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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>much of the material below has been or is being reworked into the entries <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> and <a class="existingWikiWord" href="/nlab/show/connection+on+a+smooth+principal+%E2%88%9E-bundle">connection on a smooth principal ∞-bundle</a></p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="chernweil_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Weil theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory+introduction">∞-Chern-Weil theory introduction</a></p> </li> </ul> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+an+%28%E2%88%9E%2C1%29-topos">differential cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>, <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></li> </ul> </li> </ul> <h2 id="connection">Connection</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+valued+differential+forms">∞-Lie algebroid valued differential forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connection+on+a+principal+%E2%88%9E-bundle">∞-connection on a principal ∞-bundle</a></p> </li> </ul> <h2 id="curvature">Curvature</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Gauss-Bonnet+theorem">Chern-Gauss-Bonnet theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-Chern-Weil+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <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='#Motivation'>Motivation</a></li> <ul> <li><a href='#fractional_differential_classes'>Fractional differential classes</a></li> <li><a href='#higher_differential_spin_structures'>Higher differential spin structures</a></li> </ul> <li><a href='#idea'>Idea</a></li> <li><a href='#PreparatoryConcepts'>Preparatory concepts</a></li> <li><a href='#ChernWeil'><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='#InfinityLieAlgebraConnection'><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>-Lie algebra valued connections</a></li> <li><a href='#InfChernWeil'>Curvature characteristics</a></li> <li><a href='#higher_order_chernsimons_forms'>Higher order Chern-Simons forms</a></li> <li><a href='#ChernCharacter'>Chern character</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#principal_1bundles_with_connection'>Principal 1-bundles with connection</a></li> <li><a href='#principal_2bundles_with_connection'>Principal 2-bundles with connection</a></li> <li><a href='#DiffStringStruc'>Twisted differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo>−</mo></mrow><annotation encoding="application/x-tex">String-</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math>-structures</a></li> <ul> <li><a href='#StringStructure'>The string-lifting Chern–Simons <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-bundle with connection</a></li> <li><a href='#differential_string_structures'>Differential string structures</a></li> <li><a href='#FivebraneStructure'>The Fivebrane-lifting Chern-Simons 7-bundle with connection</a></li> <li><a href='#DiffFivebraneStrucs'>Differential fivebrane structures</a></li> </ul> <li><a href='#chernsimons_theory'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons theory</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <p>In every <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> there is an <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos#ChernWeilTheory">intrinsic notion of Chern-Weil theory</a>. We discuss the concrete realization of this in the cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>s. This s the case that subsumes ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> of <a class="existingWikiWord" href="/nlab/show/SmoothMfd">smooth</a> <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> and generalizes it to <a class="existingWikiWord" href="/nlab/show/connections+on+smooth+principal+%E2%88%9E-bundles">connections on smooth principal ∞-bundles</a>.</p> <h2 id="Motivation">Motivation</h2> <p>The central motivation for the study of a higher generalization of ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> is the interest in extending the <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> for a given <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to the higher connected covers 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> through the whole <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Beyond the simply connected cover, these higher connected covers are still <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>s but fail to be (finite dimensional) <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>s. They do however have natural realizations as <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a>s. Higher Chern-Weil theory is the extension of <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> from Lie groups to such smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups. It allows the refinement of differential characteristic classes to <em>fractional</em> differential characteristic classes, that capture finer cohomological information.</p> <h3 id="fractional_differential_classes">Fractional differential classes</h3> <p>We give some examples of such <em>fractional characteristic classes</em> that occur in practice.</p> <p>It is a familiar classical fact that the first <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><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>ℬ</mi><mi>SO</mi><mo>→</mo><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> p_1 : \mathcal{B}SO \to \mathcal{B}^4 \mathbb{Z} \,, </annotation></semantics></math></div> <p>which represents the generator of the fourth <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>SO</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} SO</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> allows a division by 2 when pulled back one step along the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> to the classifying space of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, in that there is a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℬ</mi><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℬ</mi><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 2}} \\ \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4 \mathbb{Z} } \,, </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, where the top horizontal morphism represents a generator of the 4th integral cohomology of the classifying space of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> and the right vertical morphism is induced by multiplication by 2 on the additive group of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s.</p> <p>This means that 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 class="existingWikiWord" href="/nlab/show/manifold">manifold</a> with <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> exhibited by a classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat g</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℬ</mi><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>ℬ</mi><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z} \\ & {}^{\mathllap{\hat g}}\nearrow & \downarrow && \downarrow^{\mathrlap{\cdot 2}} \\ X &\stackrel{g}{\to}& \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4\mathbb{Z} } </annotation></semantics></math></div> <p>of its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>, the characteristic class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mover><mo>→</mo><mi>g</mi></mover><mi>ℬ</mi><mi>SO</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">p_1(T X) : X \stackrel{g}{\to} \mathcal{B}SO \stackrel{p_1}{\to} \mathcal{B}^4 \mathbb{Z}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> regarded as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi></mrow><annotation encoding="application/x-tex">SO</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a> contains less information than the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mover><mo>→</mo><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mover><mi>ℬ</mi><mi>Spin</mi><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1(T X) : X \stackrel{\hat g}{\to} \mathcal{B}Spin \stackrel{\frac{1}{2}p_1}{\to} \mathcal{B}^4 \mathbb{Z}</annotation></semantics></math>. For instance if the 4th cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> happens to be 2-<a class="existingWikiWord" href="/nlab/show/torsion">torsion</a>, the former class entirely vanishes, while the latter need not.</p> <p>This familiar situation poses no problem to classical <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, because both the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> as well as the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> of course have canonical structures of <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>s, so that the <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> may be applied to either. We shall write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathrm{Spin}</annotation></semantics></math> for the smooth refinement of the classifying space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi mathvariant="normal">Spin</mi></mrow><annotation encoding="application/x-tex">B \mathrm{Spin}</annotation></semantics></math>: the delooping Lie groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spin</mi></mrow><annotation encoding="application/x-tex">\mathrm{Spin}</annotation></semantics></math> or equivalently the moduli stack for smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spin</mi></mrow><annotation encoding="application/x-tex">\mathrm{Spin}</annotation></semantics></math>-principal bundles. Here and in the following the boldface indicates smooth (or otherwise cohesive) refinements. Accordingly, there is a smooth refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}\mathbf{p} : \mathbf{B} \mathrm{Spin} \to \mathbf{B}^3 U(1)</annotation></semantics></math> of the first Pontryagin class, which takes smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spin</mi></mrow><annotation encoding="application/x-tex">\mathrm{Spin}</annotation></semantics></math>-principal bundles to their first Pontryagin class. This in turn has has a further differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi mathvariant="normal">Spin</mi> <mi mathvariant="normal">conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi mathvariant="normal">conn</mi></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}{\hat {\mathbf{p}}} : \mathbf{B}\mathrm{Spin}_{\mathrm{conn}} \to \mathbf{B}^3 U(1)_{\mathrm{conn}}</annotation></semantics></math> that takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Spin</mi></mrow><annotation encoding="application/x-tex">\mathrm{Spin}</annotation></semantics></math>-principal bundles with connection to their Chern-Simons 2-gerbes with connection.