw</annotation></semantics></math>-twisted differential string structures on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> are characterized by a globally defined 3-form 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> <p>This 3-form is the globally defined connection 3-form of an appropriate <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a> equivalent to the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS(\nabla)</annotation></semantics></math> whose underlying 3-bundle is by assumption trivial: on a trivial circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-bundle every connection may be represented by a globally defined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form.</p> <p>This statement appears as (<a href="#Waldorf">Waldorf, theorem 1.3.3</a>), where circle 3-bundles are modeled as <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a>s. The explicit construction of the globally defined 3-form in this model is spelled out in lemma 3.2.4 there.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Elements in the defining <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> from def. <a class="maruku-ref" href="#DifferentialStringStructure"></a> over a given <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \in \mathbf{H}(X,\mathbf{B} Spin)_{conn}</annotation></semantics></math> are chracterized by an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\alpha] \in H^4_{diff}(X)</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a></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>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>α</mi></mrow><annotation encoding="application/x-tex"> \Omega : CS(\nabla) \stackrel{\simeq}{\to} \alpha </annotation></semantics></math></div> <p>between the corresponding <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> and the given <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>. In the case at hand, both have underlying trivial class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c(CS(\nabla)) = c(\alpha) = 0</annotation></semantics></math>.</p> <p>By the <a href="http://nlab.mathforge.org/nlab/show/ordinary%20differential%20cohomology#AbstractProperties">characteristic class exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msubsup><mi>Ω</mi> <mi>int</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>c</mi></mover><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> 0 \to \Omega^3(X)/\Omega_{int}^{3}(X) \to H^4_{diff}(X) \stackrel{c}{\to} H^4(X, \mathbb{Z}) </annotation></semantics></math></div> <p>any two classes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) \simeq H^4_{diff}(X)</annotation></semantics></math> that have trivial underlying class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 \mathbf{H}(X, \mathbf{B}^3 U(1)) \simeq H^4(X, \mathbb{Z})</annotation></semantics></math> differ by a 3-form modulo a closed 3-form with integral periods.</p> <p>Therefore both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha]</annotation></semantics></math> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CS(\nabla)] \in H^4_{diff}(X)</annotation></semantics></math> are given by a globally defined 3-form modulo an integral form: the global connection 3-form on these trivial <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundles with connection</a>.</p> </div> <h3 id="ChernWeilTheory">Construction in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-Cech cocycles</h3> <p>We use the <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> (as described there) by the local <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet]_{proj,loc}</annotation></semantics></math> to give an explicit construction of twisted differential string structures in terms of <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a>-cocycles with coefficients in <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+valued+differential+forms">∞-Lie algebra valued differential forms</a>.</p> <p>Proofs not displayed here can be found at <em><a class="existingWikiWord" href="/nlab/show/differential+string+structure+--+proofs">differential string structure – proofs</a></em> .</p> <p>Recall the following fact from <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a> (<a href="#FSS">FSS</a>).</p> <div class="num_prop" id="LieIntegrationOfDifferentialPOne"> <h6 id="proposition_2">Proposition</h6> <p>The differential fractional Pontryagin class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2} \hat \mathbf{p}_1</annotation></semantics></math> is presented in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet]_{proj}</annotation></semantics></math> by the top morphism of simplicial presheaves in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><msub><mo stretchy="false">)</mo> <mrow><mi>ChW</mi><mo>,</mo><mi>smp</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mrow><mi>ChW</mi><mo>,</mo><mi>smp</mi></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>smp</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mi>smp</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>c</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{cosk}_3\exp(\mathfrak{so})_{ChW,smp} &\stackrel{\exp(\mu, cs)}{\to}& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,smp} \\ \downarrow && \downarrow \\ \mathbf{cosk}_3\exp(\mathfrak{so})_{diff,smp} &\stackrel{\exp(\mu, cs)}{\to}& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{smp} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}Spin_{c} } \,. </annotation></semantics></math></div></div> <p>Here the middle morphism is the direct <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">L-∞ algebra cocycle</a> while the top morphisms is its restriction to coefficients for <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connections</a>.</p> <p>In order to compute the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1</annotation></semantics></math> we now find a <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of this morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mu,cs)</annotation></semantics></math> by a fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet]_{proj}</annotation></semantics></math>. By the fact that this is a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> then also the hom of any cofibrant object into this morphism, computing the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids, is a fibration, and therefore, by the general discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>, we obtain the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s as the ordinary <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>s of this fibration.</p> <h4 id="PresentationOfClassByFibration">Presentation of the differential class by a fibration</h4> <p>In order to factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mu,cs)</annotation></semantics></math> into a weak equivalence followed by a fibration, we start by considering such a factorization before differential refinement, on the underlying characteristic class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mu)</annotation></semantics></math>.</p> <p>To that end, we replace the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><mi>𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g} = \mathfrak{so}</annotation></semantics></math> by an equivalent but bigger <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">Lie 3-algebra</a> (following <a href="#SSSIII">SSSIII</a>). We need the following notation:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><mi>𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g} = \mathfrak{so}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> (the Lie algebra of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">b^2 \mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 3-algebra</a>, the single generator in degee 3 of its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> we denote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in CE(b^2 \mathbb{R})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d c = 0</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo>∈</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle \in W(\mathfrak{g})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a>, regarded as an element of the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{so}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>∈</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu := \langle -,[-,-]\rangle \in CE(\mathfrak{g})</annotation></semantics></math> the degree 3 <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cocycle</a>, identified with a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>←</mo><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) \leftarrow