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id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#for_braided_groups'>For braided <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</a></li> <li><a href='#ByLieIntegration'>For <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 obtained by Lie integration</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#InfGaugeTrafo'>1-Morphisms: integration of infinitesimal gauge transformations</a></li> <ul> <li><a href='#observation'>Observation</a></li> <li><a href='#observation_2'>Observation</a></li> </ul> <li><a href='#ordinary_connections_on_principal_1bundles'>Ordinary connections on principal 1-bundles</a></li> <li><a href='#further_examples'>Further examples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In every <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> there is an <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#ChernWeilTheory">intrinsic notion of ∞-Chern-Weil theory</a> that gives rise to a notion of <em>connection</em> on <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s. We describe here details of the realization of this general abstract structure 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.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a>, a <em>connection</em> on a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> is a structure that supports the <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism+in+Smooth%E2%88%9EGrpd">Chern-Weil homomorphism in Smooth∞Grpd</a>: it interpolates between the <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><msubsup><mi>H</mi> <mi>smooth</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in H^1_{smooth}(X,G)</annotation></semantics></math> of the bundle and the refinements to <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> of its <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>es: the <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic class</a>es.</p> <p>This generalizes the notion of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> and the ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>.</p> <p>See the <a href="http://nlab.mathforge.org/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd#Motivation">Motivation section</a> at <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a> and the page <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory+introduction">∞-Chern-Weil theory introduction</a> for more background.</p> <h2 id="definition">Definition</h2> <h3 id="for_braided_groups">For braided <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</h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> equippd with <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \in Grp(\mathbf{H})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mi>𝔾</mi></msub><mo>=</mo><msub><mi>θ</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi></mrow><annotation encoding="application/x-tex"> curv_{\mathbb{G}} = \theta_{\mathbf{B}\mathbb{G}} \;\colon\; \mathbf{B}\mathbb{G} \to \flat_{dR}\mathbf{B}^2 \mathbb{G} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> on the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G} \in Grp(\mathbf{H})</annotation></semantics></math>.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi></mrow><annotation encoding="application/x-tex"> \Omega(-,\mathbb{G}) \to \flat_{dR}\mathbf{B}^2 \mathbb{G} </annotation></semantics></math></div> <p>be the morphism out of a <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> object which is universal with the property that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\Sigma \in \mathbf{H}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, the induced <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\Sigma, \Omega(-,\mathbb{G})] \to [\Sigma, \flat_{dR}\mathbf{B}^2 \mathbb{G}] </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/1-epimorphism">1-epimorphism</a>.</p> <p>Then write <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}\mathbb{G}_{conn}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> 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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>curv</mi> <mi>𝔾</mi></msub></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}\mathbb{G}_{conn} &\to& \Omega(-,\mathbb{G}) \\ \downarrow && \downarrow \\ \mathbf{B}\mathbb{G} &\stackrel{curv_{\mathbb{G}}}{\to}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \,. </annotation></semantics></math></div> <p>We say that <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}\mathbb{G}_{conn}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <strong><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">\mathbb{G}</annotation></semantics></math>-principal <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.</strong></p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = \mathbf{B}^{n-1}U(1)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a> the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math> is presented by the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>, hence is the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack for <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a>.</p> <h3 id="ByLieIntegration">For <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 obtained by Lie integration</h3> <p>We assume that the reader is familiar with the notation and constructions discussed at <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. The following definition may be understood as a direct generalization of the description of 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>-connections as cocycles in the stack <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> as discussed at <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, in view of the characterization of <a href="Weil+algebra#CharacterizationInSmoothTopos">Weil algebra in the smooth infinity-topos</a></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> CE <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_6c9e327ae6b83b653ce127201582016624056e5f_1"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> W <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_6c9e327ae6b83b653ce127201582016624056e5f_2"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/invariant+polynomials">invariant polynomials</a> inv</strong></p> <p><a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_6c9e327ae6b83b653ce127201582016624056e5f_3"><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 class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> (<a