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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#ConsistencyConditions'>Consistency conditions</a></li> <ul> <li><a href='#ShiftedCFieldFluxQuantizationCondition'>Shifted flux quantization condition</a></li> <li><a href='#CFieldTadpoleCancellationCondition'>C-Field tadpole cancellation condition</a></li> </ul> <li><a href='#Models'>Models</a></li> <ul> <li><a href='#DFMmodel'>The DFM-model</a></li> <ul> <li><a href='#construction_via__gauge_fields'>Construction via <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 fields</a></li> <ul> <li><a href='#claim'>Claim</a></li> </ul> <li><a href='#DFMOrientationAndFractionalClasses'>Orientation and fractional classes</a></li> <li><a href='#restriction_to_the_boundary'>Restriction to the boundary</a></li> </ul> <li><a href='#description_in_chernweil_theory'>Description in <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</a></li> <ul> <li><a href='#abstract_definition'>Abstract definition</a></li> <li><a href='#CWPerspectiveGeneralProperties'>General properties</a></li> <li><a href='#CWPresentationByDifferentialFormData'>Presentation by differential form data</a></li> <li><a href='#RestrictionToBoundaryInChernWeil'>Restriction to the boundary</a></li> </ul> <li><a href='#ModelByTwistedCohomotopy'>Model by twisted Cohomotopy</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#dualitysymmetric_formulation'>Duality-symmetric formulation</a></li> <ul> <li><a href='#ReferencesCFieldGaugeAlgebra'>Supergravity C-Field gauge algebra</a></li> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The field content of 11-dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> contains a <a class="existingWikiWord" href="/nlab/show/higher+U%281%29-gauge+field">higher U(1)-gauge field</a> called the <em>supergravity C-field</em> or <em>M-theory 3-form</em> , which is locally a 3-form and globally <em>some</em> variant of a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a>.</p> <p>There have been several suggestions for what <em>precisely</em> its correct global description must be, see at <em><a href="#Models">Models</a></em> below</p> <h2 id="ConsistencyConditions">Consistency conditions</h2> <p>Several subtle consistency conditions (<a class="existingWikiWord" href="/nlab/show/quantum+anomaly+cancellation">quantum anomaly cancellation</a>-conditions) have been argued for the <a class="existingWikiWord" href="/nlab/show/charge+quantization">charge quantization</a> of the supergravity C-field:</p> <ol> <li> <p><a href="#ShiftedCFieldFluxQuantizationCondition">Shifted flux quantization condition</a></p> </li> <li> <p><a href="#CFieldTadpoleCancellationCondition">C-Field tadpole cancellation condition</a></p> </li> </ol> <h3 id="ShiftedCFieldFluxQuantizationCondition">Shifted flux quantization condition</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization+condition">shifted C-field flux quantization condition</a></em> is a <a class="existingWikiWord" href="/nlab/show/charge+quantization">charge quantization</a>-condition on the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> expected in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>. It says that the <a class="existingWikiWord" href="/nlab/show/real+cohomology">real cohomology</a> class of the <a class="existingWikiWord" href="/nlab/show/flux+density">flux density</a> (<a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a>) <a class="existingWikiWord" href="/nlab/show/differential+4-form">differential 4-form</a> <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">G_4 \in \Omega^4(X)</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> becomes integral after shifted by one quarter of the <a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a>, hence the condition that with the shifted 4-flux density defined as</p> <div class="maruku-equation" id="eq:Shiftwed"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>G</mi><mo>˜</mo></mover> <mn>4</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>G</mi> <mn>4</mn></msub><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mo>∇</mo> <mrow><mi>T</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><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"> \widetilde G_4 \;\coloneqq\; G_4 + \tfrac{1}{4}p_1(\nabla_{T X}) \;\in\; \Omega^4(X) </annotation></semantics></math></div> <p>(for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>T</mi><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\nabla_{T X}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, in particular the <a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a>) we have (using the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> to translate from <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> to <a class="existingWikiWord" href="/nlab/show/real+cohomology">real cohomology</a>) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>G</mi><mo>˜</mo></mover> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\widetilde G_4</annotation></semantics></math> represents an <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a>-class:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mover><mi>G</mi><mo>˜</mo></mover> <mn>4</mn></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo>↪</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow></mover><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\widetilde G_4] \;\in\; H^4(X, \mathbb{Z}) \overset{H^4(X, \mathbb{Z}\hookrightarrow \mathbb{R})}{\longrightarrow} H^4(X, \mathbb{R}) \,. </annotation></semantics></math></div> <p>This condition was originally argued for in (<a href="#Witten96a">Witten 96a</a>, <a href="#Witten96b">Witten 96b</a>) as a sufficient condition for ensuring that the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> for the <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> on an <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> is divisible by 2.</p> <p>Proposals for encoding this condition by a <a class="existingWikiWord" href="/nlab/show/Wu+class">Wu class</a>-shifted variant of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology+theory">stable</a> <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> were considered in <a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Hopkins-Singer 02</a>, <a href="#DiaconescuFreedMoore03">Diaconescu-Freed-Moore 03</a>, <a href="#FSS12">FSS 12</a>.</p> <p>It turns out that the shifted flux quantization condition on the C-field is naturally implied (<a href="#FSS19b">FSS1 19b, Prop. 4.12</a>) by the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math> is the differential form datum underlying, via <a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan's theorem</a>, a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in unstable <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-</a> <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> in degree 4 (<a class="existingWikiWord" href="/nlab/show/Hypothesis+H">Hypothesis H</a>).</p> <h3 id="CFieldTadpoleCancellationCondition">C-Field tadpole cancellation condition</h3> <p>In <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">compactified</a> on 8-dimensional <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{(8)}</annotation></semantics></math> (see <em><a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on 8-manifolds</a></em>) <a class="existingWikiWord" href="/nlab/show/tadpole+cancellation+condition+for+the+supergravity+C-field">tadpole cancellation condition for the supergravity C-field</a> has been argued (<a href="shifted+C-field+flux+quantization#SethiVafaWitten96">Sethi-Vafa-Witten 96</a>, <a href="shifted+C-field+flux+quantization#BeckerBecker96">Becker-Becker 96</a>, <a href="shifted+C-field+flux+quantization#DasguptaMukhi97">Dasgupta-Mukhi 97</a>) to be the condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mrow><mi>M</mi><mn>2</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>48</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>−</mo><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>8</mn></msup><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mrow><msub><mi>I</mi> <mn>8</mn></msub><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>8</mn></msup><mo stretchy="false">)</mo></mrow></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>∈</mo><mo>!</mo></mover><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \underset{ I_8(X^8) }{ \underbrace{ \tfrac{1}{48}\big( p_2 - (\tfrac{1}{2}p_1)^2 \big)[X^{8}] } } \;\;\;\; \overset{!}{\in} \mathbb{Z} \,, </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mrow><mi>M</mi><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">N_{M2}</annotation></semantics></math> is the net number of <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> in the spacetime (whose <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> appears as points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{(8)}</annotation></semantics></math>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a>/flux of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/second+Pontryagin+class">second Pontryagin class</a> combining to <a class="existingWikiWord" href="/nlab/show/I8">I8</a>, all regarded here in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>.</p> </li> </ol> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mn>8</mn></msup></mrow><annotation encoding="application/x-tex">X^{8}</annotation></semantics></math> has</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Spin%287%29-structure">Spin(7)-structure</a> (hence in particular if it is a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a>, which has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">SU(4) = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a>-structure)</li> </ul> <p>or</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sp%282%29.Sp%281%29-structure">Sp(2).Sp(1)-structure</a></li> </ul> <p>then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>χ</mi></mrow><annotation encoding="application/x-tex"> \tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Euler+class">Euler class</a> (see <a href="Spin7-manifold#CharacteristicClassesForSpinStructure">this Prop.</a> and <a href="quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure">this Prop.