CE(b^2 \mathbb{R}) : \mu </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>s; and normalized such that its continuation to a 3-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> is the image in</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> of a generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Spin</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H^3(Spin,\mathbb{Z}) \simeq \mathbb{Z}</annotation></semantics></math>;</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cs</mi><mo>∈</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cs \in W(\mathfrak{g})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a> interpolating between the two; characterized by the fact that it fits into the commuting 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>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>μ</mi></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>cs</mi></mover></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow></mover></mtd> <mtd><mi>inv</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^2 \mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^2 \mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle-,-\rangle}{\leftarrow}& inv(b^2 \mathbb{R}) & = CE(b^3 \mathbb{R}) } </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo>:</mo><mo>=</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}_\mu := \mathfrak{string}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a>.</p> </li> </ul> <div class="num_defn" id="ResolutionOfLieAlgebra"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b\mathbb{R} \to \mathfrak{g}_\mu)</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> whose <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>⊕</mo><mo stretchy="false">⟨</mo><mi>b</mi><mo stretchy="false">⟩</mo><mo>⊕</mo><mo stretchy="false">⟨</mo><mi>c</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> CE(b\mathbb{R} \to \mathfrak{g}_\mu) = (\wedge^\bullet( \mathfrak{g}^* \oplus \langle b\rangle \oplus \langle c \rangle ), d) \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> a generator in degree 2, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> a generator in degree 3, and with differential defined on generators by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><msub><mo stretchy="false">|</mo> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup></mtd></mtr> <mtr><mtd><mi>d</mi><mi>b</mi></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>μ</mi><mo>+</mo><mi>c</mi></mtd></mtr> <mtr><mtd><mi>d</mi><mi>c</mi></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d|_{\mathfrak{g}^*} & = [-,-]^* \\ d b & = - \mu + c \\ d c & = 0 \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_prop" id="FactorizationOfTheCocycle"> <h6 id="proposition_3">Proposition</h6> <p>The 3-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mi>μ</mi></mover><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R})</annotation></semantics></math> factors as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>↦</mo><mi>μ</mi><mo>,</mo><mi>b</mi><mo>↦</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mover><mi>CE</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>↦</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mover><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>μ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) \stackrel{(c \mapsto \mu, b \mapsto 0)}{\leftarrow} CE(b\mathbb{R} \to \mathfrak{g}_\mu) \stackrel{(c \mapsto c)}{\leftarrow} CE(b^2 \mathbb{R}) : \mu \,, </annotation></semantics></math></div> <p>where the morphism on the left (which is the identity when restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> and acts on the new generators as indicated) is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>.</p> </div> <p>The point of introducing the resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b \mathbb{R} \to \mathfrak{g}_\mu)</annotation></semantics></math> in the above way is that it naturally supports the obstruction theory of lifts from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> to <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> <a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">2-connection</a></p> <div class="num_prop" id="LongFiberSequenceOnLieAlgebras"> <h6 id="observation_3">Observation</h6> <p>The defining projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}_\mu \to \mathfrak{g}</annotation></semantics></math> factors through the above quasi-isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">(b \mathbb{R} \to \mathfrak{g}_\mu) \to \mathfrak{g}</annotation></semantics></math> by the canonical inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{g}_\mu \to (b \mathbb{R} \to \mathfrak{g}_\mu) \,, </annotation></semantics></math></div> <p>which dually on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi></mrow><annotation encoding="application/x-tex">CE</annotation></semantics></math>-algebras is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>t</mi> <mi>a</mi></msup><mo>↦</mo><msup><mi>t</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex"> t^a \mapsto t^a </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>b</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>b</mi></mrow><annotation encoding="application/x-tex"> b \mapsto - b </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>c</mi><mo>↦</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c \mapsto 0 \,. </annotation></semantics></math></div> <p>In total we are looking at a convenient presentation of the long <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> of the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> extension:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>b</mi><mi>ℝ</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>𝔤</mi> <mi>μ</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && (b \mathbb{R} \to \mathfrak{g}_\mu) &\to& b^2 \mathbb{R} \\ && & \nearrow & \downarrow^{\mathrlap{\simeq}} \\ b \mathbb{R} &\to& \mathfrak{g}_\mu &\to& \mathfrak{g} } \,. </annotation></semantics></math></div></div> <p>(The signs appearing here are just unimportant convention made in order for some of the formulas below to come out nice.)</p> <div class="num_prop" id="BareFibration"> <h6 id="proposition_4">Proposition</h6> <p>The image under <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the above factorization is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex"> \exp(\mu) : \mathbf{cosk}_3\exp(\mathfrak{g}) \to \mathbf{cosk}_3\exp(b \mathbb{R} \to \mathfrak{g}_\mu) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c </annotation></semantics></math></div> <p>where the first morphism is a weak equivalence followed by a fibration in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet]_{proj}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>To see that the left morphism is objectwise a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, notice that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[k]</annotation></semantics></math>-cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(b \mathbb{R} \to \mathfrak{g}_\mu)</annotation></semantics></math> consists of a triple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B,C)</annotation></semantics></math>, where <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> is a vertical flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued 1-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">U\times\Delta^k</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a vertical 2-form and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a 3-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">U\times\Delta^k</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mo>=</mo><mi>C</mi><mo>−</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d B=C-\mu(A,A,A)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d C=0</annotation></semantics></math>, since <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> is flat. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is uniquely determined by <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, and there are no conditions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. This means that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[k]</annotation></semantics></math>-cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(b \mathbb{R} \to \mathfrak{g}_\mu)</annotation></semantics></math> is identified with a pair consisting of a based <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo>→</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">f : \Delta^k \to Spin</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/vertical+differential+form">vertical 2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \in \Omega^2_{si,vert}(U \times \Delta^k)</annotation></semantics></math>, (both suitably with sitting instants perpendicular to the boundary of the simplex). Since there is no further condition on the 2-form, it can always be extended from the boundary of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-simplex to the interior (for instance simply by radially rescaling it smoothly to 0). Accordingly the <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(b \mathbb{R} \to \mathfrak{g}_\mu)(U)</annotation></semantics></math> are the same as those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})(U)</annotation></semantics></math>. The morphism between them is the identity in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and picks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B = 0</annotation></semantics></math> and is hence clearly an isomorphism on homotopy groups.</p> <p>We turn now to discussing that the second morphism is a fibration. The nontrivial degrees of the lifting problem</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>Λ</mi><mo stretchy="false">[</mo><mi>k</mi><msub><mo stretchy="false">]</mo> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda[k]_i &\to& \mathbf{cosk}_3\exp(b\mathbb{R} \to \mathfrak{g}_\mu)(U) \\ \downarrow && \downarrow \\ \Delta[k] &\to& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c(U) } </annotation></semantics></math></div> <p>are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">k = 3</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">k = 4</annotation></semantics></math>.</p> <p>Notice that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^3 \mathbb{R}/ \mathbb{Z}_c(U)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">U \to \mathbb{R}/\mathbb{Z}</annotation></semantics></math> and that the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\exp(b\mathbb{R} \to \mathfrak{g}_\mu) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c</annotation></semantics></math> sends the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,B)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow></msub><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">⟨</mo><mi>θ</mi><mo>∧</mo><mo stretchy="false">[</mo><mi>θ</mi><mo>∧</mo><mi>θ</mi><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>+</mo><mi>d</mi><mi>B</mi><mo stretchy="false">)</mo><mi>mod</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\int_{\Delta^3}(f^* \langle \theta \wedge [\theta \wedge \theta]\rangle + d B) mod \mathbb{Z}</annotation></semantics></math>.</p> <p>Our lifting problem in degree 3, has given a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo>→</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">c : U \times \Delta^3 \to \mathbb{R}/\mathbb{Z}</annotation></semantics></math> and a smooth function (with sitting instants at the subfaces) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>U</mi><mo>×</mo><msubsup><mi>Λ</mi> <mi>i</mi> <mn>3</mn></msubsup><mo>→</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">f : U \times \Lambda^3_i \to Spin</annotation></semantics></math> together with a 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/horn">horn</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msubsup><mi>Λ</mi> <mi>i</mi> <mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">U \times \Lambda^3_i</annotation></semantics></math>.</p> <p>By pullback along the standard <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> <a class="existingWikiWord" href="/nlab/show/retract">retract</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>3</mn></msup><mo>→</mo><msubsup><mi>Λ</mi> <mi>i</mi> <mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">\Delta^3 \to \Lambda^3_i</annotation></semantics></math> which is non-smooth only where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has sitting instants, we can always extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo>→</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">f' : U \times \Delta^3 \to Spin</annotation></semantics></math> with the property that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">⟨</mo><mi>θ</mi><mo>∧</mo><mo stretchy="false">[</mo><mi>θ</mi><mo>∧</mo><mi>θ</mi><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\int_{\Delta^3} (f')^* \langle \theta \wedge [\theta \wedge \theta]\rangle = 0</annotation></semantics></math>. (Following the general discussion at <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>.)</p> <p>In order to find a horn filler for the 2-form component, consider any smooth 2-form with sitting instants and non-vanishing integeral on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^2</annotation></semantics></math>, regarded as the missing face of the <a class="existingWikiWord" href="/nlab/show/horn">horn</a>. By multiplying it with a suitable smooth function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> we can obtain an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde B \in \Omega^3_{si,vert}(U \times \partial \Delta^3)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">U \times \partial \Delta^3</annotation></semantics></math> with the property that its integral over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\partial \Delta^3</annotation></semantics></math> is the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a> it remains to extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde B</annotation></semantics></math> to the interior of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^3</annotation></semantics></math> in any way, as long as it is smooth and has sitting instants.</p> <p>To that end, we can find in a similar fashion a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized family of closed 3-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with sitting instants on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^3</annotation></semantics></math>, whose integral over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^3</annotation></semantics></math> equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>. Since by sitting instants this 3-form vanishes in a neighbourhood of the boundary, the standard formula for the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> applied to it produces a 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo><mo>∈</mo><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B' \in \Omega^2_{si, vert}(U \times \Delta^3)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mo>′</mo><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d B' = C</annotation></semantics></math> that itself is radially constant at the boundary. By construction the difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><mo>−</mo><mi>B</mi><mo>′</mo><msub><mo stretchy="false">|</mo> <mrow><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup></mrow></msub></mrow><annotation encoding="application/x-tex">\tilde B - B'|_{\partial \Delta^3}</annotation></semantics></math> has vanishing surface integral. By the discussion at <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> it follows that the difference extends smoothly and with sitting instants to a closed 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat B \in \Omega^2_{si,vert}(U \times \Delta^3)</annotation></semantics></math>. Therefore the sum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B' + \hat B \in \Omega^2_{si,vert}(U \times \Delta^3) </annotation></semantics></math></div> <p>equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> when restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mi>i</mi> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\Lambda^k_i</annotation></semantics></math> and has the property that its integral over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^3</annotation></semantics></math> equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>. Together with our extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math>, this constitutes a pair that solves the lifting problem.