</p> <p>All this is still captured by the traditional (refined) Chern-Weil homomorphism. But this is no longer the case as we keep climbing up the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>. In the next step the second <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub><mo>:</mo><mi>ℬ</mi><mi>SO</mi><mo>→</mo><msup><mi>ℬ</mi> <mn>8</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">p_2 : \mathcal{B}SO \to \mathcal{B}^8 \mathbb{Z}</annotation></semantics></math> may be divided by 6 when pulled back to the classifying space of the <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> (<a href="http://ncatlab.org/schreiber/edit/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSII">SSSII</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>ℬ</mi><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>ℬ</mi> <mn>8</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mn>6</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℬ</mi><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>ℬ</mi> <mn>8</mn></msup><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{B} String &\stackrel{\frac{1}{6}p_2}{\to}& \mathcal{B}^8 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 6}} \\ \mathcal{B} SO &\stackrel{p_2}{\to}& \mathcal{B}^8 \mathbb{Z} } \,, </annotation></semantics></math></div> <p>As before, this means that if a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> admits a <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, then the characteristic class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_2(X)</annotation></semantics></math> contains less information than the fractional refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{6}p_2(X)</annotation></semantics></math> that it admits. In particular, the former may vanish if the degree 8 cohomology group 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> has 6-<a class="existingWikiWord" href="/nlab/show/torsion">torsion</a>, while the latter need not vanish.</p> <p>For purposes of ordinary cohomology this is no problem, but for the differential refinement by ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> it is: the <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> does not admit a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> structure that would make it a smooth version of the homotopy fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1</annotation></semantics></math> and hence standard <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> cannot produce the differential refinement of the fractional class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{6}p_2</annotation></semantics></math>.</p> <p>But <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 can: there is a natural smooth refinement of the <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> to a <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>: the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}String</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a>. The fractional second Pontryagin class does lift to this smooth refinement to produce a <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1) </annotation></semantics></math></div> <p>internal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a>. Since this now lives in a smooth context, it does now have a differential Chern-Weil refinement</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo><mo>→</mo><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>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{H}_{conn}(-, \mathbf{B}String) \to \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) </annotation></semantics></math></div> <p>that takes smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>s with 2-connection to degree 8-cocycles in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <p>This kind of refinement we discuss in a bit more detail in the next section.</p> <h3 id="higher_differential_spin_structures">Higher differential spin structures</h3> <p>These refined differential invariants of fractional characteristic classes are relevant in the discussion of higher differential spin structures. (See the first part of (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSII">SatiSchreiberStasheff II</a> for a review).) Ordinary <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>s on a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> may be understood as trivializations of what are called <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <a class="existingWikiWord" href="/nlab/show/Pfaffian+line+bundle">Pfaffian line bundle</a>s on the configuration space of the spinning <a class="existingWikiWord" href="/nlab/show/relativistic+particle">quantum particle</a> propagating on that manifold. (This physical origin is after all the origin of the term <em>spin structure</em> .) When these point-like super-particles are generalized to higher-dimensional <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>-<a class="existingWikiWord" href="/nlab/show/brane">brane</a>s, the trivialization of the corresponding Pfaffian line bundles correspond to <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>s for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p = 1</annotation></semantics></math> (this goes back to (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#Killingback">Killingback</a>) and (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#WittenIndexLoopSpace">Witten</a>) and has been made rigorous in (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#BunkeStringStruc">Bunke</a>) then to <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a>s for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">p = 5</annotation></semantics></math> (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSII">SatiSchreiberStasheff II</a>)).</p> <p>More precisely, the Pfaffian line bundles appearing here come equipped with a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a>, and what matters is a trivialization of these bundles as bundles with connection. This refinement translates to differential refinements of the string structures and the fivebrane structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The differential form data of a twisted <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a> constitutes what in the physics literature is called the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>. While this still can and has been captured with tools of ordinary Chern-Weil theory and ordinary differential cohomology (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#WaldorfStringConn#FreedCharge">Freed</a>, <a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#WaldorfStringConn">Waldorf</a>) it has a natural formulation in higher Chern-Weil theory. Going beyond that, the <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">magnetic dual</a> Green-Schwarz mechanism can be seen to encode a twisted differential fivebrane structure and this is not practical to be studied without some higher geometry.</p> <p>The following restates this in a bit more technical detail.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">G = Spin</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, the first nontrivial <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> is the first fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a> given by a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>ℬ</mi><mi>G</mi><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1 : \mathcal{B}G \to K(\mathbb{Z}, 4)</annotation></semantics></math> in ordinary <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>ℬ</mi><mi>Spin</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(\mathcal{B}Spin, \mathbb{Z})</annotation></semantics></math>. This induces a map</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><mi>Spin</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℬ</mi><mi>Spin</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^1(X, Spin) = H(X, \mathcal{B}Spin) \to H^4(X, \mathbb{Z}) </annotation></semantics></math></div> <p>from isomorphism classes of topological <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>s to degree 4 integral cohomology.</p> <p>If we assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> then we may consider the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>∼</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spin Bund(X)/ \sim = H(X,\mathbf{B}Spin) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>-classes of <em>smooth</em> <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>s. Here and in all of the following, the boldface in “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>” indicates a refinement, here of the bare classifying space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}G</annotation></semantics></math> to a smooth incarnation.</p> <p>Then ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> provides a refinement of the fractional Pontryagin class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X, \mathbf{B}Spin) \to H^4(X,\mathbb{Z})</annotation></semantics></math> to a map to <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{diff}^4(X)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mi>p</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>:</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>diff</mi> <mn>4</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2} \hat p_1 : H(X, \mathbf{B}Spin) \to H_{diff}^4(X) \,. </annotation></semantics></math></div> <p>The first point of passing to a <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>-refinement of this situation is that it allows to refine, in turn, these morphisms of cohomology <em>sets</em> to morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><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></mrow><annotation encoding="application/x-tex"> \frac{1}{2} \mathbf{p}_1 : \mathbf{H}(X, \mathbf{B}Spin) \to \mathbf{H}(X,\mathbf{B}^3 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><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2} \hat \mathbf{p}_1 : \mathbf{H}_{conn}(X, \mathbf{B}Spin) \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1)) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s: here <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> is the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> whose objects are smooth <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>s, and whose morphisms are smooth homomorphisms between these. Similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <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}(X,\mathbf{B}^3 U(1))</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a> whose objects are smooth <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1">circle 2-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal 3-bundles</a>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))</annotation></semantics></math> is accordingly the <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a> whose objects are <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundles with connection</a>, whose morphisms are homomorphisms between these, whose 2-morphisms are higher homotopies between those. The original morphism of cohomology sets is the <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> of this, the restriction to connected components:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}p_1 = \pi_0(\frac{1}{2}\mathbf{p}_1) \,. </annotation></semantics></math></div> <p>This refinement to cocylce <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids notably has the consequence that it allows us to produce the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s of these morphisms. To see the relevance of this, recall (from <em><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a></em> ) that the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the bare fractional Pontryagin class, which is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></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></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}(X,\mathbf{B}String) &\to& * \\ \downarrow &\swArrow_\simeq& \downarrow \\ \mathbf{H}(X,\mathbf{B}G) &\stackrel{\frac{1}{2} \mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,, </annotation></semantics></math></div> <p>defines the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid <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>String</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \mathbf{B}String)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>s 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> ( <em>smooth</em> , but not <em>differential</em> ).