</a>, respectively), hence in these cases the condition is equivalently</p> <div class="maruku-equation" id="eq:DasguptaMukhiConditionInTermsOfEuler"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mrow><mi>M</mi><mn>2</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>24</mn></mfrac></mstyle><mi>χ</mi><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>8</mn></msup><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> N_{M2} \;=\; - \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;+\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\chi[X]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</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> <h2 id="Models">Models</h2> <p>One proposal for a mathematical model of the C-field is as a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in a <a class="existingWikiWord" href="/nlab/show/Wu+class">Wu class</a>-shifted variant of <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> in degree 4:</p> <ul> <li><a href="#DFMmodel">The DFM model</a></li> </ul> <p>Another proposal is that the C-field is simply a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-</a> <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> (<a class="existingWikiWord" href="/nlab/show/Hypothesis+H">Hypothesis H</a>):</p> <ul> <li><a href="#ModelByTwistedCohomotopy">C-field charge quantization in J-twisted Cohomotopy</a></li> </ul> <h3 id="DFMmodel">The DFM-model</h3> <h4 id="construction_via__gauge_fields">Construction via <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 fields</h4> <p>In (<a href="#DiaconescuFreedMoore03">DFM, section 3</a>) the following definition is considered and argued to be a good model of 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.</p> <div class="num_note" id="NatureOfE8"> <h6 id="note">Note</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">B E_8</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <a class="existingWikiWord" href="/nlab/show/E8">E8</a> satisfy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>i</mi></msub><mi>B</mi><msub><mi>E</mi> <mn>8</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">|</mo><mi>i</mi><mo>=</mo><mn>4</mn></mtd></mtr> <mtr><mtd><mn>0</mn><mo stretchy="false">|</mo><mi>i</mi><mo>≠</mo><mn>4</mn><mo>,</mo><mi>i</mi><mo>≤</mo><mn>15</mn></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_i B E_8 = \left\{ \array{ \mathbb{Z} | i = 4 \\ 0 | i \neq 4, i \leq 15 } \right. \,. </annotation></semantics></math></div> <p>Therefore for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi><mo>≤</mo><mn>15</mn></mrow><annotation encoding="application/x-tex">dim X \leq 15</annotation></semantics></math> there is a canonical <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^1(X, E_8) \simeq H^4(X, \mathbb{Z}) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="DFMDefinitionOfCField"> <h6 id="definition">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi><mo><</mo><mn>15</mn></mrow><annotation encoding="application/x-tex">dim X \lt 15</annotation></semantics></math>. For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><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">a \in H^4(X, \mathbb{Z})</annotation></semantics></math>. choose an <a class="existingWikiWord" href="/nlab/show/E8">E8</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><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> which represents <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> under the above isomorphism.</p> <p>Write then</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>E</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{E}(X) \in Grpd </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/object">object</a>s are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mo>∇</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P,\nabla,c)</annotation></semantics></math> where</p> <ul> <li> <p><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> is one of the chosen <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>-bundles,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on <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>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><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">c \in \Omega^3(X)</annotation></semantics></math> is a degree-3 <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>:</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mo>∇</mo> <mn>2</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega : (P, \nabla_1, c_1) \to (P, \nabla_2, c_2)</annotation></semantics></math></p> <p>are parameterized by their source and target triples together with a closed 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>ℤ</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^3_{\mathbb{Z}}(X)</annotation></semantics></math> with integral <a class="existingWikiWord" href="/nlab/show/period">period</a>s, subject to the condition that</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><mo>+</mo><mi>ω</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> c_2 -c_1 = CS(\nabla_1,\nabla_2) + \omega \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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></mrow><annotation encoding="application/x-tex">CS(\nabla_1, \nabla_2)</annotation></semantics></math> is the relative <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> corresponding to the linear path of connections from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_2</annotation></semantics></math></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow></mover><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mo>∇</mo> <mn>2</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow></mover><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><msub><mo>∇</mo> <mn>3</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\omega_2 \circ \omega_1 ) : (P,\nabla_1, C_1) \stackrel{\omega_1}{\to} (P, \nabla_2, C_2) \stackrel{\omega_2}{\to} (P, \nabla_3, C_3) </annotation></semantics></math></div> <p>is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>2</mn></msub><mo>−</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>3</mn></msub><mo>−</mo><mo>∇</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega_1 + \omega_2 + \langle (\nabla_2-\nabla_1)\wedge(\nabla_3-\nabla2) \rangle \,. </annotation></semantics></math></div></li> </ul> </div> <p>See (<a href="#DiaconescuFreedMoore03">DFM, (3.22), (3.23)</a>).</p> <p>Here we think 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> as equipped with a <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian structure</a> and <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> and think of each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mo>∇</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P,\nabla,C)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{E}(X)</annotation></semantics></math> as inducing an degree-4 cocycle in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> with <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 4-form</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> <mrow><mo>∇</mo><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mi>tr</mi><msub><mi>F</mi> <mo>∇</mo></msub><mo>∧</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>tr</mi><msub><mi>R</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>R</mi> <mi>ω</mi></msub><mo>+</mo><mi>d</mi><mi>c</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G}_{\nabla,c} = tr F_\nabla \wedge F_\nabla - \frac{1}{2} tr R_\omega \wedge R_\omega + d c \,. </annotation></semantics></math></div> <p>Notice that with the normalization implicit here the second terms is one half of the image of something in integral cohomology. So this is not itself a differential character, but can be regarded as “shifted differential character”: a trivialization of the trivial 5-character with global connection 4-form given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>tr</mi><msub><mi>R</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>R</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2} tr R_\omega \wedge R_\omega</annotation></semantics></math>. See <a href="#DFMOrientationAndFractionalClasses">below</a> for more on this.</p> <div class="num_prop" id="DFModelHomotopyGroups"> <h6 id="claim">Claim</h6> <p>The above groupoid has <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>≃</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 \simeq H^4_{diff}(Y)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">(</mo><mo>∇</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(-,(\nabla,c)) \simeq H^2(Y, U(1))</annotation></semantics></math> .</p> </li> </ul> </div> <p>The first, the set of connected components (gauge equivalence classes 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>-fields) is isomorphic to the set of <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> in degree 4 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>. In fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> is naturally a <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over this abelian group: the torsor 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><mi>tr</mi><msup><mi>R</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\frac{1}{2}tr R^2</annotation></semantics></math>-shifted differential characters.</p> <p>The second, the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, is that of flat <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>s.</p> <h4 id="DFMOrientationAndFractionalClasses">Orientation and fractional classes</h4> <p>Ordinarily, given a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Spin \times E_8</annotation></semantics></math>-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">P \to Y</annotation></semantics></math> with 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><mi>λ</mi><mo>:</mo><mo>=</mo><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"> \lambda := \frac{1}{2}p_1(P) </annotation></semantics></math></div> <p>and second <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mo>=</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a := c_2(P) </annotation></semantics></math></div> <p>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 is supposed to have a <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> class in <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>dR</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>λ</mi> <mi>dR</mi></msub><mo>∈</mo><msubsup><mi>H</mi> <mi>dR</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a_{dR} + \frac{1}{2} \lambda_{dR} \in H_{dR}^4(Y) \,. </annotation></semantics></math></div> <p>Since in general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>=</mo><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">\lambda = \frac{1}{2}p_1(P)</annotation></semantics></math> is not further divisible in integral cohomology, this means that this cannot be the curvature of any <a class="existingWikiWord" href="/nlab/show/differential+character">differential character</a>/<a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a>/<a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a>, since these are necessarily the images in de Rham cohomology of their integral classes.