</p> <p>The extension problem in degree 4 amounts to a similar construction: by coskeletalness the condition is that for a given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">c : U \to \mathbb{R}/\mathbb{Z}</annotation></semantics></math> and a given vertical 2-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">U \times \partial \Delta^3</annotation></semantics></math> such that its integral equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>, as well as a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>U</mi><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo>→</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">f : U \times \partial \Delta^3 \to Spin</annotation></semantics></math>, we can extend the 2-form and the function along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mo>∂</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo>→</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">U \times \partial \Delta^3 \to U \times \Delta^3</annotation></semantics></math>. The latter follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mi>Spin</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi_2 Spin = 0</annotation></semantics></math> which guarantees a continuous filler (with sitting instants), and using the <a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a> to make this smooth. We are left with the problem of extending the 2-form, which is the same problem we discussed above after the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde B</annotation></semantics></math>.</p> </div> <p>We now proceed to extend this factorization to the exponentiated differential coefficients (see <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a>).</p> <div class="num_prop" id="PresentationByFibration"> <h6 id="proposition_5">Proposition</h6> <p><strong>(presentation of the differential class by a fibration)</strong></p> <p>Under <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> the <a href="#FactorizationOfTheCocycle">above factorization</a> of the Lie algebra cocycle<br />maps to the factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>ChW</mi><mo>,</mo><mi>ch</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \exp(\mu, cs) : \mathbf{cosk}_3 \exp(\mathfrak{g})_{ChW} \stackrel{\simeq}{\to} \mathbf{cosk}_3 \exp((b \mathbb{R} \to \mathfrak{g}_\mu))_{ChW} \to \mathbf{B}^3 U(1)_{ChW,ch} </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mu,cs)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math>, where the first morphism is a weak equivalence and the second a fibration.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The following proof makes use of details discussed at <em><a class="existingWikiWord" href="/nlab/show/differential+string+structure+--+proofs">differential string structure – proofs</a></em> .</p> <p>We discuss that the first morphism is an equivalence. Clearly it is injective on homotopy groups: if a sphere 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>-data cannot be filled, then also adding the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,C)</annotation></semantics></math>-data does not yield a filler. So we need to check that it is also surjective on homotopy groups: if the <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>-data can be filled, then also the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,C)</annotation></semantics></math>-data has a filler. Since the curvature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is horizontal it is already extended. We may extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> in any smooth way to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">U \times \Delta^k</annotation></semantics></math> (for instance by rescaling it smoothly to zero at the center of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-simplex) and then take the equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">d B = - CS(A) + C + H</annotation></semantics></math> to define the extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>We now check that the second morphism is a fibration. It is itself the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo>→</mo><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo stretchy="false">/</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mrow><mi>ChW</mi><mo>,</mo><mi>ch</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{cosk}_{3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{ChW} \to \exp(b^2 \mathbb{R})_{ChW}/\mathbb{Z} \stackrel{\int_{\Delta^\bullet}}{\to} \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,ch} \,. </annotation></semantics></math></div> <p>Here the second morphism is a degreewise surjection of simplicial abelian groups, hence a degreewise surjection under the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> functor, hence is itself already a projective fibration. Therefore it is sufficient to show that the first morphism here is a fibration.</p> <p>In degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">k = 3</annotation></semantics></math> the lifting problems</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>Λ</mi><mo stretchy="false">[</mo><mi>k</mi><msub><mo stretchy="false">]</mo> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>smp</mi><mo>,</mo><mi>ChW</mi></mrow></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mrow><mi>smp</mi><mo>,</mo><mi>ChW</mi></mrow></msub><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda[k]_i &\to & \exp(b\mathbb{R} \to \mathfrak{g}_{\mu})_{smp,ChW}(U) \\ \downarrow && \downarrow \\ \Delta[k] &\to& \exp(b^2 \mathbb{R})_{smp,ChW}/\mathbb{Z}(U) } </annotation></semantics></math></div> <p>may all be equivalently reformulated as lifting against a <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>k</mi></msup><mo>↪</mo><msup><mi>D</mi> <mi>k</mi></msup><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">D^k \hookrightarrow D^k \times [0,1]</annotation></semantics></math> by using the sitting instants of all forms.</p> <p>We have then a 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>si</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>D</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in \Omega^3_{si}(U \times D^{k-1}\times [0,1])</annotation></semantics></math> with horizontal curvature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G} \in \Omega^4(U)</annotation></semantics></math> and differential form data <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>D</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">U \times D^{k-1}</annotation></semantics></math> given. We may always extend <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> along the cylinder direction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> (its vertical part is equivalently a based smooth function to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> which we may extend constantly). <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> has to be horizontal so it is to be constantly extended along the cylinder.</p> <p>We can then use the kind of formula that proves the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> to extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>:</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>k</mi></msup><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>k</mi></msup><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Psi : (D^k \times [0,1]) \times [0,1] \to (D^k \times [0,1])</annotation></semantics></math> be a smooth contraction. Then while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(H - CS(A) + C)</annotation></semantics></math> may be non-vanishing, by horizonatlity of their <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>s we still have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>t</mi></msub></mrow></msub><msubsup><mi>Ψ</mi> <mi>t</mi> <mo>*</mo></msubsup><mi>d</mi><mo stretchy="false">(</mo><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota_{\partial_t} \Psi_t^* d(H - CS(A) + C)</annotation></semantics></math> vanishes (since the contraction vanishes).</p> <p>Therefore the 2-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msub><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>t</mi></msub></mrow></msub><msubsup><mi>Ψ</mi> <mi>t</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde B := \int_{[0,1]} \iota_{\partial_t} \Psi_t^*(H - CS(A) + C) </annotation></semantics></math></div> <p>satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \tilde B = (H - CS(A) + C)</annotation></semantics></math>. It may however not coincide with our given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t = 0</annotation></semantics></math>. But the difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>−</mo><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B - \tilde B|_{t = 0}</annotation></semantics></math> is a closed form on the left boundary of the cylinder. We may find some closed 2-form on the other boundary such that the integral around the boundary vanishes. Then the argument from the proof of the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the <a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a> applies and we find an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> to a closed 2-form on the interior. The sum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mo>=</mo><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><mo>+</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex"> \hat B := \tilde B + \lambda </annotation></semantics></math></div> <p>then still satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>=</mo><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d \hat B = H - CS(A) - C</annotation></semantics></math> and it coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> on the left boundary.