href="Weil+algebra#FreedHopkins13">Freed-Hopkins 13</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_6c9e327ae6b83b653ce127201582016624056e5f_4"><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><msubsup><mi>Ω</mi> <mfrac linethickness="0"><mrow><mi>li</mi></mrow><mrow><mi>cl</mi></mrow></mfrac> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Ω</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><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><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Ω</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\simeq& \Omega^\bullet(\mathbf{B}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G) } </annotation></semantics></math></div></div> <p>We discuss now connections on those <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 which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">G \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> is an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">smooth ∞-group</a> that arises from <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><mover><mo>↪</mo><mi>CE</mi></mover></mrow><annotation encoding="application/x-tex">\mathfrak{g} \in L_\infty \stackrel{CE}{\hookrightarrow} </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dgAlg">dgAlg</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">{}^{op}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> over the <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>s and of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> with <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{g})</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{g})</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, write <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>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>dgAlg</mi></mrow><annotation encoding="application/x-tex">\Omega^\bullet(X) \in dgAlg</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> of smooth <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s. 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></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> let <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> be the standard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> regarded as a smooth <a class="existingWikiWord" href="/nlab/show/manifold+with+corners">manifold with corners</a> in the standard way. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>si</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{si}(X \times \Delta^k)</annotation></semantics></math> for the sub-<a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> of differential forms with sitting instants perpendicular to the boundary of the simplex, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{si,vert}(X\times \Delta^k)</annotation></semantics></math> for the further sub-dg-algebra of <a class="existingWikiWord" href="/nlab/show/vertical+differential+form">vertical differential form</a>s with respect to the canonical projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times \Delta^k \to X</annotation></semantics></math>.</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>A morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>←</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/dgAlg">dgAlg</a> we call an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+valued+differential+forms">L-∞ algebra valued differential form</a> with values in <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>, dually a morphism of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>inn</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A : T X \to inn(\mathfrak{g}) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a> to the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">inner automorphism ∞-Lie algebra</a>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> is the composite of morphisms of <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mi>A</mi></mover><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mover><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>:</mo><msub><mi>F</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \mathfrak{g}^*[2] : F_{A} </annotation></semantics></math></div> <p>that injects the shifted generators into the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>.</p> <p>Precisely if the curvatures vanish does the morphism factor through the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</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> <mi>A</mi></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><msub><mi>A</mi> <mi>flat</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>A</mi></mover></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> (F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right) </annotation></semantics></math></div> <p>in which case we call <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> <strong>flat</strong>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>s 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> are the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mi>A</mi></mover><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><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><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{\langle F_{(-)} \rangle}{\leftarrow} inv(\mathfrak{g}) : \langle F_A\rangle \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>→</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">inv(\mathfrak{g}) \to W(\mathfrak{g})</annotation></semantics></math> is the inclusion of the <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a>s.</p> </div> <p>We define now <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>smooth</mi></msub><mo>↪</mo></mrow><annotation encoding="application/x-tex">{}_{smooth} \hookrightarrow </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>s and <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s between them.