</p> <p>See (<a href="#DiaconescuFreedMoore03">DFM 03, section 12.1</a>, <a href="#FSS12">FSS 12</a>)</p> <h4 id="restriction_to_the_boundary">Restriction to the boundary</h4> <p>By <a href="#DiaconescuFreedMoore03">DFM, section 12</a> on a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> with boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>∂</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X = \partial Y</annotation></semantics></math> we are to impose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><msub><mo stretchy="false">|</mo> <mrow><mo>∂</mo><mi>Y</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">C|_{\partial Y} = 0</annotation></semantics></math>.</p> <p>See the discussion <a href="#RestrictionToBoundaryInChernWeil">below</a> for how this reproduces the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> for heterotic <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> on the boundary.</p> <h3 id="description_in_chernweil_theory">Description in <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</h3> <p>Some remarks on ways to regard 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 from the point of view of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a>.</p> <h4 id="abstract_definition">Abstract definition</h4> <p>We shall consider the sum of two <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> fields, whose curvature is the image in de Rham cohomology of the proper integral class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>a</mi><mo>−</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">2 a - \lambda </annotation></semantics></math></p> <p>Recall from the discussion at <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a> that in the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>:</mo><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> the circle 3-bundles with local 3-form connection over an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">Y \in \mathbf{H}</annotation></semantics></math> (for instance a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, or an <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>) are <a class="existingWikiWord" href="/nlab/show/object">object</a>s in the <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-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>Y</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}(Y, \mathbf{B}^3 U(1))</annotation></semantics></math> that is 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><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>Y</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> <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><mo stretchy="false">↓</mo></mtd></mtr> <mtr><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>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1)) &\to& H^4_{dR}(Y) \\ \downarrow && \downarrow \\ \mathbf{H}(Y, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <p>(Recall from the discussion there that if desired one may pass to the canonical presentation of this by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> over <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and that in this explicit presentation we may replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4_{dR}(Y)</annotation></semantics></math> with the more familiar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^4_{cl}(Y)</annotation></semantics></math>. )</p> <p>We consider now the analog of this definition for the universal curvature form on <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> replaced by the difference of the differentially refined second <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> of <a class="existingWikiWord" href="/nlab/show/E8">E8</a> and the first fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a> of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>. The resulting <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 we tentatively call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C Field(Y)</annotation></semantics></math>, though we shall have to discuss to which extend this faithfully models 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, and which aspects of it.</p> <div class="num_defn" id="AbstractDefOfCField"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">Y \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">C Field(Y) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> be 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><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><mo stretchy="false">↓</mo></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><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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 \\ \mathbf{H}(Y, \mathbf{B} (Spin \times E_8)) &\stackrel{(2\mathbf{c}_2)_{dR}- (\frac{1}{2}\mathbf{p}_1)_{dR}}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,. </annotation></semantics></math></div></div> <div class="num_note" id="PastingDecompOfAbstractDefOfCField"> <h6 id="note_2">Note</h6> <p>By its <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#ChernWeilTheory">intrinsic definition</a> we have that the differential characteristic class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">(\mathbf{c}_2)_{dR}</annotation></semantics></math> is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mn>8</mn></msub><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub></mrow></mover><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><mover><mo>→</mo><mi>curv</mi></mover><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex"> (\mathbf{c}_2)_{dR} : \mathbf{B}E_8 \stackrel{\mathbf{c}_2}{\to} \mathbf{B}^3 U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^4 U(1) </annotation></semantics></math></div> <p>of the smooth refinement of the second <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> with the <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#CurvatureCharacteristics">universal curvature form</a> on <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>. Similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">(\frac{1}{2}\mathbf{p}_2)_{dR}</annotation></semantics></math>.</p> </div> <p>Therefore we may either compute the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> in def. <a class="maruku-ref" href="#AbstractDefOfCField"></a> directly, or in two consecutive steps. Both methods lead to their insights.</p> <p>In</p> <ul> <li><a href="#CWPerspectiveGeneralProperties">General properties</a></li> </ul> <p>we consider general abstract consequences of the above definition, mainly making use of the factorization. In</p> <ul> <li><a href="#CWPresentationByDifferentialFormData">Presentation by differential form data</a></li> </ul> <p>we find a presentation by <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> of the direct <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>.</p> <p>In the first approach <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> on the <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s never appear explicitly. In the second approach they appear as <a class="existingWikiWord" href="/nlab/show/pseudo-connection">pseudo-connection</a>s, or as genuine connections whose morphisms are however allowed to shift them arbitrarily. This means that these connections are purely auxiliary data that serve to present the required homotopies. They do not survive in cohomology. This is as in the <a href="#DFMmodel">DFM model</a> above.</p> <p>Finally in</p> <ul> <li><a href="#RestrictionToBoundaryInChernWeil">Restriction to the boundary</a></li> </ul> <p>we comment how genuine <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>-connections may appear inside the second presentation of 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>-model.</p> <h4 id="CWPerspectiveGeneralProperties">General properties</h4> <p>This implies by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>s that 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 from def. <a class="maruku-ref" href="#AbstractDefOfCField"></a> may be decomposed into two consecutive pullbacks of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mover><mi>χ</mi><mo stretchy="false">^</mo></mover></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> <mtd><mover><mo>→</mo><mi>ω</mi></mover></mtd> <mtd><msubsup><mi>H</mi> <mi>dR</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> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mn>2</mn><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><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></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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(X) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) &\stackrel{\omega}{\to}& H^4_{dR}(X) \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}(E_8 \times Spin(10,1))) &\stackrel{2\mathbf{c}_2- \frac{1}{2}\mathbf{p}_2}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,, </annotation></semantics></math></div> <p>where on the right we find the <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#DifferentialCohomology">defining pullback</a> for (the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a> of) <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <p>=–</p> <p>This implies the following structure and properties.