</p> <p>Notice that here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde B</annotation></semantics></math> indeed has sitting instants: since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS(A)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> have sitting instants they are constant on their value at the boundary in a neighbourhood perpendicular to the boundary, which means for these 3-forms in the degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\leq 3</annotation></semantics></math> that they <em>vanish</em> in a neighbourhood of the boundary, hence that the above integral is towards the boundary over a vanishing integrand.</p> <p>In degree 4 the nature of the lifting problem</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>Λ</mi><mo stretchy="false">[</mo><mn>4</mn><msub><mo stretchy="false">]</mo> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mrow><mi>ChW</mi><mo>,</mo><mi>ch</mi></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda[4]_i &\to& \mathbf{cosk}_3\exp(b\mathbb{R} \to \mathfrak{g}_\mu)(U) \\ \downarrow && \downarrow \\ \Delta[4] &\to& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,ch} } </annotation></semantics></math></div> <p>starts out differently, due to the presence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{cosk}_3</annotation></semantics></math>, but it then ends up amounting to the same kind of argument:</p> <p>We have four functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">U \to \mathbb{R}/\mathbb{Z}</annotation></semantics></math> which we may realize as the <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> of a 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>∖</mo><msub><mi>δ</mi> <mi>i</mi></msub><mi>Δ</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \times (\partial \Delta[4] \setminus \delta_i \Delta[3])</annotation></semantics></math>, and we have a lift to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B,C, H)</annotation></semantics></math>-data on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>∖</mo><msub><mi>δ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \times (\partial \Delta[4]\setminus \delta_i(\Delta[3]))</annotation></semantics></math> (the boundary of the 4-simplex minus one of its 3-simplex faces).</p> <p>We observe that we can</p> <ul> <li> <p>always extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> smoothly to the remaining 3-face such that its <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> there reproduces the signed difference of the four given functions corresponding to the other faces (choose any smooth 3-form with sitting instants and with non-vanishing integral and rescale smoothly);</p> </li> <li> <p>fill the <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>-data horizonatlly due to the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Spin</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi_2 (Spin) = 0</annotation></semantics></math>.</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-form is already horizontal, hence already filled.</p> </li> </ul> <p>Moreover, by the fact that the 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> already is defined on all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>∖</mo><msub><mi>δ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[4] \setminus \delta_i(\Delta[3])</annotation></semantics></math> its fiber integral over the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[3]</annotation></semantics></math> coincides with the fiber integral of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">H - CS(A) + C</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>∖</mo><msub><mi>δ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[4] \setminus \delta_i (\Delta[3])</annotation></semantics></math>). But by the fact that we have lifted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>vert</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow></msub></mrow><annotation encoding="application/x-tex">\mu(A_{vert}) = CS(A)|_{\Delta^3}</annotation></semantics></math> is an integral cocycle, it follows that this equals the fiber integral of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C - CS(A)</annotation></semantics></math> over the remaining face.</p> <p>Use then as above the vertical Poincare lemma-formula to find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde B</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">U \times \Delta^3</annotation></semantics></math> with sitting instants that satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mo>=</mo><mi>H</mi><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d B = H - CS(A) + C</annotation></semantics></math> there. Then extend the closed difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>−</mo><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><msub><mo stretchy="false">|</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">B - \tilde B|_{0}</annotation></semantics></math> to a closed smooth 2-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^3</annotation></semantics></math>. As before, the difference</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mo>=</mo><mover><mi>B</mi><mo stretchy="false">˜</mo></mover><mo>+</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex"> \hat B := \tilde B + \lambda </annotation></semantics></math></div> <p>is an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> that constitutes a lift.</p> </div> <h4 id="ExplicitCocycles">Explicit Cech cocycles</h4> <div class="num_cor" id="PresentationBySimplicialPresheaves"> <h6 id="corollary">Corollary</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, for any choice of differentiaby <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> with corresponding cofibrant presentation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>C</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">\hat X = C(\{C_i\})\in [CartSp_{smooth}^{op}, sSet]_{proj}</annotation></semantics></math> we have that the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>s of <a href="#DifferentialStringStructure">twisted different String structures</a>s are presented by the ordinary <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>s of the morphism of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [CartSp^{op}, sSet](\hat X,\exp(\mu,cs)) : [CartSp^{op}, sSet](\hat X, \mathbf{cosk}_3 \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{ChW}) \to [CartSp^{op}, sSet](\hat X, \mathbf{B}^3 U(1)_{ChW}) \,. </annotation></semantics></math></div> <p>over any basepoints in the connected components of the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> on the right, which correspond to the elements<br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mover><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\hat \mathbf{C}_3] \in H_{diff}^4(X)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet]_{proj}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet](\hat X,\exp(\mu,cs))</annotation></semantics></math> is a fibration because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>cs</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mu,cs)</annotation></semantics></math> is and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat X</annotation></semantics></math> is cofibrant.</p> <p>It follows from the discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> that the ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>String</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</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><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ String_{diff,tw}(X) &\to& H_{diff}^4(X) \\ \downarrow && \downarrow \\ [CartSp^{op}, sSet](\hat X, \mathbf{cosk}_3 \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{ChW}) &\to& [CartSp^{op}, sSet](\hat X, \mathbf{B}^3 U(1)_{ChW}) } </annotation></semantics></math></div> <p>is a presentation for the defining <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{diff,tw}(X)</annotation></semantics></math>, as defined <a href="#DifferentialStringStructure">above</a>.</p> </div> <p>We unwind the explicit expression for a twisted differential string structure under this equivalence.