</p> <p>We can now replace the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\mathbf{p}_1</annotation></semantics></math> by its differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1</annotation></semantics></math> and obtain an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{diff}(X)</annotation></semantics></math> that differentially refines the 2-groupoid <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>String</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,\mathbf{B}String)</annotation></semantics></math> of String-structures as the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>String</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ String_{diff}(X) &\to& * \\ \downarrow &\swArrow_\simeq& \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}G) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) } \,. </annotation></semantics></math></div> <p>This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>String</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String_{diff}(X)</annotation></semantics></math> we may call the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of <em><a class="existingWikiWord" href="/nlab/show/differential+string-structures">differential string-structures</a></em> . A cocycle in there is naturally identified with a tuple consisting of</p> <ul> <li> <p>a smooth <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><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+2-gerbe">Chern-Simons 2-gerbe</a> with connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS(\nabla)</annotation></semantics></math> induced by this;</p> </li> <li> <p>a choice of trivialization of this Chern-Simons 2-gerbe – this is the <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> in the middle of the above pullback diagram.</p> </li> </ul> <p>We may think of this as a refinement of <a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a>es: the first Pontryagin <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo>∧</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle F_\nabla \wedge F_\nabla \rangle</annotation></semantics></math> itself is constrained to vanish, and so the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> 3-connection itself constitutes cohomological data.</p> <p>So far this uses mostly just a little bit of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> or at least some <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. The first glimpse of something beyond ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> appearing is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid <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>String</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,\mathbf{B}String)</annotation></semantics></math> which may be thought of as the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of <em>smooth</em> <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>s.</p> <p>But suppose we fix an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X, \mathbf{B}String)</annotation></semantics></math> is nontrivial. Then we can continue the proceed to higher degrees:</p> <p>the next topological characteristic class is the second fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub><mo>:</mo><mi>ℬ</mi><mi>String</mi><mo>→</mo><msup><mi>ℬ</mi> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^7 U(1)</annotation></semantics></math>. Since the <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> does not have the structure of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, this cannot be refined to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> using ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>. However, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Weil theory it can:</p> <p>we may obtain a differential refinement</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} </annotation></semantics></math></div> <p>that maps smooth <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>s with 2-connectins to their <a href="#FivebraneStructure">Chern-Simons circle 7-bundle with connection</a>. This is an example of the higher version of the <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a>.</p> <p>And naturally we are then entitled to form its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s and produce the <a class="existingWikiWord" href="/nlab/show/n-groupoid">7-groupoid</a> of <em><a href="#DiffFivebraneStrucs">differential fivebrane structures</a></em> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fivebrane</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fivebrane_{diff}(X)</annotation></semantics></math>. For that notice (see <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a>) that the homotopy fiber of the smooth but non-differential cocycles</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>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</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>7</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}(X, \mathbf{B}Fivebrane) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}(X, \mathbf{B}String) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{H}(X, \mathbf{B}^7 U(1)) } </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/n-groupoid">7-groupoid</a> of smooth <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a>s 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>. Its differential refinement</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Fivebrane</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</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{ Fivebrane_{diff}(X) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{H}_{conn}(X, \mathbf{B}String) &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1)) } </annotation></semantics></math></div> <p>we may therefore call the 7-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fivebrane</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fivebrane_{diff}(X)</annotation></semantics></math> of <em>differential fivebrane structures</em> . Cocycles in here are naturally identified with tuples of</p> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-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>, equipped with a <a href="#InfinityLieAlgebraConnection">2-connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>;</p> </li> <li> <p>the Chern-Simons <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 7-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CS</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS_7(\nabla)</annotation></semantics></math> with connection induced by it;</p> </li> <li> <p>a choice of trivialization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CS</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS_7(\nabla)</annotation></semantics></math>.</p> </li> </ul> <p>These are the kind of structures that <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 studies.</p> <h2 id="idea">Idea</h2> <p>Ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> is about refinements of <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>es of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> (equivalently of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}G</annotation></semantics></math> of that Lie group) from ordinary <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>.</p> <p>Under <em><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</em> we want to understand the generalization of this to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>: where <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>s are generalized to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a>s, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>s are generalized to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>s and <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s to <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s.</p> <p>So <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 produces <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>-refinements of <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>es of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a>, equivalently of the corresponding <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}G</annotation></semantics></math>.</p> <p>Ordinary Chern-Weil theory is formulated in the context of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>. We need to widen this context somewhat in order that it can accomodate the relevant higher structures and so we place ourselves in the context of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoids">∞-Lie groupoids</a>.</p> <p>In every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos that admits a notion of differential cohomology, there is a general abstract notion of refinement of <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>es in <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> to <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+forms">curvature characteristic classes</a> in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <p>The main construction in ∞-Chern-Weil theory is a concrete <em>model</em> or <em>presentation</em> of this abstract operation. This model is constructed in terms of <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of objects in <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a>. This construction is the higher analog of the <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a>. Its crucial intermediate step is the definition and construction of <em><a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connections</a></em> on <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s.