</p> <div class="num_note" id="DifferentialCharacterOfCField"> <h6 id="notedefinition">Note/Definition</h6> <p>By the above there exists canonically a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>χ</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><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><mover><mo>→</mo><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub></mrow></mover><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 \chi : C Field(X) \to \mathbf{H}(X,\mathbf{B}^3 U(1)) \stackrel{\tau_{\leq 0}}{\to} H^4_{diff}(X) </annotation></semantics></math></div> <p>that maps <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 configurations to <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> in degree 4, whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>χ</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(\hat \chi)</annotation></semantics></math> is the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>dR</mi></msub><mo>:</mo><mo>=</mo><mi>curv</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>2</mn></msub><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{c}_2)_{dR}- (\frac{1}{2}\mathbf{p}_2)_{dR} := curv(2\mathbf{c}_2 - \frac{1}{2}\mathbf{p}_2)</annotation></semantics></math> in de Rham cohomology of the second Chern-class of some <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>-bundle.</p> <p>The differential cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>χ</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat \chi(C)</annotation></semantics></math> has all the general properties that make its <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a> over <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> worldvolumes be well-defined. (Apart from the coefficient of <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>, this is the only requirement from which <a href="#DiaconescuFreedMoore03">DFM</a> deduce their model.)</p> </div> <p>The following proposition describes the first two <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s of the <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C Field(Y)</annotation></semantics></math>.</p> <div class="num_prop" id="HomotopyGroupsOfCField"> <h6 id="proposition">Proposition</h6> <p>Over a fixed <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/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>Spin</mi></msub></mrow><annotation encoding="application/x-tex">P_{Spin}</annotation></semantics></math> we have a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> (of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed sets</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>C</mi><msub><mi>Field</mi> <mrow><msub><mi>P</mi> <mi>Spin</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>dR</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mrow><mn>2</mn><mi>ℤ</mi></mrow></msub><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> * \to H^3(Y, U(1)) \to \pi_0 C Field_{P_{Spin}}(Y) \to H^4_{dR}(Y)_{2 \mathbb{Z}} \to * </annotation></semantics></math></div> <p>and</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>C</mi><msub><mi>Field</mi> <mrow><msub><mi>P</mi> <mi>Spin</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 C Field_{P_{Spin}}(Y)</annotation></semantics></math> is the group of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">([c], f) \in H^2(X, U(1)) \times C^\infty(X, E_8)</annotation></semantics></math> 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> is a smooth refinement under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><msub><mo>≃</mo> <mn>14</mn></msub><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_8 \simeq_{14} B^2 U(1) \simeq K(\mathbb{Z},3)</annotation></semantics></math> of the integral image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[c]</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Notice that we have the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>s</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><msub><mi>Field</mi> <mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>χ</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><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> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mover><mi>χ</mi><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>Y</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> <mtd><mover><mo>→</mo><mi>ω</mi></mover></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><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></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><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mn>2</mn><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>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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_{\omega(\hat \chi(-)) = 0}(Y) &\to& \mathbf{H}(Y, \mathbf{\flat} \mathbf{B}^3 U(1)) &\to& {*} \\ \downarrow && \downarrow && \downarrow^{\mathrlap{0}} \\ C Field(Y) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1)) &\stackrel{\omega}{\to}& H^4_{dR}(Y) \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(Y, \mathbf{B}E_8) &\stackrel{2\mathbf{c}_2}{\to}& \mathbf{H}(Y, \mathbf{B}^3 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)) } \,, </annotation></semantics></math></div> <p>where the top right square is discussed at <a href="http://nlab.mathforge.org/nlab/show/cohesive+%28infinity%2C1%29-topos#DifferentialCohomology">cohesive (∞,1)-topos – Differential cohomology</a>. By the discussion at <a href="http://nlab.mathforge.org/nlab/show/smooth%20infinity-groupoid#StrucFlat">smooth ∞-groupoid – Flat cohomology</a> we have that <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>Y</mi><mo>,</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><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>3</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 \mathbf{H}(Y, \mathbf{\flat} \mathbf{B}^3 U(1)) \simeq H^3(Y,U(1))</annotation></semantics></math>, where on the right we have <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (for instance realized as <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>). Finally observe that <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>Y</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>E</mi></mstyle> <mn>8</mn></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</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">\pi_0 \mathbf{H}(Y, \mathbf{E}_8) \simeq \pi_0 \mathbf{H}(Y.\mathbf{B}^3 U(1))</annotation></semantics></math>, by the <a href="#NatureOfE8">above remark</a>. Therefore after passing to <a class="existingWikiWord" href="/nlab/show/0-connected">connected components</a> by applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(-)</annotation></semantics></math> we get on cohomology</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mn>0</mn></msub><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mover><mi>χ</mi><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>ω</mi></mover></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><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></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></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ H^3(Y, U(1)) & \stackrel{\cdot 2}{\to}& H^3(Y, U(1)) &\to& {*} \\ \downarrow && \downarrow && \downarrow^{\mathrlap{0}} \\ \pi_0 C Field(Y) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}^4(Y) &\stackrel{\omega}{\to}& H^4_{dR}(Y) \\ \downarrow && \downarrow && \downarrow \\ H^1(Y, E_8) & \stackrel{\cdot 2}{\to}& H^4(Y, \mathbb{Z}) &\stackrel{curv}{\to}& H^4_{dR}(Y) } </annotation></semantics></math></div> <p>by reasoning as discussed at <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>. In parallel to the familiar <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><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>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>dR</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>ℤ</mi></msub><mo>→</mo><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> * \to H^3(Y, U(1)) \to H^4_{diff}(Y) \to H^4_{dR}(Y)_{\mathbb{Z}} \to * \,. </annotation></semantics></math></div> <p>this therefore implies also the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>C</mi><mi>Field</mi><mo>→</mo><msubsup><mi>H</mi> <mi>dR</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mrow><mn>2</mn><mi>ℤ</mi></mrow></msub><mo>→</mo><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> * \to H^3(Y, U(1)) \to \pi_0 C Field \to H^4_{dR}(Y)_{2 \mathbb{Z}} \to * \,. </annotation></semantics></math></div> <p>Next we redo the entire discussion after applying the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>-construction to everything. Using that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Q</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Q</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega \mathbf{H}(Y, \mathbf{B}Q) \simeq \mathbf{H}(Y, \Omega \mathbf{B}Q) \simeq \mathbf{H}(Y, Q) </annotation></semantics></math></div> <p>on general grounds (see <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> for details) and that also</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega (\mathbf{\flat}\mathbf{B}^n U(1)) \simeq \mathbf{\flat}\mathbf{B}^{n-1} U(1) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega (\mathbf{\flat}_{dR}\mathbf{B}^n U(1)) \simeq \mathbf{\flat}_{dR}\mathbf{B}^{n-1} U(1) </annotation></semantics></math></div> <p>(since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{\flat}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/right+adjoint">right</a> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a>s – by the discussion at <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> – and hence commute with the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> that defines <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>), we have then the looped <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ω</mi><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>P</mi> <mi>Spin</mi></msub></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>Ω</mi><mover><mi>χ</mi><mo stretchy="false">^</mo></mover></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>flat</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>ω</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mn>2</mn><mi>Ω</mi><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>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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{ \Omega C Field(Y)_{P_{Spin}} &\stackrel{\Omega \hat \chi}{\to}& \mathbf{H}_{flat}(Y, \mathbf{B}^2 U(1)) &\stackrel{\omega}{\to}& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{H}(Y, E_8) &\stackrel{2\Omega \mathbf{c}_2}{\to}& \mathbf{H}(Y, \mathbf{B}^2 U(1)) &\stackrel{curv}{\to}& \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)) } \,. </annotation></semantics></math></div> <p>Observe that <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> here is a smooth but <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> object: so that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(Y, E_8) \simeq H^0(Y, E_8) = C^\infty(Y, E_8) </annotation></semantics></math></div> <p>is the set of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">Y \to E_8</annotation></semantics></math> (to be thought of as the the set of <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s from the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to itself).