</p> <p>Any twisting cocycle is in the above presentation given by a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne</a>-cocycle (as discussed at <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>3</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>3</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hat \mathbf{H}_3 = ((H_3)_i, \cdots) </annotation></semantics></math></div> <p>with local connection 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>3</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H_3)_i \in \Omega^3(U_i)</annotation></semantics></math> and globally defined <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 4-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>4</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G}_4 \in \Omega^4(X)</annotation></semantics></math>.</p> <div class="num_note" id="UnwindingTheLocalData"> <h6 id="note_2">Note</h6> <p>A differential string structure 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> twisted by this cocycles is on patches <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>←</mo><mover><mi>W</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(U_i) \leftarrow \tilde W(b\mathbb{R}\to \mathfrak{g}_\mu) </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/dgAlg">dgAlg</a>, subject to some horizontality constraints. The components of this are over each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> a collection of differential forms of the following structure</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msub><mi>F</mi> <mi>ω</mi></msub><mo>=</mo></mtd> <mtd><mi>d</mi><mi>ω</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><mi>ω</mi><mo>∧</mo><mi>ω</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><msub><mi>H</mi> <mn>3</mn></msub><mo>=</mo></mtd> <mtd><mo>∇</mo><mi>B</mi><mo>:</mo><mo>=</mo><mi>d</mi><mi>B</mi><mo>+</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>C</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>𝒢</mi> <mn>4</mn></msub><mo>=</mo></mtd> <mtd><mi>d</mi><msub><mi>C</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><mi>d</mi><msub><mi>F</mi> <mi>ω</mi></msub><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><mi>ω</mi><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mi>d</mi><msub><mi>H</mi> <mn>3</mn></msub><mo>=</mo></mtd> <mtd><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">⟩</mo><mo>−</mo><msub><mi>𝒢</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd><mi>d</mi><msub><mi>𝒢</mi> <mn>4</mn></msub><mo>=</mo></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>←</mo><mrow><mtable><mtr><mtd><msup><mi>t</mi> <mi>a</mi></msup></mtd> <mtd><mo>↦</mo><msup><mi>ω</mi> <mi>a</mi></msup></mtd></mtr> <mtr><mtd><msup><mi>r</mi> <mi>a</mi></msup></mtd> <mtd><mo>↦</mo><msubsup><mi>F</mi> <mi>ω</mi> <mi>a</mi></msubsup></mtd></mtr> <mtr><mtd><mi>b</mi></mtd> <mtd><mo>↦</mo><mi>B</mi></mtd></mtr> <mtr><mtd><mi>c</mi></mtd> <mtd><mo>↦</mo><msub><mi>C</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><mi>h</mi></mtd> <mtd><mo>↦</mo><msub><mi>H</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><mi>g</mi></mtd> <mtd><mo>↦</mo><msub><mi>𝒢</mi> <mn>4</mn></msub></mtd></mtr></mtable></mrow></mover><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msup><mi>r</mi> <mi>a</mi></msup><mo>=</mo></mtd> <mtd><mi>d</mi><msup><mi>t</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>t</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>t</mi> <mi>c</mi></msup></mtd></mtr> <mtr><mtd><mi>h</mi><mo>=</mo></mtd> <mtd><mi>d</mi><mi>b</mi><mo>+</mo><mi>cs</mi><mo>−</mo><mi>c</mi></mtd></mtr> <mtr><mtd><mi>g</mi><mo>=</mo></mtd> <mtd><mi>d</mi><mi>c</mi></mtd></mtr> <mtr><mtd><mi>d</mi><msup><mi>r</mi> <mi>a</mi></msup><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>t</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>r</mi> <mi>a</mi></msup></mtd></mtr> <mtr><mtd><mi>d</mi><mi>h</mi><mo>=</mo></mtd> <mtd><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo>−</mo><mi>g</mi></mtd></mtr> <mtr><mtd><mi>d</mi><mi>g</mi><mo>=</mo></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ F_\omega =& d \omega + \frac{1}{2}[\omega \wedge \omega] \\ H_3 =& \nabla B := d B + CS(\omega) - C_3 \\ \mathcal{G}_4 =& d C_3 \\ d F_\omega =& - [\omega \wedge F_\omega] \\ d H_3 =& \langle F_\omega \wedge F_\omega\rangle - \mathcal{G}_4 \\ d \mathcal{G}_4 =& 0 } \right)_i \;\;\;\; \stackrel{ \array{ t^a & \mapsto \omega^a \\ r^a & \mapsto F^a_\omega \\ b & \mapsto B \\ c & \mapsto C_3 \\ h & \mapsto H_3 \\ g & \mapsto \mathcal{G}_4 } }{\leftarrow}| \;\;\;\; \left( \array{ r^a =& d t^a + \frac{1}{2}C^a{}_{b c} t^b \wedge t^c \\ h = & d b + cs - c \\ g =& d c \\ d r^a =& - C^a{}_{b c} t^b \wedge r^a \\ d h =& \langle -,-\rangle - g \\ d g =& 0 } \right) \,. </annotation></semantics></math></div></div> <p>Here we are indicating on the right the generators and their relation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)</annotation></semantics></math> and on the left their images and the images of the relations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U_i)</annotation></semantics></math>. This are first the definitions of the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a>s themselves and then the <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> satisfied by these.</p> <h3 id="differential_string_structures_and_fermionic_string_quantum_anomalies">Differential string structures and fermionic string quantum anomalies</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Pfaffian+line+bundle">Pfaffian line bundle</a> controlling the <a class="existingWikiWord" href="/nlab/show/fermionic+path+integral">fermionic path integral</a> of the <a class="existingWikiWord" href="/nlab/show/heterotic+superstring">heterotic superstring</a> propagating on target <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> trivializes precisely if the target has a (geometric) string structure.</p> <p>One shows that the <a class="existingWikiWord" href="/nlab/show/Pfaffian+line+bundle">Pfaffian line bundle</a> on the worldsheet is <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> as a bundle with connection with the <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of the differential string structure on the target space to the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math>. So the target space having a (differential) string structure is a sufficient condition for the cancellation of the <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a>.</p> <p>(First argued in <a href="#Killingback">Killingback</a>, later made precise in (<a href="#Bunke">Bunke</a>)).</p> <h3 id="InheteroticSugra">The Green-Schwarz mechanism in heterotic supergravity</h3> <p>We discuss the application of twisted differential string structures in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> and <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>.</p> <p>Local differential form data as in note <a class="maruku-ref" href="#UnwindingTheLocalData"></a> above is known in <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">theoretical physics</a> in the context of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> for 10-dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>.</p> <p>In this context</p> <ul> <li> <p><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> is called the <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>;</p> </li> <li> <p>the components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>3</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((H_3)_i, \cdots)</annotation></semantics></math> of the above cocycle are known as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>𝒢</mi><mo stretchy="false">^</mo></mover> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\hat \mathcal{G}_4</annotation></semantics></math>-twisted <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>.