</p> <p>This model itself is after all built on concrete familiar constructions in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> and can be studied and appreciated in itself without recourse to the higher topos theory that we claim it provides a model for. The so inclined reader can ignore all the general abstract discussion in the following and concentrate on the concrete differential geometry.</p> <p>ere is how this entry here proceeds.</p> <p>A warmup for the full theory that connects to classical constructions is given at</p> <ul> <li><a href="#PreparatoryConcepts">Introduction</a></li> </ul> <p>Then in</p> <ul> <li><a href="#ChernWeil">∞-Chern-Weil theory</a></li> </ul> <p>we discuss the general definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connections and of the Chern-Weil homomorphism and discuss some general properties. Then we turn to discussing</p> <ul> <li><a href="#Examples">Examples</a></li> </ul> <h2 id="PreparatoryConcepts">Preparatory concepts</h2> <p>General <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, as described below, is naturally formulated in the context of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">(infinity,1)-topos</a>-theory and some of its aspects can only be understood from that perspective.</p> <p>However, unwinding the abstract higher topos theoretic concepts in terms of 1-categoriecal models yields concrete structures in familiar contexts of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> that connect to various classical and familiar concepts. Since a full appreciation of the abstract formulation benefits from having a feeling for how these concrete models work out, the reader may at this point wish to look into some such basic aspects. These may be found behind the following link</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinity-Chern-Weil+theory+--+preparatory+concepts">infinity-Chern-Weil theory – preparatory concepts</a>.</li> </ul> <h2 id="ChernWeil"><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</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>,</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">G,A</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s in an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>, respectively, every <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">c : \mathbf{B}G \to A</annotation></semantics></math> serves to pull back the <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos#GroupalCurvature">canonical intrinsic curvature form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A</annotation></semantics></math> to an <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos#deRham">intrinsic differential form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mi>A</mi></msub><mo>∘</mo><mi>c</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">curv_A\circ c : \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}A</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> regarded naturally as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a>, we show that the ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s may be understood as a concrete <em>model</em> for this simple abstract situation, which applies to those characteristic classes <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> that happen to be in the image of the <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#IntegrationOfCocycles">Lie intgeration of Lie algebra cocycles</a>.</p> <p>More generally, this construction applies for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a> with <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> a characteristic class on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> that arises from Lie integration of a cocycle in the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>.</p> <p>The ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> uses a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> as an intermediate tool for interpolating from a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> to its curvature characteristic, represented by the <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle F_\nabla \rangle</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>∇</mo></msub></mrow><annotation encoding="application/x-tex">F_\nabla</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> and <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 stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle - \rangle</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>. The choice of connection in this construction may be understood as providing a correspondence space in the following construction.</p> <p>We know from the discussion of abelian differential cohomology <a href="#AbGerbe">above</a> that the intrinsic morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><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>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}/\mathbb{Z} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} = \infty LieGrpd</annotation></semantics></math> is modeled in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj,cov}</annotation></semantics></math> by the correspondence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>Γ</mi> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><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><msub><mi>ℝ</mi> <mi>simp</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">/</mo><msub><mi>Γ</mi> <mi>simp</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n \mathbb{R}/\Gamma_{diff,simp} &\stackrel{curv}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{simp} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n \mathbb{R}/\Gamma_{simp} } \,. </annotation></semantics></math></div> <p>If we write</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><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R})) : \mathbf{cosk}_{n+1}( (U,[k]) \mapsto \left\{ \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(b^{n-1}\mathbb{R}) } \right\} ) </annotation></semantics></math></div> <p>and so forth, then this correspondence is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(b^{n-1}\mathbb{R}\to *) } \,. </annotation></semantics></math></div> <p>If now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z}</annotation></semantics></math> is modeled by the Lie integration of a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> on a Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mover><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Γ</mi><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex"> \mathbf{cosk}_{k+1} \exp(\mathfrak{g}) \stackrel{\exp(\mu)}{\to} \exp(b^{n-1}\mathbb{R})/\Gamma = \exp(b^{n-1}\mathbb{R} \to *)/\Gamma </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq n-1</annotation></semantics></math>, then the total intrinsic differential form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>Γ</mi><mo>→</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>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\Gamma \to \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}</annotation></semantics></math> is modeled by the zig-zag of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><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><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{cosk}_{k+1}\exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to *) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]</annotation></semantics></math>. In order to compute with such zig-zags of morphisms, in particular in order to compute <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s, it is helpful to complete this to a single correspondence. There is a fairly evident choice for the tip of this total corresponence, namely</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \exp(\mathfrak{g} \to inn(\mathfrak{g})) \,. </annotation></semantics></math></div> <p>It remains to complete the square and extend the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo lspace="0em" rspace="thinmathspace">mathb</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mu : \mathfrak{g} \to b^{n-1}\mathb{R}</annotation></semantics></math> to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔤</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g} \to inn(\mathfrak{g})) \to (b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R}))</annotation></semantics></math>. This is accomplished by an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>μ</mi></msub><mo>:</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>μ</mi></msub><mo>,</mo><msub><mi>cs</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow></mover><mi>inn</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \langle -\rangle_\mu : inn(\mathfrak{g}) \stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to} inn(b^{n-1}\mathbb{R}) \to b^n \mathbb{R} </annotation></semantics></math></div> <p>which is in transgression with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, witnessed by the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cs</mi> <mi>μ</mi></msub></mrow><annotation encoding="application/x-tex">cs_\mu</annotation></semantics></math>. Using this, we obtain the total diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>μ</mi></msub><mo>,</mo><msub><mi>cs</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">mathrak</mo><mi>g</mi><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><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) &\stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{cosk}_{k+1}\exp(\mathrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}\to *) } \,. </annotation></semantics></math></div> <p>By the fact that this commutes, we have that the correspondence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><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><msub><mi>ℝ</mi> <mi>simp</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>simp</mi></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>:</mo><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ \mathbf{B}G_{diff,simp} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp} \\ \downarrow \\ \mathbf{B}G_{simp} } \right) \;\; := \;\; \left( \array{ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) &\to& \exp(* \to b^n \mathbb{R}) \\ \downarrow \\ \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to *) } \right) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj,cov}</annotation></semantics></math> models the intrinsic curvature characteristic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><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>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}</annotation></semantics></math>.