</p> </div> <h4 id="CWPresentationByDifferentialFormData">Presentation by differential form data</h4> <p>In order to compute 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C Field(X)</annotation></semantics></math> more explicitly, we follow the discussion at <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>, where presentations of this pullback in terms of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> arising from <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> is given.</p> <p>Write now</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>:</mo><mo>=</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo>×</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{g} := \mathfrak{e}_8 \times \mathfrak{e}_8 \times \mathfrak{so}(10,1) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>:</mo><mo>=</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G := E_8 \times E_8 \times Spin(10,1)</annotation></semantics></math> and write</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>=</mo><msub><mi>μ</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow></msub><mo>+</mo><msub><mi>μ</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow></msub><mo>−</mo><msub><mi>μ</mi> <mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \mu := \mu_{\mathfrak{e}_8} + \mu_{\mathfrak{e}_8} - \mu_{\mathfrak{so}(10,1)} </annotation></semantics></math></div> <p>for the sum of the canonical <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cocycles</a> in <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> with the respective <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a>s.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔢</mi> <mn>8</mn></msub><mo>×</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> \mathfrak{e}_8 \times \mathfrak{so}(10,1) \to \mathfrak{g} </annotation></semantics></math></div> <p>for the canonical <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> embedding Write</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>c</mi></mstyle><mo>:</mo><mo>=</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo>×</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>Δ</mi></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{c} := & \mathbf{cosk}_3( \exp(\mathfrak{e}_8 \times \mathfrak{so}(10,1)) ) &\stackrel{\Delta}{\to}& \mathbf{cosk}_3( \exp(\mathfrak{g}) ) & \stackrel{\exp(\mu)}{\to} & \mathbf{B}^3 U(1)_c \\ \downarrow^{\simeq} \\ \mathbf{B}G } </annotation></semantics></math></div> <p>for the corresponding smooth <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>. See <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+homomorphism">∞-Chern-Weil homomorphism</a> for details. By the discussion there we present <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> by</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"><mrow><msub><mi>cosk</mi> <mn>3</mn></msub></mrow></mstyle><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>diff</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>4</mn></msup><msub><mi>ℝ</mi> <mi>dR</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>E</mi> <mn>8</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{diff} &\to& \mathbf{B}^4 \mathbb{R}_{dR} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}E_8 } \,. </annotation></semantics></math></div> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a> we have that the top morphism is a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> in the global projective <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><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> (there it is shown that the analogous morphism out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><msub><mi>cosk</mi> <mn>3</mn></msub></mrow></mstyle><mi>exp</mi><mo stretchy="false">(</mo><mi>b</mi><mi>ℝ</mi><mo>→</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{e}_8)_{ChW}</annotation></semantics></math> is a fibration, but then so is this one, because the components on the left are the same but with fewer conditions on them, so that the lifts that existed before still exist here).</p> <p>Over some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and <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><mo>∈</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">[k] \in \Delta</annotation></semantics></math> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>ℝ</mi></msup><mo>→</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">\exp(b^\mathbb{R} \to \mathfrak{g}_\mu)_{diff}</annotation></semantics></math> is given by differential form data</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>A</mi></msub><mo>=</mo></mtd> <mtd><mi>d</mi><mi>A</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><mi>A</mi><mo>∧</mo><mi>A</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><msub><mi>C</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>A</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>H</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>H</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><mi>d</mi><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><mi>A</mi><mo>∧</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mi>d</mi><msub><mi>C</mi> <mn>3</mn></msub><mo>=</mo></mtd> <mtd><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><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>A</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>A</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><mo>+</mo></mtd></mtr> <mtr><mtd><mi>c</mi><mo>=</mo></mtd> <mtd><mi>d</mi><mi>b</mi><mo>+</mo><mi>cs</mi><mo>−</mo><mi>h</mi></mtd></mtr> <mtr><mtd><mi>g</mi><mo>=</mo></mtd> <mtd><mi>d</mi><mi>h</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>c</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></mrow><annotation encoding="application/x-tex"> \left( \array{ F_A =& d A + \frac{1}{2}[A \wedge A] \\ C_3 =& \nabla B := d B + CS(A) - H_3 \\ \mathcal{G}_4 =& d H_3 \\ d F_A =& - [A \wedge F_A] \\ d C_3 =& \langle F_A \wedge F_A\rangle - \mathcal{G}_4 \\ d \mathcal{G}_4 =& 0 } \right)_i \;\;\;\; \stackrel{ \array{ t^a & \mapsto A^a \\ r^a & \mapsto F^a_A \\ 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 + \\ c = & d b + cs - h \\ g =& d h \\ d r^a =& - C^a{}_{b c} t^b \wedge r^a \\ d c =& \langle -,-\rangle - g \\ d g =& 0 } \right) </annotation></semantics></math></div> <p>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>. Here, recall, <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> takes values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo>×</mo><mi>𝔢</mi><mo>×</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g} = \mathfrak{e}_8 \times \mathfrak{e} \times \mathfrak{so}(10,1)</annotation></semantics></math>, so that for instance the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_4</annotation></semantics></math>-curvature is in detail given by</p> <div class="maruku-equation" id="eq:TheFourCurvature"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>4</mn></msub><mo>=</mo><mi>d</mi><msub><mi>H</mi> <mn>3</mn></msub><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><msubsup><mi>A</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow> <mi>L</mi></msubsup></mrow></msub><mo>∧</mo><msub><mi>F</mi> <mrow><msubsup><mi>A</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow> <mi>L</mi></msubsup></mrow></msub><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><msubsup><mi>A</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow> <mi>R</mi></msubsup></mrow></msub><mo>∧</mo><msub><mi>F</mi> <mrow><msubsup><mi>A</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow> <mi>R</mi></msubsup></mrow></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo>∧</mo><msub><mi>F</mi> <mi>ω</mi></msub><mo stretchy="false">⟩</mo><mo>−</mo><mi>d</mi><msub><mi>C</mi> <mn>3</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G}_4 = d H_3 = \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle + \langle F_{A^R_{\mathfrak{e}_8}} \wedge F_{A^R_{\mathfrak{e}_8}} \rangle - \langle F_{\omega} \wedge F_{\omega} \rangle - d C_3 \,, </annotation></semantics></math></div> <p>where <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> denotes the <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> be a differentiably <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>. We hit all connected components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \mathbf{B}G)</annotation></semantics></math> by considering in</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><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo><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><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex"> [CartSp^{op}, sSet](C(U_i), \exp(b \mathbb{R} \to \mathfrak{g}_\mu))_{diff} </annotation></semantics></math></div> <p>those cocycles that</p> <ul> <li> <p>involve genuine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> (as opposed to the more general <a class="existingWikiWord" href="/nlab/show/pseudo-connection">pseudo-connection</a>s that are also contained);</p> </li> <li> <p>have a globally defined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">C_3</annotation></semantics></math>-form.</p> </li> </ul> <p>Write therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mo>∇</mo><mo>,</mo><msub><mi>C</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P, \nabla, C_3)</annotation></semantics></math> for such a cocycle.