</p> </li> </ul> <p>In this application the twisting cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>𝒢</mi><mo stretchy="false">^</mo></mover> <mn>4</mn></msub><mo>∈</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat \mathcal{G}_4 \in H^4_{diff}(X)</annotation></semantics></math> is itself the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> of a <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> with local connection form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>𝔲</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \Omega^1(U, \mathfrak{u})</annotation></semantics></math>. Therefore in this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>3</mn></msub><mo>=</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_3 = CS(A)</annotation></semantics></math> and the above local form data becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>3</mn></msub><mo>=</mo><mi>d</mi><mi>B</mi><mo>+</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>−</mo><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_3 = d B + CS(\omega) - CS(A) </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>d</mi><msub><mi>H</mi> <mn>3</mn></msub><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d H_3 = \langle F_\omega \wedge F_\omega \rangle - \langle F_A \wedge F_A \rangle \,. </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">H_3</annotation></semantics></math> is the would-be <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a>, this is the first higher <a class="existingWikiWord" href="/nlab/show/Maxwell+equation">Maxwell equation</a> that exhibits</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>mag</mi></msub><mo>:</mo><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> j_{mag} := \langle F_\omega \wedge F_\omega \rangle - \langle F_A \wedge F_A \rangle </annotation></semantics></math></div> <p>as the <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> distribution that twists this 2-bundle. This may be interpreted as the magnetic charge density of a classical background density of magnetic <a class="existingWikiWord" href="/nlab/show/fundamental+brane">fivebranes</a>. For more details on this see <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>.</p> <p>More precisely, the twisted differential string structure of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> in heterotic <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> for fixed gauge bundles are therefore given by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>GSBackground</mi> <mrow><mi>fixed</mi><mi>gauge</mi><mi>bundle</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>π</mi> <mn>0</mn></msub><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ GSBackground_{fixed gauge bundle}(X) &\to& \pi_0 \mathbf{H}_{conn}(X, \mathbf{B}U) \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{c}_2}} \\ \mathbf{H}_{conn}(X,\mathbf{B}Spin) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^3 U(1)) } \,. </annotation></semantics></math></div> <p>Clearly, if we take into account also gauge transformations of the gauge bundle, we should replace this by the full</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>GSBackground</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ GSBackground(X) &\to& \mathbf{H}_{conn}(X, \mathbf{B}U) \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{c}_2}} \\ \mathbf{H}_{conn}(X,\mathbf{B}Spin) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^3 U(1)) } \,. </annotation></semantics></math></div> <p>The look of this diagram makes manifest how in this situation we are looking at the structures that homotopically cancel the differential classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}_2</annotation></semantics></math> against each other.</p> <p>More discussion of this is in (<a href="#SSSIII">SSSIII</a>).</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))</annotation></semantics></math> is abelian, we may consider the corresponding <a href="http://ncatlab.org/nlab/show/fiber%20sequence#MayerVietoris">Mayer-Vietoris</a> sequence by realizing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GSBackground</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GSBackground(X)</annotation></semantics></math> equivalently as the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the difference of differential cocycles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>−</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1 - \hat \mathbf{c}_2</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>GSBackground</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>−</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>4</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ GSBackground(X) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}Spin \times \mathbf{B}U) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1-\hat \mathbf{c}_2}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^4 U(1)) } \,. </annotation></semantics></math></div> <p>Indeed, the above explicit presentation by simplicial presheaves generalizes immediately to describe this case, realizing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-twisted differential string structures equivalently as differential “untwisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-twisted-string-structures”.</p> <p>We may usefully formalize this further by defining the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">String^{\mathbf{c}_2}</annotation></semantics></math>-2-group to be the homotopy fiber</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>String</mi> <mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} String^{\mathbf{c}_2} &\to& * \\ \downarrow && \downarrow \\ \mathbf{G}(Spin \times U) &\stackrel{\frac{1}{2}\mathbf{p}_1 - \mathbf{c}_2}{\to}& \mathbf{B}^3 U(1) } \,. </annotation></semantics></math></div> <p>We have then that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GSBackground</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GSBackground(X)</annotation></semantics></math> is the 3-groupoid of <em>untwisted</em> differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>String</mi> <mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}String^{\mathbf{c}_2}</annotation></semantics></math>-structures.</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>GSBackground</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mn>0</mn></msup></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>4</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ GSBackground(X) &\to& * \\ \downarrow && \downarrow^{0} \\ \mathbf{H}_{conn}(\mathbf{B} (Spin \times U)) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \mathbf{c}_2}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^4 U(1)) } \,. </annotation></semantics></math></div> <p>More on this in (<a href="#FiSaSc">FiSaSc</a>).</p> <p>This is supposed to be (see section 12 of (<a href="#DFM">DFM</a>)) the restriction to the boundary of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>, which is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>H</mi> <mi>dR</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>d</mi><msub><mi>C</mi> <mn>3</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>4</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C Field(Y) &\to& H^4_{dR}(Y) \\ \downarrow && \downarrow^{\mathrlap{d C_3}} \\ \mathbf{H}(Y,\mathbf{B} (Spin \times U)) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \mathbf{c}_2}{\to}& \mathbf{H}(Y, \mathbf{B}^4 U(1)) } \,. </annotation></semantics></math></div> <p>where <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 11-dimensional with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">parital</mo><mi>Y</mi><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\parital Y = X</annotation></semantics></math>. Notice that here in the bottom left we have bundles <em>without</em> connection, or equivalently (when computing the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> by an ordinary pullback along a fibration) with <a class="existingWikiWord" href="/nlab/show/pseudo-connection">pseudo-connection</a>s.</p> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> under a shift of connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>1</mn></msub><mo>↦</mo><msub><mo>∇</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_1 \mapsto \nabla_2</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field transforms as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>+</mo><mi>CS</mi><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo>,</mo><msub><mo>∇</mo> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C_2 = C_1 + CS(\nabla_1, \nabla_2) \,, </annotation></semantics></math></div> <p>where on the right we have the relative <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a>. This vanishes precisely on the genuine gauge transformations. Hence as we restrict from 11-dimensions to 10, two things happen:</p> <ol> <li> <p>the supergravity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field vanishes,</p> </li> <li> <p>the gauge bundles develop genuine connections.