</p> <p>We may identify cocycles with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{diff}</annotation></semantics></math> as <em>(pseudo)-<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>-connections</em> on the underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>-cocycle. If their curvature is represented by a cocycle in <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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}</annotation></semantics></math> which is given by a globally defined form, then these are <em>genuine</em> <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>-connections. In either case, they serve as an intermediate step in computing the curvature characteristics.</p> <h3 id="InfinityLieAlgebraConnection"><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>-Lie algebra valued connections</h3> <blockquote> <p>The content of this section is at <a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">connection on an infinity-bundle</a>.</p> </blockquote> <h3 id="InfChernWeil">Curvature characteristics</h3> <div class="un_def"> <h6 id="definition">Definition</h6> <p><strong>(Chern-Weil curvature characteristics)</strong></p> <p>Let <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 stretchy="false">⟩</mo><mo>:</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>b</mi> <mi>p</mi></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\langle -\rangle : inn(\mathfrak{g}) \to b^{p} \mathbb{R}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> on the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">Lie n-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>. Postcomposition with the corresponding diagram of dg-algebras</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>p</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CE(\mathfrak{g}) &\leftarrow& 0 \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& CE(b^p \mathbb{R}) } </annotation></semantics></math></div> <p>induces a morphism of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">⟩</mo><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub></mrow><annotation encoding="application/x-tex"> \langle F_{(-)} \rangle : \mathbf{B}G_{diff} \to \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp} </annotation></semantics></math></div> <p>into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">coskeleton</a> of the model for the de Rham coefficient object <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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR}\mathbf{B}^{p+1}\mathbb{R}</annotation></semantics></math> discussed <a href="#U1FromLieIntegration">above</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>:</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">\nabla : \hat X \to \mathbf{B}G_{diff}</annotation></semantics></math> a connection, we call the induced intrinsic de Rham cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">⟩</mo><mo>:</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>∇</mo></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub><mover><mo>→</mo><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow></mover><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub></mrow><annotation encoding="application/x-tex"> \langle F_\nabla \rangle : \hat X \stackrel{\nabla}{\to} \mathbf{B}G_{diff} \stackrel{\langle -\rangle}{\to} \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp} </annotation></semantics></math></div> <p>the Chern-Weil <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> with respect to <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 stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -\rangle</annotation></semantics></math>.</p> </div> <div class="un_lemma"> <h6 id="lemma">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>:</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>↪</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">\nabla : \hat X \to \mathbf{B}G_{conn} \hookrightarrow \mathbf{B}G_{diff}</annotation></semantics></math> a genuine connection, the induced curvature characteristic forms are globally defined closed forms, in that their cocycle factors through the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^{p+1}_{cl}(-)</annotation></semantics></math> of closed <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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math>-forms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">⟩</mo></mrow></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">⟩</mo></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}G_{conn} &\stackrel{\langle F_{(-)}\rangle}{\to}& \Omega^{p+1}_{cl}(-) \\ & \nearrow & \downarrow && \downarrow \\ \hat X &\stackrel{}{\to}& \mathbf{B}G_{diff} &\stackrel{\langle F_{(-)}\rangle}{\to}& \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1} \mathbb{R}_{simp} } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>for given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,[k])</annotation></semantics></math> notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">⟩</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\langle F_{\nabla}\rangle(U,[k]) \in \Omega^\bullet(U\times \Delta^k)</annotation></semantics></math> is closed and for <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> a genuine connection has no leg along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^k</annotation></semantics></math>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\partial_t</annotation></semantics></math> a vector field along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^k</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>t</mi></msub></mrow></msub><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\iota_{\partial_t} \langle F_A\rangle = 0</annotation></semantics></math>. Therefore the <a class="existingWikiWord" href="/nlab/show/Lie+derivative">Lie derivative</a> along a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\partial_t</annotation></semantics></math> along the simplex vanishes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>t</mi></msub><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo><mo>=</mo><mi>d</mi><msub><mi>ι</mi> <mi>t</mi></msub><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo><mo>+</mo><msub><mi>ι</mi> <mi>t</mi></msub><mi>d</mi><mo stretchy="false">⟨</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"> \mathcal{L}_t \langle F_A\rangle = d \iota_t \langle F_A\rangle + \iota_t d \langle F_A\rangle = 0 \,. </annotation></semantics></math></div></div> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>As for the groupal case <a href="#AbGerbesConnection">above</a>, we hence find that the genuine <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>-connections are selected among all pseudo-connections as those whose curvature characteristic has a 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> cocycle representative.</p> </div> <p>So a genuine <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>-Lie algebra valued connection is a cocycle with values in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-coskeleton of the simplicial presheaf of diagrams, which over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">U,[k]</annotation></semantics></math> assigns the set of diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>Δ</mi><mi>k</mi><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>cocycle</mi><mspace width="thickmathspace"></mspace><mi>for</mi><mspace width="thickmathspace"></mspace><mi>underlying</mi><mspace width="thickmathspace"></mspace><mi>G</mi><mo>−</mo><mi>principal</mi><mspace width="thickmathspace"></mspace><mn>∞</mn><mo>−</mo><mi>bundle</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>connection</mi><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mi>curvature</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>curvature</mi><mspace width="thickmathspace"></mspace><mi>characteristic</mi><mspace width="thickmathspace"></mspace><mi>forms</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(U \times \Delta k)_{vert} &\leftarrow& CE(\mathfrak{g}) &&& cocycle\;for\;underlying\;G-principal\;\infty-bundle \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(X) &\leftarrow& CE(inn(\mathfrak{g})) = W(\mathfrak{g}) &&& connection\;and\;curvature \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\leftarrow& inv(\mathfrak{g}) &&& curvature\;characteristic\;forms } \,, </annotation></semantics></math></div> <p>(with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U \times \Delta^k)_{vert}</annotation></semantics></math> the dg-algebra of <a class="existingWikiWord" href="/nlab/show/vertical+differential+form">vertical differential form</a>s on the bundle <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><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U \times \Delta^k \to U</annotation></semantics></math>), where the top morphism encodes the cocycle for the underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>τ</mi> <mi>n</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = \tau_n\exp(\mathfrak{g})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, where the middle morphism encodes the connection data and the bottom morphism the <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>s.</p> <p>Such <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>-Lie algebra valued connections were introduced in <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSI">SSSI</a> and further studied in <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSIII">SSSIII</a>.</p> <h3 id="higher_order_chernsimons_forms">Higher order Chern-Simons forms</h3> <p>See at <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> the section <a href="http://ncatlab.org/nlab/show/Chern-Simons+form#HigherOrderChernSimonsForms">In ∞-Chern-Weil theory</a>.