</p> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]</annotation></semantics></math> for the simplicial presheaf given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>{</mo><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><msub><mi>A</mi> <mi>vert</mi></msub></mrow></mover><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \exp(\mathfrak{g}) : (U,[k]) \mapsto \left\{ \Omega^\bullet_{si,vert}(U \times\Delta^k) \stackrel{A_{vert}}{\leftarrow} CE(\mathfrak{g}) \right\} </annotation></semantics></math></div> <p>(the untruncated <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</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>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo>∈</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{diff} \in [CartSp_{smooth}^{op}, sSet]</annotation></semantics></math> for the simplicial presheaf given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo>:</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>A</mi> <mi>vert</mi></msub></mrow></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mi>si</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>A</mi></mover></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp(\mathfrak{g})_{diff} : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times\Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right\} \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo>∈</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{ChW} \in [CartSp_{smooth}^{op}, sSet]</annotation></semantics></math> for the simplicial presheaf given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo>:</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msubsup><mi>Ω</mi> <mrow><mi>si</mi><mo>,</mo><mi>vert</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>A</mi> <mi>vert</mi></msub></mrow></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mi>si</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>A</mi></mover></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</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></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo></mrow></mover></mtd> <mtd><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp(\mathfrak{g})_{ChW} : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times\Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) } \right\} \,. </annotation></semantics></math></div> <p>Define the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{conn}</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>↦</mo><mrow><mo>{</mo><msubsup><mi>Ω</mi> <mi>si</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mover><mo>←</mo><mi>A</mi></mover><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∀</mo><mi>v</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>:</mo><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \exp(\mathfrak{g})_{conn}(U) : [k] \mapsto \left\{ \Omega^\bullet_{si}(U \times \Delta^k) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \;\; | \;\; \forall v \in \Gamma(T \Delta^k) : \iota_v F_A = 0 \right\} </annotation></semantics></math></div></div> <p>Here on the right we have in each case the <a class="existingWikiWord" href="/nlab/show/set">set</a>s of horizontal morphisms in <a class="existingWikiWord" href="/nlab/show/dgAlg">dgAlg</a> that make <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a>s in <a class="existingWikiWord" href="/nlab/show/dgAlg">dgAlg</a> as indicated, with the vertical morphisms being the canonical projections and inclusions. The functoriality in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f : K \to U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>l</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\rho : [k] \to [l]</annotation></semantics></math> is by the evident precomposition with the pullback of differential forms <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><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>K</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U \times \Delta^k) \stackrel{(f,id)^*}{\to} \Omega^\bullet(K \times \Delta^k)</annotation></semantics></math> and <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>l</mi></msup><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>ρ</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U \times \Delta^l) \stackrel{(id,\rho)^*}{\leftarrow} \Omega^\bullet(U, \times \Delta^k)</annotation></semantics></math>.</p> <div class="un_prop" id="SequenceOfInclusionsOfCoefficients"> <h6 id="proposition">Proposition</h6> <p>There are canonical morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op},sSet]</annotation></semantics></math> between these objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \exp(\mathfrak{g})_{conn} &\hookrightarrow& \exp(\mathfrak{g})_{ChW} &\hookrightarrow& \exp(\mathfrak{g})_{diff} \\ && && \downarrow \\ && && \exp(\mathfrak{g}) } \,, </annotation></semantics></math></div> <p>where the horizontal morphisms are <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> and the vertical morphism is over each <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> an equivalence of <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> (it is a weak equivalence between fibrant objects in the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>).</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub><mo>↪</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>dff</mi></msub></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{ChW} \hookrightarrow \exp(\mathfrak{g})_{dff}</annotation></semantics></math> is clear. The weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo>→</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{diff} \to \exp(\mathfrak{g})</annotation></semantics></math> is discussed at <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> (but is also directly verified).</p> <p>To see the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>↪</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{conn} \hookrightarrow \exp(\mathfrak{g})_{ChW}</annotation></semantics></math> we need to check that the horizonality condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\iota_v F_A = 0</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued form <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> for all <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> tangent to the simplex implies that all the <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle F_A\rangle</annotation></semantics></math> are <em>basic forms</em> that “descend to <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>”, hence that are in the image of the inclusion <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 stretchy="false">)</mo><mo>→</mo><msubsup><mi>Ω</mi> <mi>si</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U) \to \Omega^\bullet_{si}(U \times \Delta^k)</annotation></semantics></math>.