</p> </li> </ol> <h3 id="relation_to_string_2connections">Relation to string 2-connections</h3> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></em> we have that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G := \mathbf{cosk}_{n+1} \exp(\mathfrak{g}) </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth Lie n-group</a> obtained from it by <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, the coefficient for <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connections</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G_{conn} := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})_{conn} \,, </annotation></semantics></math></div> <p>where on the very right we have the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>{</mo><mi>A</mi><mo>:</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi>is</mi><mspace width="thickmathspace"></mspace><mi>vertically</mi><mspace width="thickmathspace"></mspace><mi>flat</mi><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><msub><mi>F</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mi>is</mi><mspace width="thickmathspace"></mspace><mi>horizontal</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp(\mathfrak{g}) : (U,[k]) \mapsto \left\{ A : CE(\mathfrak{g}) \to \Omega^\bullet(U \times \Delta^n) | A\;is\;vertically\;flat\; and\;F_A\;is \; horizontal \right\} \,. </annotation></semantics></math></div> <p>(See <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+homomorphism">∞-Chern-Weil homomorphism</a> for details).</p> <div class="num_prop" id="StringConnectionsFromDiffStringStructzres"> <h6 id="proposition_6">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of entirely untwisted differential string structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (the twist being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0 \in H^4_{diff}(X)</annotation></semantics></math>) is equivalent to that of <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>s with <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">2-connection</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi><mo>=</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>String</mi><mn>2</mn><msub><mi>Bund</mi> <mo>∇</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> String_{diff, tw = 0}(X) \simeq String 2Bund_{\nabla}(X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By the above discussion of <a href="#ExplicitCocycles">Cech cocycles</a> we compute <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi><mo>=</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{diff, tw = 0}(X)</annotation></semantics></math> as the ordinary fiber of the morphism of simplicial presheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [CartSp^{op}, sSet]( C(\{U_i\}), \mathbf{cosk}_3 \exp(b \mathbb{R} \to \mathfrak{g}_\mu)) \to [CartSp^{op}, sSet]( C(\{U_i\}), \mathbf{B}^3 U(1)_{diff}) </annotation></semantics></math></div> <p>over the identically vanishing cocycle.</p> <p>In terms of the component formulas spelled out in the <a href="#InheteroticSugra">above discussion</a> of the GS-mechanism, this amounts to restricting to those cocycles for which in each degree the equations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> C = 0 </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>G</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> G = 0 </annotation></semantics></math></div> <p>holds.</p> <p>Comparing this to the explicit formulas for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(b \mathbb{R} \to \mathfrak{g}_\mu)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{conn}</annotation></semantics></math> in the <a href="#PresentationOfClassByFibration">above</a> we see that these cocycles are exactly those that factor through the canonical inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{g}_\mu \to (b \mathbb{R} \to \mathfrak{g}_\mu) </annotation></semantics></math></div> <p>from observation <a class="maruku-ref" href="#LongFiberSequenceOnLieAlgebras"></a> of the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> into the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> Lie 3-algebra of the extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">b \mathbb{R} \to \mathfrak{g}_\mu \to \mathfrak{g}</annotation></semantics></math> that defines it.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+differential+cohomology">nonabelian differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/differential+spin+structure">differential spin structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <strong>differential string structure</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chiral+Dolbeault+complex">chiral Dolbeault complex</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/Hypothesis+H">Hypothesis H</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted differential c-structure</a></p> <ul> <li> <p><strong>differential string structure</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+T-duality">differential T-duality</a></p> </li> </ul> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>A discussion of differential string structures in terms of <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a>s is given in</p> <ul> <li id="Waldorf"><a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>String Connections and Chern-Simons Theory</em>, Trans. Amer. Math. Soc. <strong>365</strong> (2013), 4393-4432, (<a href="http://arxiv.org/abs/0906.0117">arXiv:0906.0117</a>, <a href="https://doi.org/10.1090/S0002-9947-2013-05816-3">doi:10.1090/S0002-9947-2013-05816-3</a>)</li> </ul> <p>The description of the <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> is in section 3 of</p> <ul> <li id="DFM">E. Diaconescu, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-theory 3-form and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-gauge theory</em> [<a href="http://arxiv.org/abs/hep-th/0312069">arXiv:hep-th/0312069</a>]</li> </ul> <p>The local data for the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+valued+differential+forms">∞-Lie algebra valued differential forms</a> for the description of twisted differential string structures as above was given in</p> <ul> <li id="SSSIII"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>: <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em>, Comm. Math. Phys. <strong>315</strong> 1 (2012) 169-213 [<a href="https://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>, <a href="https://doi.org/10.1007/s00220-012-1510-3">doi:10.1007/s00220-012-1510-3</a>]</li> </ul> <p>The full Čech-Deligne cocycles induced by this (but not yet the homotopy fibers over them) were discussed in</p> <ul> <li id="FiorenzaSatiSchreiber12"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/%C4%8Cech+Cocycles+for+Differential+Characteristic+Classes">Čech Cocycles for Differential Characteristic Classes</a></em>, Advances in Theoretical and Mathematical Physics, <strong>16</strong> 1 (2012) 149-250 [<a href="https://arxiv.org/abs/1011.4735">arXiv:1011.4735</a>, <a href="https://doi.org/10.1007/BF02104916">doi:10.1007/BF02104916</a>]</li> </ul> <p>A comprehensive discussion including all the formal background and the applications is attempted at</p> <ul> <li id="dcct"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, section 4.2 of: <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> <p>The translation between the <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a>-picture of <a href="#Waldorf">Waldorf</a> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>-piucture of <a href="#SSSIII">Sati et al.</a>, <a href="#FiorenzaSatiSchreiber12">Fiorenza et al.</a> and <a href="#dcct">dcct</a> is worked out in some detail in:</p> <ul> <li>Alessandra Capotosti, <em><a class="existingWikiWord" href="/nlab/show/From+String+structures+to+Spin+structures+on+loop+spaces">From String structures to Spin structures on loop spaces</a></em>, Ph.D. thesis, Università degli Studi Roma Tre, Rome (April 2016) [<a href="http://www.matfis.uniroma3.it/Allegati/Dottorato/TESI/capotosti/PhD%20Thesis%202016%20A%20Capotosti.pdf">thesis pdf</a>, <a class="existingWikiWord" href="/nlab/files/Capotosti-FromStringStructures.pdf" title="pdf">pdf</a>, talk slides: <a class="existingWikiWord" href="/nlab/files/Capotosti-FromStringStruc-Slides.pdf" title="pdf">pdf</a>]</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> cancellation in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> has been first discussed in</p> <ul> <li id="Killingback">Killingback, <em>World-sheet anomalies and loop geometry</em> Nuclear Physics B Volume 288, 1987, Pages 578-588</li> </ul> <p>and given a rigorous treatment in</p> <ul> <li id="Bunke"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <em>String structures and trivialisations of a Pfaffian line bundle</em> (<a href="http://arxiv.org/abs/0909.0846">arXiv</a>)</li> </ul> <p>More discussion on the relation to spin structures on smooth loop space is in</p> <ul> <li>Alessandra Capotosti, <em><a class="existingWikiWord" href="/nlab/show/From+String+structures+to+Spin+structures+on+loop+spaces">From String structures to Spin structures on loop spaces</a></em>, Ph.D. thesis, Università degli Studi Roma Tre, Rome, April 2016</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 17, 2024 at 16:51:13. 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