</p> <h3 id="ChernCharacter">Chern character</h3> <p>Above we have considered <a href="#InfinityLieAlgebraConnection">∞-Lie algebra valued connections</a> and their <a href="#InfChernWeil">curvature characteristic forms</a>. We now wish to show how these model the intrinsic <a class="existingWikiWord" href="/schreiber/show/Chern+character+in+an+%28%E2%88%9E%2C1%29-topos">Chern character in an (∞,1)-topos</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>R</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ch_{\mathbf{B}G} : \mathbf{B}G \to \mathbf{\Pi}(\mathbf{B}G) \to \mathbf{\Pi}(\mathbf{B}G)\otimes R \,. </annotation></semantics></math></div> <p>Since our ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is assumed to be <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected</a> we have in addition to the notion of <a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower in an (∞,1)-category</a> the notion of <a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower in an (∞,1)-topos</a>. Both notions are dual to each other: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> any object and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>1</mn></mrow></msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Pi}(A) \to \cdots \to \tau_{\leq 2}\mathbf{\Pi}(A) \to \tau_{\leq 1}\mathbf{\Pi}(A) \to \tau_{\leq 0}\mathbf{\Pi}(A) = * </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">intrinsic Postnikov tower</a> of its <a class="existingWikiWord" href="/nlab/show/path+%E2%88%9E-groupoid">path ∞-groupoid</a>, the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite 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>⋮</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mstyle mathvariant="bold"><mi>π</mi></mstyle> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mstyle mathvariant="bold"><mi>π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>3</mn></mrow></msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>1</mn></mrow></msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots && && && \vdots \\ \downarrow && && && \downarrow \\ A_2 && &\to& \cdots &\to& \mathbf{B}\mathbf{\pi}_3(A) &\to& * \\ \downarrow && && && && \downarrow \\ A_1 && &\to& \cdots && && \mathbf{B}\mathbf{\pi}_2(A) &\to& * \\ \downarrow && && && && \downarrow && \downarrow \\ A &\to& \mathbf{\Pi}A &\to& \cdots &\to& \tau_{\leq 3} \mathbf{\Pi}A &\to& \tau_{\leq 2} \mathbf{\Pi}A &\to& \tau_{\leq 1} \mathbf{\Pi}A } \,, </annotation></semantics></math></div> <p>defines the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</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><mi>⋯</mi><mo>→</mo><msub><mi>A</mi> <mn>3</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>0</mn></msub><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> * \to \cdots \to A_3 \to A_2 \to A_1 \to A_0 = A </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Since our <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is assumed to be even <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected</a>, the Postnikov tower of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}(A)</annotation></semantics></math> is the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">LConst : \infty Grpd \to \mathbf{H}</annotation></semantics></math> of the ordinary <a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(A)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>. Accordingly, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mstyle mathvariant="bold"><mi>π</mi></mstyle> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>LConst</mi><msup><mi>B</mi> <mi>n</mi></msup><msub><mi>π</mi> <mi>n</mi></msub><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathbf{\pi}_n(A) = LConst B^n \pi_n \Pi(A)</annotation></semantics></math></p> <p>The point now is that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a> we may form smooh refinements of these discrete extensions: every discrete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n+1}\mathbb{Z}</annotation></semantics></math> we want to refine to a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math>. By the discussion at <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#GeomReal">geometric realization</a>, both have equivalent underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(\mathbf{B}^{n+1}\mathbb{Z}) \simeq \Pi(\mathbf{B}^n U(1)) \simeq K(\mathbb{Z},n+1) \,. </annotation></semantics></math></div> <p>For every direct summand abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> in one of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>π</mi></mstyle> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\pi}_n(A)</annotation></semantics></math> we can ask for a refinement of the cocycle from coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n \mathbb{Z}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z}</annotation></semantics></math>. This does not change the geometric realization, up to equivalence, but does change the smooth structure. And it allows to refine to differential coefficients by postcomposing further with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo>→</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>ℝ</mi></mrow><annotation encoding="application/x-tex">curv : \mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR}\mathbf{B}^{n}\mathbb{R}</annotation></semantics></math>.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><mn>∞</mn><mi>Lie</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">A = \mathbf{B}Spin \in \mathbf{H} = \infty Lie Grpd</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, we refine the internal Whitehead tower to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots \\ \mathbf{B}Fivebrane &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1) } \,, </annotation></semantics></math></div> <p>where the deloopings of the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> and the <a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a> appear.</p> <p>The result of such a smooth refinement is that we may apply the intrinsic curvature classes and the intrinsic de Rham theorem to obtain cocycles in realified cohomology, for instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><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><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>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}Spin \to \mathbf{B}^3 U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^4 U(1) = \mathbf{\flat}_{dR} \mathbf{B}^4 \mathbb{R} \,. </annotation></semantics></math></div> <p>If we have a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>-principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math>, we may form over it the covering circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-group bundles on which these higher cocycles naturally live</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>P</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>P</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <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></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P_2 &\to& \mathbf{B}Fivebrane &\to& * \\ \downarrow && \downarrow && \downarrow \\ P_1 &\to& \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& * \\ \downarrow && \downarrow && && \downarrow \\ X &\to& \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1) } \,. </annotation></semantics></math></div> <p>Here for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> an ordinary space, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>τ</mi> <mn>0</mn></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X = \tau_0 X</annotation></semantics></math>, the higher circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-group principal bundes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">P_k</annotation></semantics></math> have the property that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub><msub><mi>P</mi> <mi>k</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\tau_0 P_k = X</annotation></semantics></math>. Therefore the 0-truncation of the entire composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>8</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> P_1 \to \mathbf{B}^7 U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^8 \mathbb{R} </annotation></semantics></math></div> <p>defines a closed 8-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. This is the curvature characeristic form given by the Chern-Weil homomorphism in this degree. Its refinement to Deligne cohomology in this construction lives naturally not 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>, but on the covering <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>(…)</p> <h2 id="Examples">Examples</h2> <h3 id="principal_1bundles_with_connection">Principal 1-bundles with connection</h3> <p>We spell out here how the general theory of <a href="#InfinityLieAlgebraConnection">∞-Lie algebra valued connection</a> reduces to the standard notion of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> on ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s and how the <a href="#InfChernWeil">∞-Chern-Weil homomorphism</a> reduces on these to the ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a>.</p> <p>Let <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> be a (finite dimensional) <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>2</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the simply connected <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> integrating <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>.</p> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>The coefficient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> of <a href="#InfinityLieAlgebraConnection">genuine ∞-Lie algebra connections</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> is weakly equivalent to the simplicial presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><mi>Ξ</mi><mo stretchy="false">[</mo><mi>G</mi><mo>×</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munder><mover><mo>→</mo><mrow><msub><mi>Ad</mi> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi> <mn>1</mn></msub><mi>d</mi><msubsup><mi>p</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G_{conn} \stackrel{\simeq}{\to} \Xi[G\times \Omega^1(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1(-,\mathfrak{g})] </annotation></semantics></math></div> <p>that assigns objectwise the <a class="existingWikiWord" href="/nlab/show/groupoid+of+Lie-algebra+valued+1-forms">groupoid of Lie-algebra valued 1-forms</a>.</p> <p>This is moreover isomorphic to the simplicial presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><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><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cdots = [CartSp^{op},sSet](\mathbf{P}_1(-),\mathbf{B}G]) </annotation></semantics></math></div> <p>of morphisms out of the <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a>.</p> <p>The flat coefficient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{\flat}\mathbf{B}G</annotation></semantics></math> is modeled by the subobject</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">[</mo><mi>G</mi><mo>×</mo><msubsup><mi>Ω</mi> <mi>flat</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munder><mover><mo>→</mo><mrow><msub><mi>Ad</mi> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi> <mn>1</mn></msub><mi>d</mi><msubsup><mi>p</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover></mover><msubsup><mi>Ω</mi> <mi>flat</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Xi[G\times \Omega^1_{flat}(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1_{flat}(-,\mathfrak{g})] </annotation></semantics></math></div> <p>of groupoids of Lie-algebra valued forms with vanishing <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form.</p> <p>This is isomorphic to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><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><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cdots = [CartSp^{op},sSet](\mathbf{\Pi}_1(-),\mathbf{B}G]) </annotation></semantics></math></div> <p>of morphism out of the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The statements about morphisms out of the path groupoid are discussed in detail in <a href="http://arxiv.org/abs/0705.0452">SchrWalI</a>.