</p> <p>For this it is sufficient to show that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma(T \Delta^k)</annotation></semantics></math> we have</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></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_v \langle F_A \rangle = 0</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></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">\mathcal{L}_v \langle F_A \rangle = 0</annotation></semantics></math></p> </li> </ol> <p>where in the second line we have the <a class="existingWikiWord" href="/nlab/show/Lie+derivative">Lie derivative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{L}_v</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>.</p> <p>The first condition is evidently satisfied if already <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\iota_v F_A = 0</annotation></semantics></math>. The second condition follows with <a class="existingWikiWord" href="/nlab/show/Cartan+calculus">Cartan calculus</a> and using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo 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">d_{dR} \langle F_A\rangle = 0</annotation></semantics></math> (which holds as a consequence of the definition of <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><msub><mi>ℒ</mi> <mi>v</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>v</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>v</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}_v \langle F_A \rangle = d \iota_v \langle F_A \rangle + \iota_v d \langle F_A \rangle = 0 \,. </annotation></semantics></math></div></div> <div class="un_lemma"> <h6 id="remark">Remark</h6> <p>For a general <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">F_A</annotation></semantics></math> themselves are not necessarily closed (rather they satisfy the <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a>), hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>s: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.</p> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>sSet</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\mathbf{cosk}_{n+1} : sSet \to sSet</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/simplicial+coskeleton">simplicial coskeleton</a> functor. Its prolongation to simplicial presheaves we denote here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\tau_n</annotation></semantics></math> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>n</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \tau_n \exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet] </annotation></semantics></math></div> <p>etc. This is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></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> <mi>n</mi></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"> \tau_n \exp(\mathfrak{g}) = \mathbf{B}G </annotation></semantics></math></div> <p>of the universal <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</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> to an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">smooth n-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <div class="un_def"> <h6 id="definition_4">Definition</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \in [CartSp_{smooth}^{op}, sSet]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\hat X \to X</annotation></semantics></math> any cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> in the local projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> (see <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> for details), we say that the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/hom+object">hom object</a></p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mi>τ</mi> <mi>n</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g}))</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</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>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mi>τ</mi> <mi>n</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})_{diff})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</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> equipped with <strong>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>-connection</strong>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mi>τ</mi> <mi>n</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})_{conn})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</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> equipped with <strong><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>-connection</strong>.</p> </li> </ul> </div> <div class="un_remark"> <h6 id="remark_2">Remark</h6> <p>In view of this definition we may read the <a href="#SequenceOfInclusionsOfCoefficients">above</a> sequence of morpisms of coefficient objects as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>genuine</mi><mspace width="thickmathspace"></mspace><mi>connections</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>ChW</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>pseudo</mi><mo>−</mo><mi>connection</mi><mspace width="thickmathspace"></mspace><mi>with</mi><mspace width="thickmathspace"></mspace><mi>global</mi><mspace width="thickmathspace"></mspace><mi>curvature</mi><mspace width="thickmathspace"></mspace><mi>characteristics</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>pseudo</mi><mo>−</mo><mi>connections</mi></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><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>bare</mi><mi>bundles</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \exp(\mathfrak{g})_{conn} &&& genuine\;connections \\ \downarrow \\ \exp(\mathfrak{g})_{ChW} &&& pseudo-connection\;with\;global\;curvature\;characteristics \\ \downarrow \\ \exp(\mathfrak{g})_{diff} &&& pseudo-connections \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) &&& bare bundles } \,, </annotation></semantics></math></div> <p>As we shall see in more detail below, the components of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connection in terms of the above diagrams we may think of as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><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></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>A</mi> <mi>vert</mi></msub></mrow></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>gauge</mi><mspace width="thickmathspace"></mspace><mi>transformation</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>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>A</mi></mover></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>𝔤</mi><mo>−</mo><mi>valued</mi><mspace width="thickmathspace"></mspace><mi>form</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><mover><mo>←</mo><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy="false">⟩</mo></mrow></mover></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></mrow><annotation