</p> <p>For gauge transformations between two such pairs, parameterized by the above form data patchwise 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>1</mn></msup></mrow><annotation encoding="application/x-tex">U \times \Delta^1</annotation></semantics></math>, the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_4</annotation></semantics></math> vanishes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^1</annotation></semantics></math> implies the <a href="http://nlab.mathforge.org/nlab/show/infinity-Chern-Weil+theory+introduction#InfGaugeTrafo">infinitesmal gauge transformation</a> law</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>C</mi><mo>=</mo><msub><mi>d</mi> <mi>U</mi></msub><msub><mi>ω</mi> <mi>t</mi></msub><mo>+</mo><msub><mi>ι</mi> <mi>t</mi></msub><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mover><mi>A</mi><mo stretchy="false">^</mo></mover></msub><mo>∧</mo><msub><mi>F</mi> <mover><mi>A</mi><mo stretchy="false">^</mo></mover></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \frac{d}{d t} C = d_U \omega_t + \iota_t \langle F_{\hat A} \wedge F_{\hat A}\rangle \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>,</mo><msub><mi>𝔢</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat A\in \Omega^1(U \times \Delta^1, \mathfrak{e}_8)</annotation></semantics></math> is the shift of the 1-forms. This integrates to</p> <div class="maruku-equation" id="eq:CGaugeTransformation"><span class="maruku-eq-number">(4)</span><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>d</mi><mi>ω</mi><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 + d \omega + CS(\nabla_1,\nabla_2) \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow></msub><msub><mi>ω</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\omega := \int_{\Delta^1} \omega_t</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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><mo>=</mo><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mover><mo>∇</mo><mo stretchy="false">^</mo></mover></msub><mo>∧</mo><msub><mi>F</mi> <mover><mo>∇</mo><mo stretchy="false">^</mo></mover></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">CS(\nabla_1, \nabla_2) = \int_{\Delta^1} \langle F_{\hat \nabla} \wedge F_{\hat \nabla}\rangle </annotation></semantics></math> is the relative <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> corresponding to the shift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-connection.</p> </li> </ul> <h4 id="RestrictionToBoundaryInChernWeil">Restriction to the boundary</h4> <p>We have seen that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mi>Field</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C Field(Y)</annotation></semantics></math> is the 3-groupoid of those <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cocycles</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> with coefficients in <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>diff</mi></msub></mrow><annotation encoding="application/x-tex">\exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{diff}</annotation></semantics></math> such that the curvature 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></mrow><annotation encoding="application/x-tex">\mathcal{G}_4</annotation></semantics></math> has a fixed globally defined value.</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/subobject">subobject</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>b</mi><mi>ℝ</mi><mo>−</mo><msub><mi>𝔤</mi> <mi>μ</mi></msub><msubsup><mo stretchy="false">)</mo> <mi>diff</mi> <mrow><mi>C</mi><mo>=</mo><mn>0</mn></mrow></msubsup><mo>↪</mo><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>diff</mi></msub></mrow><annotation encoding="application/x-tex"> \exp(b \mathbb{R} - \mathfrak{g}_\mu)_{diff}^{C = 0} \hookrightarrow \exp(b \mathbb{R} - \mathfrak{g}_\mu)_{diff} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>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> on those <a class="existingWikiWord" href="/nlab/show/object">object</a>s and <a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>s for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">C = 0</annotation></semantics></math>.</p> <p>By the <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> law <a class="maruku-eqref" href="#eq:CGaugeTransformation">(4)</a></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>d</mi><mi>ω</mi><mo>+</mo><mi>CS</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C_2 = C_1 + d \omega + CS(A_1, A_2) </annotation></semantics></math></div> <p>this means that this picks those <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s for which the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> vanishes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow></msub><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><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> CS(A_1,A_2) = \int_{\Delta^1} \langle F_{A} \wedge F_{A}\rangle = 0 \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi>A</mi> <mi>U</mi></msub><mo>+</mo><mi>λ</mi><mi>d</mi><mi>t</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = A_U + \lambda d t \in \Omega^1(U \times \Delta^1, \mathfrak{g})</annotation></semantics></math> is the 1-form datum (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> the canonical coordinate on the 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></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">\Delta^1 = [0,1]</annotation></semantics></math>).</p> <div class="num_note" id="NatureOfGaugedCSForm"> <h6 id="note_3">Note</h6> <p>In the literature often the relative <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> is considered for “ungauged” paths of connections: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda = 0</annotation></semantics></math> in the above formula, hence for a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-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> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">U \times \Delta^1</annotation></semantics></math> with no leg along the simplex (only depending on the simplex coordinate). Here, however, it is crucially important that we consider the general “gauged” paths.</p> </div> <p>Notice that on the <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a> and <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a> <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">\mathfrak{e}_8</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</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> is non-degenerate and positive definite (or negative definite, depending on convention). The latter condition means that this integral vanishes precisely if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>t</mi></msub></mrow></msub><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><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota_{\partial_t} \langle F_A \wedge F_A \rangle = 0 \,. </annotation></semantics></math></div> <p>This is the case on paths for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>t</mi></msub><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\iota_t F_A = 0 </annotation></semantics></math>, but this are exactly the paths that induce genuine <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math>, where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>A</mi><mo>=</mo><msub><mi>d</mi> <mi>U</mi></msub><mi>λ</mi><mo>+</mo><mo stretchy="false">[</mo><mi>λ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{d}{d t} A = d_U \lambda + [\lambda , A] \,. </annotation></semantics></math></div> <p>This means that cocycles with coefficients in this subobject for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">C = 0</annotation></semantics></math> are cocycles as described at <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>, exhibiting the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> on the heterotic boundary, witnessed by the restriction of the curvature equation <a class="maruku-eqref" href="#eq:TheFourCurvature">(3)</a> to vanishing <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</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msubsup><mi>H</mi> <mn>3</mn> <mi>L</mi></msubsup><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><msubsup><mi>A</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow> <mi>L</mi></msubsup></mrow></msub><mo>∧</mo><msub><mi>F</mi> <mrow><msubsup><mi>A</mi> <mrow><msub><mi>𝔢</mi> <mn>8</mn></msub></mrow> <mi>L</mi></msubsup></mrow></msub><mo stretchy="false">⟩</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><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></mrow><annotation encoding="application/x-tex"> d H_3^L = \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle - \frac{1}{2}\langle F_{\omega} \wedge F_{\omega} \rangle </annotation></semantics></math></div> <h3 id="ModelByTwistedCohomotopy">Model by twisted Cohomotopy</h3> <p>It turns out that the <a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization+condition">shifted C-field flux quantization condition</a> is naturally implied (<a href="#FSS19b">FSS1 19b, Prop. 4.12</a>) by the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math> is the differential form datum underlying, via <a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan's theorem</a>, a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in unstable <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-</a> <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> in degree 4 (<a class="existingWikiWord" href="/nlab/show/Hypothesis+H">Hypothesis H</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (“<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>-field”)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a> (“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field”)</p> </li> <li> <p><strong>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</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/C-field+tadpole+cancellation">C-field tadpole cancellation</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><a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">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> <div> <p><strong>Table of <a class="existingWikiWord" href="/nlab/show/brane">branes</a> appearing in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>/<a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong> (for classification see at <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em>).