</p> </div> <div class="un_cor"> <h6 id="corollary">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and <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> a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> we have a natural equivalence of groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi><msub><mi>Bund</mi> <mo>∇</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}G_{conn}) \simeq G Bund_\nabla(X) </annotation></semantics></math></div> <p>with the groupoid of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles with connection on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>For <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 stretchy="false">⟩</mo><mo>:</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>b</mi> <mi>p</mi></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\langle - \rangle : inn(\mathfrak{g}) \to b^p \mathbb{R}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>, the <a href="#InfChernWeil">induced morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>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><mo lspace="0em" rspace="thinmathspace">U</mo> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">⟩</mo></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [CartSp^{op}, sSet](C(\U_i\}), \mathbf{B}G_{diff}) \stackrel{\langle F_{(-)}\rangle}{\to} \Omega^{p+1}_{cl}(X) </annotation></semantics></math></div> <p>is that of the ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a>.</p> </div> <p>We have seen that a refinement of the Chern-Weil homomorphism is available. The above morphism extends to a morphism</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>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">⟩</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>τ</mi> <mn>1</mn></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \langle F_{(-)}\rangle : \mathbf{H}(-, \mathbf{B}G) \to \mathbf{H}(-, \tau_{1} \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R} ) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} = \infty LieGrpd</annotation></semantics></math> represented 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><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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>2</mn></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [CartSp^{op},sSet](-, \mathbf{B}G_{diff}) &\to& [CartSp^{op},sSet](-, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R}) \\ \downarrow^{\simeq} \\ [CartSp^{op},sSet](-, \mathbf{B}G) } \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a smooth manifold with good cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>2</mn></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [CartSp^{op},sSet](C\{U_i\}, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R})</annotation></semantics></math> is the groupoid whose objects are closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p+1</annotation></semantics></math>-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and whose morphisms are given by <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>-forms modulo exact forms.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math> range over a set of generators for all <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a>s. 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>H</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msub><mi>τ</mi> <mn>1</mn></msub><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \mathbf{H}(-,\mathbf{B}G) \to \prod_i \tau_1\mathbf{\flat}_{dR}\mathbf{B}^{n_i} \mathbb{R} </annotation></semantics></math></div> <p>is an approximation to the intrinsic Chern-character. We may consider its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>s over a given set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math> of curvature characteristic forms.</p> <p>Assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>,</mo><mo>∇</mo><mo>′</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">\nabla, \nabla' : C(\{U_i\}) \to \mathbf{B}G_{diff}</annotation></semantics></math> are two genuine connections with coinciding curvature characteristic classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Q</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Q_i\}</annotation></semantics></math>. Then in the homotopy fiber they are coboundant cocycles precisely if all the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CS</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mo>∇</mo><mo>,</mo><mo>∇</mo><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS_i(\nabla,\nabla')</annotation></semantics></math> vanish modulo an exact form.</p> <p>This equivalence relation is that which defines <a class="existingWikiWord" href="/nlab/show/Simons-Sullivan+structured+bundle">Simons-Sullivan structured bundle</a>s. Their <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> completion yields <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a>.</p> <h3 id="principal_2bundles_with_connection">Principal 2-bundles with connection</h3> <p>(…)</p> <p>Let <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> be a Lie <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> coming from a <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g}_2 \to \mathfrak{g}_1)</annotation></semantics></math>. Then we have two candidate Lie 2-groups integrating this: on the one hand the <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> coming from the <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_2 \to G_1)</annotation></semantics></math> that integrates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g}_2 \to\mathfrak{g}_1)</annotation></semantics></math> degreewise as ordinary Lie algebras, and on the other hand <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cosk</mi> <mi>k</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cosk_k\exp(\mathfrak{g})</annotation></semantics></math>.</p> <div class="un_prop"> <h6 id="proposition_2">Proposition</h6> <p>The morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>2</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau_2 \exp(\mathfrak{g}) \to \mathbf{B}(G_2 \to G_1) </annotation></semantics></math></div> <p>given by evaluating 2-dimensional <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> is a weak equivalence.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Use the 3-dimensional nonabelian Stokes theorem from the appendix of <a href="http://arxiv.org/abs/0802.0663">SchrWalII</a>.</p> </div> <div class="un_cor"> <h6 id="corollary_2">Corollary</h6> <p>The object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}(G_2 \to G_1)_{conn}</annotation></semantics></math> assigns to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/2-groupoid+of+Lie+2-algebra+valued+forms">2-groupoid of Lie 2-algebra valued forms</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>.</p> </div> <p>This is described in detail in <a href="http://arxiv.org/abs/0802.0663">SchrWalII</a>, subject to the extra constraint that the 2-form curvature vanishes.</p> <div class="un_cor"> <h6 id="corollary_3">Corollary</h6> <p>A genuine connection on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_2 \to G_1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> with given cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)</annotation></semantics></math> is a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)_{conn}</annotation></semantics></math> given as follows:</p> <ol> <li>on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> a pair of forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_i \in \Omega^1(U_i, \mathfrak{g}_1)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>𝔤</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_i \in \Omega^2(U_i, \mathfrak{g}_2)</annotation></semantics></math>;</li> </ol> <p>1 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j</annotation></semantics></math> a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>,</mo><msub><mi>G</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_{i j} \in C^\infty(U_{i}\cap U_j , G_1)</annotation></semantics></math> and a 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>,</mo><msub><mi>𝔤</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_{i j} \in \Omega^1(U_i \cap U_j, \mathfrak{g}_2)</annotation></semantics></math> such that …</p> <ol> <li>and so forth</li> </ol> </div> <p>This is described in detail in <a href="">SchrWalIII</a>, subject to the extra constraint that the 2-form curvature vanishes.</p> <p>(…)</p> <h3 id="DiffStringStruc">Twisted differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo>−</mo></mrow><annotation encoding="application/x-tex">String-</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math>-structures</h3> <p>We discuss now in detail refined Chern-Weil morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hat \mathbf{c} : \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^n U(1)) </annotation></semantics></math></div> <p>that send <a href="#InfinityLieAlgebraConnection">∞-connections</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s to <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> that represent a given <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)</annotation></semantics></math> with coefficients in the <a href="#http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1">circle n-groupoid</a>.</p> <p>Specifically, we consider the first two steps in the <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#SmoothWhitehead">smooth refinement of the Whitehead tower</a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> that are controled by <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a>.