encoding="application/x-tex"> \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& gauge\;transformation \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& \mathfrak{g}-valued\;form \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;forms } </annotation></semantics></math></div></div> <div class="un_remark"> <h6 id="remark_3">Remark</h6> <p>In full <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a> the fundamental object of interest is really <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{diff}</annotation></semantics></math> – the object of <a class="existingWikiWord" href="/nlab/show/pseudo-connection">pseudo-connection</a>s, which serves as the correspondence object for an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-anafunctor">∞-anafunctor</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})</annotation></semantics></math> that presents the differential characteristic classes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})</annotation></semantics></math>. From an abstract point of view the other objects only serve the purpose of picking particularly nice representatives.</p> <p>This distinction is important: over objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> that are not <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s but for instance <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>s, the genuine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-connections for general higher <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> do <em>not</em> exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative.</p> </div> <h2 id="examples">Examples</h2> <h3 id="InfGaugeTrafo">1-Morphisms: integration of infinitesimal gauge transformations</h3> <p>The 1-<a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{conn}(U)</annotation></semantics></math> may be thought of as <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued forms. We unwind what these look like concretely.</p> <div class="un_defn"> <h6 id="definition_5">Definition</h6> <p>Given a 1-morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})(X)</annotation></semantics></math>, represented by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued forms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>←</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(U \times \Delta^1) \leftarrow W(\mathfrak{g}) : A </annotation></semantics></math></div> <p>consider the unique decomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi>A</mi> <mi>U</mi></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>vert</mi></msub><mo>:</mo><mo>=</mo><mi>λ</mi><mo>∧</mo><mi>d</mi><mi>s</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A = A_U + ( A_{vert} := \lambda \wedge d s) \; \; \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">A_U</annotation></semantics></math> the horizonal differential form component and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">s : \Delta^1 = [0,1] \to \mathbb{R}</annotation></semantics></math> the canonical <a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a>.</p> <p>We call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> the <strong>gauge parameter</strong> . This is a function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^1</annotation></semantics></math> with values in 0-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> 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>, plus 1-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> 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> a <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a>, plus 2-forms for a Lie 3-algebra, and so forth.</p> </div> <p>We describe now how this enccodes a gauge transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><msub><mi>A</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lambda : A_0(s=0) \stackrel{}{\to} A_U(s = 1) \,. </annotation></semantics></math></div> <div class="un_lemma"> <h6 id="observation">Observation</h6> <p>We have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>s</mi></mrow></mfrac><msub><mi>A</mi> <mi>U</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>U</mi></msub><mi>λ</mi><mo>+</mo><mo stretchy="false">[</mo><mi>λ</mi><mo>∧</mo><mi>A</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>λ</mi><mo>∧</mo><mi>A</mi><mo>∧</mo><mi>A</mi><mo stretchy="false">]</mo><mo>+</mo><mi>⋯</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>s</mi></msub></mrow></msub><msub><mi>F</mi> <mi>A</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \frac{d}{d s} A_U = (d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots) + \iota_{\partial_s} F_A \,, </annotation></semantics></math></div> <p>where the sum is over all higher brackets of the <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is the result of applying the contraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><mo>∂</mo><mi>s</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\iota_{\partial s}</annotation></semantics></math> to the defining equation for the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">F_A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> using the nature of the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>A</mi><mo>+</mo><mo stretchy="false">[</mo><mi>A</mi><mo>∧</mo><mi>A</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>A</mi><mo>∧</mo><mi>A</mi><mo>∧</mo><mi>A</mi><mo stretchy="false">]</mo><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> F_A = d_{dR} A + [A \wedge A] + [A \wedge A \wedge A] + \cdots </annotation></semantics></math></div> <p>and inserting the above decomposition for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </div> <div class="un_def"> <h6 id="definition_6">Definition</h6> <p>Define the <strong><a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> of the gauge parameter</strong> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>A</mi></msub><mi>λ</mi><mo>:</mo><mo>=</mo><mi>d</mi><mi>λ</mi><mo>+</mo><mo stretchy="false">[</mo><mi>A</mi><mo>∧</mo><mi>λ</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>A</mi><mo>∧</mo><mi>A</mi><mo>∧</mo><mi>λ</mi><mo stretchy="false">]</mo><mo>+</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_A \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots \,. </annotation></semantics></math></div></div> <p>In this notation we have</p> <ul> <li> <p>the general identity</p> <div