</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th>in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></th><th><a class="existingWikiWord" href="/nlab/show/charge">charge</a>d under <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></th><th>has <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-2%29-brane">D(-2)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D5+super+Yang-Mills+theory">D=5 super Yang-Mills theory</a> with <a class="existingWikiWord" href="/nlab/show/Khovanov+homology">Khovanov homology</a> <a class="existingWikiWord" href="/nlab/show/observables">observables</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D7+super+Yang-Mills+theory">D=7 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n+1)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-1%29-brane">D(-1)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;">2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> with <a class="existingWikiWord" href="/nlab/show/BH+entropy">BH entropy</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/N%3D4+D%3D4+super+Yang-Mills+theory">N=4 D=4 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-brane">D7-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D9-brane">D9-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%29-string">(p,q)-string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/D25-brane">D25-brane</a>)</td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/bosonic+string+theory">bosonic string theory</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">type I, II, heterotic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string">string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string+theory">little string theory</a></td></tr> <tr><td style="text-align: left;"><strong>D-brane for <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-brane">A-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B-brane">B-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11D SuGra</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M9-brane">M9-brane</a>/<a class="existingWikiWord" href="/nlab/show/O-plane">O9-plane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-wave">M-wave</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M2-brane">topological M2-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M5-brane">topological M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/S-brane">S-brane</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SM2-brane">SM2-brane</a>,<br /><a class="existingWikiWord" href="/nlab/show/membrane+instanton">membrane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane+instanton">M5-brane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane+instanton">D3-brane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong> on <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+field">self-dual</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">3-brane in 6d</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <h3 id="general">General</h3> <p>The C-field in <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a> originates as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>μ</mi><mi>ν</mi><mi>ρ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{\mu\nu\rho}</annotation></semantics></math>-field in</p> <ul> <li id="CremmerJuliaScherk78"><a class="existingWikiWord" href="/nlab/show/Eugene+Cremmer">Eugene Cremmer</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, <a class="existingWikiWord" href="/nlab/show/Jo%C3%ABl+Scherk">Joël Scherk</a>, <em>Supergravity in theory in 11 dimensions</em>, Phys. Lett. 76B (1978) 409 (<a href="https://doi.org/10.1016/0370-2693(78)90894-8">doi:10.1016/0370-2693(78)90894-8</a>)</li> </ul> <p>Re-derivation in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a>:</p> <ul> <li id="DAuriaFre82"><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, (5.11b) of: <em><a class="existingWikiWord" href="/nlab/files/GeometricSupergravity.pdf" title="Geometric Supergravity in D=11 and its hidden supergroup">Geometric Supergravity in D=11 and its hidden supergroup</a></em>, Nuclear Physics B201 (1982) 101-140 (<a href="https://doi.org/10.1016/0550-3213(82)90376-5">doi:10.1016/0550-3213(82)90376-5</a>, <a class="existingWikiWord" href="/nlab/files/GeometricSupergravityErrata.pdf" title="errata">errata</a>)</li> </ul> <p>Review in:</p> <ul> <li id="CastellaniDAuriaFre"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, III.8.53 of: <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Miemiec">André Miemiec</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Schnakenburg">Igor Schnakenburg</a>, Section 3.1.3 of: <em>Basics of M-Theory</em>, Fortsch. Phys. 54 (2006) 5-72 (<a href="http://arxiv.org/abs/hep-th/0509137">arXiv:hep-th/0509137</a>, <a href="https://doi.org/10.1002/prop.200510256">doi:10.1002/prop.200510256</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization+condition">shifted C-field flux quantization condition</a> was originally proposed in</p> <ul> <li id="Witten96a"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>On Flux Quantization In M-Theory And The Effective Action</em>, J. Geom. Phys. 22:1-13, 1997 (<a href="https://arxiv.org/abs/hep-th/9609122">arXiv:hep-th/9609122</a>)</p> </li> <li id="Witten96b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Five-Brane Effective Action In M-Theory</em>, J. Geom. Phys. 22:103-133, 1997 (<a href="https://arxiv.org/abs/hep-th/9610234">arXiv:hep-th/9610234</a>)</p> </li> </ul> <p>Proposals to model the condition by a <a class="existingWikiWord" href="/nlab/show/Wu+class">Wu class</a>-shifted variant of <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>:</p> <ul> <li id="DiaconescuFreedMoore03"><a class="existingWikiWord" href="/nlab/show/Duiliu-Emanuel+Diaconescu">Duiliu-Emanuel Diaconescu</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</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>, chapter in <a class="existingWikiWord" href="/nlab/show/Haynes+Miller">Haynes Miller</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a> (eds.) <em>Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues</em>, Cambridge University Press 2007 [<a href="http://arxiv.org/abs/hep-th/0312069">arXiv:hep-th/0312069</a>, <a href="https://doi.org/10.1017/CBO9780511721489">doi:10.1017/CBO9780511721489</a>]</li> </ul> <p>picked up e.g. in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Wijnholt">Martin Wijnholt</a>: <em>The M-Theory Three-Form and Singular Geometries</em> [<a href="https://arxiv.org/abs/2310.05838">arXiv:2310.05838</a>]</li> </ul> <p>A related model of the C-field in terms of <a class="existingWikiWord" href="/nlab/show/nonabelian+bundle+2-gerbe">nonabelian</a> <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbes">bundle 2-gerbes</a>:</p> <ul> <li id="AschieriJurčo04"><a class="existingWikiWord" href="/nlab/show/Paolo+Aschieri">Paolo Aschieri</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <em>Gerbes, M5-Brane Anomalies 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>, JHEP 0410:068 (2004) [<a href="https://arxiv.org/abs/hep-th/0409200">arXiv:hep-th/0409200</a>, <a href="https://doi.org/10.1088/1126-6708/2004/10/068">doi:10.1088/1126-6708/2004/10/068</a>]</li> </ul> <p>Further discussion ofthe <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> of the supergravity C-field, and its cancellation:</p> <ul> <li id="FreedMoore04"><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Setting the <a class="existingWikiWord" href="/nlab/show/quantum+integrand">quantum integrand</a> of M-theory</em>, Communications in Mathematical Physics <strong>263</strong> 1 (2006) 89-132, [<a href="http://arxiv.org/abs/hep-th/0409135">arXiv:hep-th/0409135</a>, <a href="https://doi.org/10.1007/s00220-005-1482-7">doi:10.1007/s00220-005-1482-7</a>]</li> </ul> <p>A summary and review of this:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Anomalies, Gauss laws, and Page charges in M-theory</em> (<a href="http://arxiv.org/abs/hep-th/0409158">arXiv:hep-th/0409158</a>)</li> </ul> <p>The discussion in <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted nonabelian differential cohomology</a> is given in</p> <ul> <li id="FSS12"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+moduli+3-stack+of+the+C-field">The moduli 3-stack of the C-field</a></em>, Communications in Mathematical Physics <strong>333</strong> 1 (2015) 117-151, [<a href="http://arxiv.org/abs/1202.2455">arXiv:1202.2455</a>, <a href="http://link.springer.com/article/10.1007%2Fs00220-014-2228-1">doi:10.1007/s00220-014-2228-1</a>]</p> </li> <li id="FiSaSc"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory</a></em> Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229?321 (<a href="http://arxiv.org/abs/1201.5277">arXiv:1201.5277</a>, <a href="https://dx.doi.org/10.4310/ATMP.2014.v18.n2.a1">doi:10.4310/ATMP.2014.v18.n2.a1</a>)</p> </li> </ul> <p>Discussion with <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a> of the C-field in <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> (<a class="existingWikiWord" href="/schreiber/show/Hypothesis+H">Hypothesis H</a>):</p> <ul> <li id="FSS19b"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation">Twisted Cohomotopy implies M-theory anomaly cancellation</a></em>, Communications in Mathematical Physics <strong>377</strong> (2020) 1961-2025 [<a href="https://arxiv.org/abs/1904.10207">arXiv:1904.10207</a>, <a href="https://doi.org/10.1007/s00220-020-03707-2">doi:10.1007/s00220-020-03707-2</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization">Twisted Cohomotopy implies M5 WZ term level quantization</a></em>, Comm. Math. Phys. <strong>384</strong> (2021) 403–432 [<a href="https://arxiv.org/abs/1906.07417">arXiv:1906.07417</a>, <a href="https://doi.org/10.1007/s00220-021-03951-0">doi:10.1007/s00220-021-03951-0</a>]</p> </li> </ul> <p>surveyed in</p> <ul> <li id="FSS20"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, §12 of: <em><a class="existingWikiWord" href="/schreiber/show/The+Character+Map+in+Twisted+Non-Abelian+Cohomology">The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalized</a></em>, World Scientific (2023) [<a href="https://arxiv.org/abs/2009.11909">arXiv:2009.11909</a>, <a href="https://doi.org/10.1142/13422">doi:10.1142/13422</a>]</li> </ul> <p>and in the generality of <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>, in <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+Cohomotopy+implies+orientifold+tadpole+cancellation">Equivariant Cohomotopy implies orientifold tadpole cancellation</a></em>, J. Geometry and Physics, <strong>156</strong> (2020) 103775 [<a href="https://arxiv.org/abs/1909.12277">arXiv:1909.12277</a>, <a href="https://doi.org/10.1016/j.geomphys.2020.103775">doi:10.1016/j.geomphys.2020.103775</a>]</p> </li> <li id="SatiSchreiber20"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, pp. 99 of: <em><a class="existingWikiWord" href="/schreiber/show/Proper+Orbifold+Cohomology">Proper Orbifold Cohomology</a></em> [<a href="https://arxiv.org/abs/2008.01101">arXiv:2008.01101</a>]</p> </li> </ul> <p>Discussion of the dual 6-form field to the 3-form C-field (required notably in the context of <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eugene+Cremmer">Eugene Cremmer</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <em>Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities</em>, Nucl. Phys. B <strong>535</strong> (1998) 242-292 [<a href="http://arxiv.org/abs/hep-th/9806106">arXiv:hep-th/9806106</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Mees+de+Roo">Mees de Roo</a>, <a class="existingWikiWord" href="/nlab/show/Olaf+Hohm">Olaf Hohm</a>, <em>Can dual gravity be reconciled with E11?