</p> <p>The smooth Whitehead tower of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a> starts as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \mathbf{B}Spin \to \mathbf{B} SO \to \mathbf{B}O \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} SO</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}O</annotation></semantics></math> classified by the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}O \to \mathbf{B} \mathbb{Z}_2</annotation></semantics></math> that sends an elemen <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>O</mi></mrow><annotation encoding="application/x-tex">k \in O</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">+1</annotation></semantics></math> if it is in the connected component of the identity and to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math> if it is not. This means we have an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} SO &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 } \; </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} Spin</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> is the is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} SO</annotation></semantics></math> classified by the <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} SO \to \mathbf{B}^2 \mathbb{Z}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \; </annotation></semantics></math></div></li> </ul> <p>Since these two steps are controled by the <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a>-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math> they have no nontrivial refinement to differential cohomology. The next step however is controled by what in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is the first <span class="newWikiWord">fractional Pontryagin class<a href="/nlab/new/fractional+Pontryagin+class">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>ℬ</mi><mi>Spin</mi><mo>→</mo><msup><mi>ℬ</mi> <mn>4</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1 : \mathcal{B} Spin \to \mathcal{B}^4 \mathbb{Z}</annotation></semantics></math> and which lifts through the <a class="existingWikiWord" href="/schreiber/show/path+%E2%88%9E-groupoid">path ∞-groupoid</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mn>∞</mn><mi>LieGrpd</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi : \infty LieGrpd \to \infty Grpd</annotation></semantics></math> to a characteristic class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a> (as discussed there) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2} p_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)</annotation></semantics></math> with coefficients in the smooth <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1">circle 3-groupoid</a>. This cocycle does arise as the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mu)</annotation></semantics></math> of the canonical <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra 3-cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>:</mo><mi>𝔰𝔬</mi><mo>→</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mu = \langle -,[-,-]\rangle: \mathfrak{so} \to b^2 \mathbb{R}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal 3-bundle</a> that this classifies is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} String</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) } \,. </annotation></semantics></math></div> <p>Notice that the fact that this is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> implies that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">X\in \mathbf{H} = \infty LieGrpd</annotation></semantics></math> also</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><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><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><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{ \mathbf{H}(X,\mathbf{B} String) &\to& * \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B} Spin) &\to& \mathbf{H}(X,\mathbf{B}^3 U(1)) } \,, </annotation></semantics></math></div> <p>which exhibits the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,\mathbf{B}String)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>s.</p> <p>As we refine in this diagram the bottom morphism to differential cohomology, we obtain correspondingly differential string structures.</p> <h4 id="StringStructure">The string-lifting Chern–Simons <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-bundle with connection</h4> <p>We describe the special case of the general <a href="#InfChernWeil"><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 homomorphism</a> for <a href="#InfinityLieAlgebraConnection"><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>-Lie algebra valued connections</a> corresponding to the <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1\colon \mathbf{B}Spin \to \mathbf{B}^3 U(1)</annotation></semantics></math>: 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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin</annotation></semantics></math>. The <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>-differential cocycle that it produces from a given <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> is the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+2-gerbe">Chern–Simons 2-bundle</a> with connection whose class is the obstruction for the existence of a <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>.</p> <p>The content of this subsection is at <a class="existingWikiWord" href="/nlab/show/Chern-Simons+2-gerbe">Chern–Simons 2-gerbe</a> in <a href="Chern-Simons+2-gerbe#InInfChernWeil">the section on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern–Weil theory</a>.</p> <h4 id="differential_string_structures">Differential string structures</h4> <p>The content of this section is at <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>.</p> <h4 id="FivebraneStructure">The Fivebrane-lifting Chern-Simons 7-bundle with connection</h4> <p>The content of this section is at <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle">Chern-Simons circle 7-bundle</a>.</p> <h4 id="DiffFivebraneStrucs">Differential fivebrane structures</h4> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mi>p</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{6}\hat p_2 : \mathbf{H}_{conn}(-,\mathbf{B}String) \to \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1)) </annotation></semantics></math></div> <p>be the differential refinement of the second fractional Pontryagin class discussed <a href="#FivebraneStructure">above</a>.</p> <p><strong>Definition</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">X \in \mathbf{H} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of <strong>differential fivebrane-structures</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fivebrane</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fivebrane_{diff}(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub><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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo><mo>→</mo><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>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{6}p_2(X) : \mathbf{H}(X,\mathbf{B}String) \to \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))</annotation></semantics></math>.</p> <p>More generally, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of <strong>twisted differential fivebrane structures</strong> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fivebrane</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fivebrane_{diff,tw}(X)</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Fivebrane</mi> <mrow><mi>diff</mi><mo>,</mo><mi>tw</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>H</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mi>p</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</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{ Fivebrane_{diff,tw}(X) &\to& H_{diff}(X,\mathbf{B}^7 U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}_{conn}(X,\mathbf{B}String) &\stackrel{\frac{1}{6}\hat p_2}{\to}& \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1)) } \,. </annotation></semantics></math></div> <p>In terms of the underlying <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>-Lie algebra valued local connection data, i.e. before Lie integration in the above sense , this has been considered in <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSIII">SSSIII</a></p> <p>(…)</p> <h3 id="chernsimons_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons theory</h3> <p>The refined higher Chern-Weil homomorphism takes values in <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>. Each of these comes with a notion of higher <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional curves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>n</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma_n \to X</annotation></semantics></math>. The map that takes a <a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">connection on an infinity-bundle</a> to this holonomy is a generalization of the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>.</p> <p>Therefore the higher Chern-Weil homomorphism defines a class of <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theories">quantum field theories</a> that we call</p> <p><a class="existingWikiWord" href="/schreiber/show/infinity-Chern-Simons+theory">infinity-Chern-Simons theory</a>.</p> <p>See there for more details.</p> <p>Special noteworthy cases are</p> <ul> <li> <p>the class of <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-models">AKSZ sigma-models</a> (<a href="#FRS">FRS11</a>)</p> </li> <li> <p>higher <a class="existingWikiWord" href="/nlab/show/Chern-Simons+supergravity">Chern-Simons supergravity</a></p> <p>(see <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a>)</p> </li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><strong>∞-Chern-Weil theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+models+and+general+theory">Principal ∞-bundles – models and general theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/Cech+cocycles+for+differential+characteristic+classes">Cech cocycles for differential characteristic classes</a></p> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/Twisted+differential+structures">Twisted differential structures</a></p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/Higher+Chern-Weil+Derivation+of+AKSZ+Sigma-Models">Higher Chern-Weil Derivation of AKSZ Sigma-Models</a></li> </ul> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>An explicit presentation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Weil homomorphism in terms of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> and the application to <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>s and <a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a>s is considered in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Cech+Cocycles+for+Differential+characteristic+Classes">Cech Cocycles for Differential characteristic Classes</a></em></li> </ul> <p>The special case that gives the <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a> is discussed in</p> <ul id="FRS"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/AKSZ+Sigma-Models+in+Higher+Chern-Weil+Theory">AKSZ Sigma-Models in Higher Chern-Weil Theory</a></em></li> </ul> <p>A general abstract account is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> <p>For a commented list of related literature see</p> <p><a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references">differential cohomology in cohesive topos – references</a></p> </body></html> </div> <div class="revisedby"> <p> Last revised on February 13, 2025 at 18:25:55. See the <a href="/nlab/history/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd" 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/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd/41" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd" accesskey="S" class="navlink" id="history" rel="nofollow">History (41 revisions)</a> <a href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd/cite" style="color: black">Cite</a> <a href="/nlab/print/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>