class="maruku-equation" id="eq:ShiftedGaugeTrafo"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>s</mi></mrow></mfrac><msub><mi>A</mi> <mi>U</mi></msub><mo>=</mo><mo>∇</mo><mi>λ</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>A</mi></msub><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex"> \frac{d}{d s} A_U = \nabla \lambda + (F_A)_s </annotation></semantics></math></div></li> <li> <p>the <strong>horizontality</strong> or <strong><a class="existingWikiWord" href="/nlab/show/rheonomy">rheonomy</a></strong> constraint or <strong><a class="existingWikiWord" href="/nlab/show/Ehresmann+connection">second Ehresmann condition</a></strong> <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>s</mi></msub></mrow></msub><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\iota_{\partial_s} F_A = 0</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a></p> <div class="maruku-equation" id="eq:GaugeTrafo"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>s</mi></mrow></mfrac><msub><mi>A</mi> <mi>U</mi></msub><mo>=</mo><mo>∇</mo><mi>λ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{d}{d s} A_U = \nabla \lambda \,. </annotation></semantics></math></div></li> </ul> <p>This is known as the equation for <strong>infinitesimal <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s</strong> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra valued form.</p> <div class="un_lemma"> <h6 id="observation_2">Observation</h6> <p>By <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>vert</mi></msub></mrow><annotation encoding="application/x-tex">A_{vert}</annotation></semantics></math> – and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> – defines an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\lambda)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a> that integrates <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 unique solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_U(s = 1)</annotation></semantics></math> of the above <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">s = 1</annotation></semantics></math> for the initial values <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_U(s = 0)</annotation></semantics></math> we may think of as the result of acting on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_U(0)</annotation></semantics></math> with the gauge transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\lambda)</annotation></semantics></math>.</p> </div> <h3 id="ordinary_connections_on_principal_1bundles">Ordinary connections on principal 1-bundles</h3> <div class="un_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(connections on ordinary bundles)</strong></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">\mathfrak{g}</annotation></semantics></math> an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> with 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> and for <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> the <a class="existingWikiWord" href="/nlab/show/groupoid+of+Lie+algebra-valued+forms">groupoid of Lie algebra-valued forms</a> we have an equivalence</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>1</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \tau_1 \exp(\mathfrak{g})_{conn} \simeq \mathbf{B}G_{conn} </annotation></semantics></math></div> <p>betweenn the 1-truncated coefficient object 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>-valued <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 the coefficient objects for ordinary <a class="existingWikiWord" href="/nlab/show/connections+on+a+bundle">connections on a bundle</a> (see there).</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Notice that the sheaves of <a class="existingWikiWord" href="/nlab/show/object">object</a>s on both sides are manifestly isomorphic, both are the sheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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></mrow><annotation encoding="application/x-tex">\Omega^1(-,\mathfrak{g})</annotation></semantics></math>.</p> <p>On <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s, we have by the <a href="#InfinitesimalGaugeTransformations">above</a> for a form <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> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>←</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U \times \Delta^1) \leftarrow W(\mathfrak{g}) : A</annotation></semantics></math> decomposed into a horizontal and a verical pice as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi>A</mi> <mi>U</mi></msub><mo>+</mo><mi>λ</mi><mo>∧</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">A = A_U + \lambda \wedge d t</annotation></semantics></math> that the condition <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><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\iota_{\partial_t} F_A = 0</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>s</mi></mrow></mfrac><mi>A</mi><mo>=</mo><msub><mi>d</mi> <mi>U</mi></msub><mi>λ</mi><mo>+</mo><mo stretchy="false">[</mo><mi>λ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{\partial}{\partial s} A = d_U \lambda + [\lambda, A] \,. </annotation></semantics></math></div> <p>For any initial value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(0)</annotation></semantics></math> this has the unique solution</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>A</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mi>d</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Ad</mi> <mrow><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>θ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} A(t) & = g(t)^{-1} (A + d_{U}) g(t) \\ & = Ad_{g(t)}(A) + g(t)^* \theta \end{aligned} </annotation></semantics></math></div> <p>(with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</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>), where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \in C^\infty([0,1], G)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>s</mi></mrow></mfrac><mrow><mo>(</mo><msub><mi>g</mi> <mo stretchy="false">(</mo></msub><mi>t</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mi>d</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mi>d</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mi>λ</mi><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>λ</mi><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mi>d</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \frac{\partial}{\partial s} \left( g_(t)^{-1} (A + d_{U}) g(t) \right) \\ & = g(t)^{-1} (A + d_{U}) \lambda g(t) - g(t)^{-1} \lambda (A + d_{U}) g(t) \end{aligned} </annotation></semantics></math></div> <p>(where for ease of notaton we write actions as if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> were a <a class="existingWikiWord" href="/nlab/show/matrix+Lie+group">matrix Lie group</a>).