</em>, Phys. Lett. B <strong>675</strong> (2009) 371-376 [<a href="http://arxiv.org/abs/0903.4384">arXiv:0903.4384</a>]</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/discrete+torsion">discrete torsion</a> (<a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariance</a>) for <a class="existingWikiWord" href="/nlab/show/circle+n-bundle">circle 3-bundles</a> describing the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, <em>Analogues of Discrete Torsion for the M-Theory Three-Form</em>, Phys.Rev. D68 (2003) 126004 (<a href="https://arxiv.org/abs/hep-th/0008170">arXiv:hep-th/0008170</a>)</p> </li> <li> <p>Shigenori Seki, <em>Discrete Torsion and Branes in M-theory from Mathematical Viewpoint</em>, Nucl.Phys. B606 (2001) 689-698 (<a href="https://arxiv.org/abs/hep-th/0103117">arXiv:hep-th/0103117</a>)</p> </li> </ul> <p>and applied to discussion of <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> (<a class="existingWikiWord" href="/nlab/show/ABJM+model">ABJM model</a>) in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ofer+Aharony">Ofer Aharony</a>, Oren Bergman, Daniel Louis Jafferis, <em>Fractional M2-branes</em>, JHEP 0811:043,2008 (<a href="https://arxiv.org/abs/0807.4924">arXiv:0807.4924</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mauricio+Romo">Mauricio Romo</a>, <em>Aspects of ABJM orbifolds with discrete torsion</em>, J. High Energ. Phys. (2011) 2011 (<a href="https://arxiv.org/abs/1011.4733">arXiv:1011.4733</a>)</p> </li> </ul> <h3 id="dualitysymmetric_formulation">Duality-symmetric formulation</h3> <div> <p>Formulation of the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a> in <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a> on fields including a flux density <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">G_7</annotation></semantics></math> <em>a priori</em> independent of the flux density <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>:</p> <ul> <li id="CastellaniDAuriaFré91"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, ch III.8.3-III.8.5 in vol 2 of: <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991) [<a href="https://doi.org/10.1142/0224">doi:10.1142/0224</a>, <a href="https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html">epdf</a>, ch III.8: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChIII8.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(Using the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a>.)</p> </blockquote> </li> <li> <p>Antonio Candiello, <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, §6 in: <em>Duality in supergravity theories</em>, Nuclear Physics B <strong>412</strong> 3 (1994) 479-501 [<a href="https://doi.org/10.1016/0550-3213(94)90389-1">doi:10.1016/0550-3213(94)90389-1</a>]</p> <blockquote> <p>(These authors seem not to be aware of <a href="#CastellaniDAuriaFré91">CDF91, III.8</a> and, contrary to the result there, conclude that it is <em>not possible</em> without introducing non-local relations.)</p> </blockquote> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> for <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a> with an <em>a priori</em> independent dual C-field field and introduction of the “<a class="existingWikiWord" href="/nlab/show/duality-symmetric+higher+gauge+theory">duality-symmetric</a>” terminology:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Nathan+Berkovits">Nathan Berkovits</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <em>Duality-Symmetric Eleven-Dimensional Supergravity and its Coupling to M-Branes</em>, Nucl. Phys. B <strong>522</strong> (1998) 214-233 [<a href="https://doi.org/10.1016/S0550-3213(98)00102-3">doi:10.1016/S0550-3213(98)00102-3</a>, <a href="https://arxiv.org/abs/hep-th/9711055">arXiv:hep-th/9711055</a>]</p> </li> <li id="CremmerJuliaLuPope"> <p><a class="existingWikiWord" href="/nlab/show/Eugene+Cremmer">Eugene Cremmer</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, Section 2 of <em>Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities</em>, Nucl.Phys. B <strong>535</strong> (1998) 242-292 [<a href="https://doi.org/10.1016/S0550-3213(98)00552-5">doi:10.1016/S0550-3213(98)00552-5</a>, <a href="https://arxiv.org/abs/hep-th/9806106">arXiv:hep-th/9806106</a>]</p> </li> <li id="BandosNurmagambetovSorokin04"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+Nurmagambetov">Alexei Nurmagambetov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, Section 2 of: <em>Various Faces of Type IIA Supergravity</em>, Nucl. Phys. B <strong>676</strong> (2004) 189-228 [<a href="https://doi.org/10.1016/j.nuclphysb.2003.10.036">doi:10.1016/j.nuclphysb.2003.10.036</a>, <a href="https://arxiv.org/abs/hep-th/0307153">arXiv:hep-th/0307153</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexei+J.+Nurmagambetov">Alexei J. Nurmagambetov</a>, <em>The Sigma-Model Representation for the Duality-Symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math> Supergravity</em>, eConf C0306234 (2003) 894-901 [<a href="https://arxiv.org/abs/hep-th/0312157">arXiv:hep-th/0312157</a>, <a href="https://inspirehep.net/literature/635585">inspire:635585</a>]</p> </li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization">shifted C-field flux quantization</a>:</p> <ul> <li><a href="duality-symmetric+higher+gauge+theory#SS23">SS23, §3.4</a></li> </ul> </div><div> <h4 id="ReferencesCFieldGaugeAlgebra">Supergravity C-Field gauge algebra</h4> <p>Identifying the super-graded gauge algebra of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">C-field</a> in <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a> (with non-trivial <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>v</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">[v_3, v_3] = -v_6</annotation></semantics></math>):</p> <ul> <li id="CremmerJuliaLuPope"> <p><a class="existingWikiWord" href="/nlab/show/Eugene+Cremmer">Eugene Cremmer</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, Equation (2.6) of <em>Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities</em>, Nucl.Phys. B <strong>535</strong> (1998) 242-292 [<a href="https://doi.org/10.1016/S0550-3213(98)00552-5">doi:10.1016/S0550-3213(98)00552-5</a>, <a href="https://arxiv.org/abs/hep-th/9806106">arXiv:hep-th/9806106</a>]</p> </li> <li> <p>I. V. Lavrinenko, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+N.+Pope">Christopher N. Pope</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+S.+Stelle">Kellogg S. Stelle</a>, (3.4) in: <em>Superdualities, Brane Tensions and Massive IIA/IIB Duality</em>, Nucl. Phys. B <strong>555</strong> (1999) 201-227 [<a href="https://doi.org/10.1016/S0550-3213(99)00307-7">doi:10.1016/S0550-3213(99)00307-7</a>, <a href="https://arxiv.org/abs/hep-th/9903057">arXiv:hep-th/9903057</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jussi+Kalkkinen">Jussi Kalkkinen</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+S.+Stelle">Kellogg S. Stelle</a>, (75) of: <em>Large Gauge Transformations in M-theory</em>, J. Geom. Phys. <strong>48</strong> (2003) 100-132 [<a href="https://doi.org/10.1016/S0393-0440(03)00027-5">doi:10.1016/S0393-0440(03)00027-5</a>, <a href="https://arxiv.org/abs/hep-th/0212081">arXiv:hep-th/0212081</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+A.+Bandos">Igor A. Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+J.+Nurmagambetov">Alexei J. Nurmagambetov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+P.+Sorokin">Dmitri P. Sorokin</a>, (86) in: <em>Various Faces of Type IIA Supergravity</em>, Nucl.Phys. B <strong>676</strong> (2004) 189-228 [<a href="https://doi.org/10.1016/j.nuclphysb.2003.10.036">doi:10.1016/j.nuclphysb.2003.10.036</a>, <a href="https://arxiv.org/abs/hep-th/0307153">arXiv:hep-th/0307153</a>]</p> </li> </ul> <p>Identification as an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra"><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>-algebra</a> (a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, in this case):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, (4.9) in: <em>Geometric and topological structures related to M-branes</em>, in <em>Superstrings, Geometry, Topology, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras</em>, Proc. Symp. Pure Math. <strong>81</strong> (2010) 181-236 [<a href="http://www.ams.org/books/pspum/081">ams:pspum/081</a>, <a href="http://arXiv.org/abs/1001.5020">arXiv:1001.5020</a>]</li> </ul> <p>and identificatoin with the rational <a class="existingWikiWord" href="/nlab/show/Whitehead+L-infinity+algebra">Whitehead <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>-algebra</a> (the <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational Quillen model</a>) of the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> (cf. <em><a class="existingWikiWord" href="/schreiber/show/Hypothesis+H">Hypothesis H</a></em>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, (13) in: <em>Mysterious Triality and M-Theory</em> [<a href="https://arxiv.org/abs/2212.13968">arXiv:2212.13968</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, (22) in: <em><a class="existingWikiWord" href="/schreiber/show/Flux+Quantization+on+Phase+Space">Flux Quantization on Phase Space</a></em> [<a href="https://arxiv.org/abs/2312.12517">arXiv:2312.12517</a>]</p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on August 23, 2024 at 11:33:50. See the <a href="/nlab/history/supergravity+C-field" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/supergravity+C-field" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2600/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/supergravity+C-field/61" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/supergravity+C-field" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/supergravity+C-field" accesskey="S" class="navlink" id="history" rel="nofollow">History (61 revisions)</a> <a href="/nlab/show/supergravity+C-field/cite" style="color: black">Cite</a> <a href="/nlab/print/supergravity+C-field" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/supergravity+C-field" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>