</p> <p>This implies that the endpoints of the path 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>-valued 1-forms are related by the usual cocycle condition 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>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mi>d</mi> <mi>U</mi></msub><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A(1) = g(1)^{-1} (A + d_U) g(1) \,. </annotation></semantics></math></div> <p>In the same fashion one sees that given 2-cell in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})(U)</annotation></semantics></math> and any 1-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> at one vertex, there is a unique lift to a 2-cell in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{g})_{conn}</annotation></semantics></math>, obtained by parallel transporting the form around. The claim then follows from the previous statement of <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>1</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">\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G</annotation></semantics></math>.</p> </div> <h3 id="further_examples">Further examples</h3> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued differential form in the sense described here is precisely an <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra+valued+form">Lie 2-algebra valued form</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">b^{n-1}\mathbb{R}</annotation></semantics></math>-valued differential form is the same as an ordinary differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form.</p> </li> <li> <p>What is called an “extended soft group manifold” in the literature on the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> is precisely a collection 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>-Lie algebroid valued forms with values in a super <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 such as the</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>/<a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> (for 11-dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>). The way <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> and <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a> are read off from “extded soft group manifolds” in this literature is – apart from this difference in terminology – precisely what is described above.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a 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+a+gerbe">connection on a gerbe</a> / <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+gerbe">connection on a bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+3-bundle">connection on a 3-bundle</a> / <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+2-gerbe">connection on a bundle 2-gerbe</a></p> </li> <li> <p><strong>connection on an ∞-bundle</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connection">flat ∞-connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjusted+Weil+algebra">adjusted Weil algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a>:</th><th>standard <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th>groupoid version of <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>:</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_1"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_2"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_3"><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} \mathbb{G}_{conn}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> without top-degree connection form</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <blockquote> <p>See also the rereferences at <em><a class="existingWikiWord" href="/nlab/show/nonabelian+differential+cohomology">nonabelian differential cohomology</a></em>.</p> </blockquote> <p>The local differential form data 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 was introduced in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="https://golem.ph.utexas.edu/category/2007/10/obstructions_to_nbundle_lifts.html">Obstructions to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Bundle Lifts Part II</a></em> (Oct 2007) [diagram: <a class="existingWikiWord" href="/nlab/files/Schreiber-PrincipalInfinityConnections-2007.pdf" title="pdf">pdf</a>]</p> </li> <li id="SSSI"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/L-infinity+algebra+connections">L-∞ algebra connections and applications to String- and Chern-Simons n-transport</a></em>, in: <em>Quantum Field Theory</em>, Birkhäuser (2009) 303-424 (<a href="http://arxiv.org/abs/0801.3480">arXiv:0801.3480</a>, <a href="https://doi.org/10.1007/978-3-7643-8736-5_17">doi:10.1007/978-3-7643-8736-5_17</a>)</p> </li> <li id="SSSIII"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em>, Communications in Mathematical Physics, October 2012, Volume 315, Issue 1, pp 169-213 (<a href="http://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>, <a href="https://link.springer.com/article/10.1007/s00220-012-1510-3">doi:10.1007/s00220-012-1510-3</a>)</p> </li> </ul> <p>The global description was then introduced 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> – An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie theoretic construction</em>, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (<a href="http://arxiv.org/abs/1011.4735">arXiv:1011.4735</a>, <a href="http://projecteuclid.org/euclid.atmp/1358950853">euclid:atmp/1358950853</a>)</li> </ul> <p>A more comprehensive account is in sections 3.9.6, 3.9.7 of</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 further developments see the references at <em><a class="existingWikiWord" href="/nlab/show/adjusted+Weil+algebra">adjusted Weil algebra</a></em>.</p> <p>Approach via splitting of higher <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroids">Atiyah Lie algebroids</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <em>Infinitesimal Higher Symmetries and Higher Connections</em>, <a href="CQTS#BunkMar24">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (27 Mar 2024) [slides:<a class="existingWikiWord" href="/nlab/files/Bunk-InfinitesimalHigherSymmetries.pdf" title="pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 2, 2024 at 06:49:33. 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