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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> <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> / <strong>connection on a bundle gerbe</strong></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><a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> </ul> <hr /> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#for_line_bundle_gerbes'>For line bundle gerbes</a></li> <li><a href='#for_principal_bundle_gerbes'>For principal bundle gerbes</a></li> </ul> <li><a href='#surface_transport'>Surface transport</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#further_references'>Further references</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>connection on a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></em> is a slight variant of a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a>-realization of a degree 3 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> cocycle.</p> <blockquote> <p>old content, needs to be polished</p> </blockquote> <p>Like a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on a locally trivialized bundle is encoded in a Lie algebra-valued connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, the connection on a bundle gerbe gives rise to a Lie-algebra valued <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (this shift in degree is directly related to the step from second to third integral cohomology). This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form is sometimes addressed as the <em>curving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form</em> of a bundle gerbe.</p> <p>But there is more data necessary to describe a connection on a bundle gerbe. The details of the definition – which is evident for line bundle gerbes but more involved for principal bundle gerbes – can be naturally derived from a functorial concept of parallel surface transport, just like connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-forms on bundles can be derived from parallel line transport.</p> <h3 id="definitions">Definitions</h3> <h4 id="for_line_bundle_gerbes">For line bundle gerbes</h4> <p>A connection (also known as “connection and curving”) on a line bundle gerbe</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mover><mo>→</mo><mi>p</mi></mover><msup><mi>Y</mi> <mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mover><mo>→</mo><mo>→</mo></mover><mi>Y</mi><mover><mo>→</mo><mi>π</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> B \stackrel{p}{\to} Y^{[2]} \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X </annotation></semantics></math></div> <p>is</p> <ul> <li> <p>a 2-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B \in \Omega^2(Y) </annotation></semantics></math></div></li> <li> <p>a connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> on the line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><msup><mi>Y</mi> <mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">B \to Y^{[2]}</annotation></semantics></math></p> </li> <li> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>B</mi><mspace width="thickmathspace"></mspace><mo>−</mo><mspace width="thickmathspace"></mspace><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>B</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>F</mi> <mo>∇</mo></msub></mrow><annotation encoding="application/x-tex"> \pi_1^*B \; -\; p_2^*B \;=\; F_\nabla </annotation></semantics></math></div></li> <li> <p>together with an extension of the bundle gerbe product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> to an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mo>∇</mo></msub><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>p</mi> <mn>12</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊗</mo><msubsup><mi>p</mi> <mn>23</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><msubsup><mi>p</mi> <mn>13</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mu_\nabla \;:\; p_{12}^* (B,\nabla) \;\; \otimes p_{23}^* (B,\nabla) \;\to\; p_{13}^* (B,\nabla) </annotation></semantics></math></div> <p>of line bundles with connection.</p> </li> </ul> <p>Notice that this condition ensures that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">d B</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> which agrees on double intersections</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>d</mi><mi>B</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>d</mi><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p_1^* d B \;\; = \;\; p_2^* d B \,. </annotation></semantics></math></div> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">d B</annotation></semantics></math> actually descends to a 3-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The <strong>curvature</strong> associated with the connection on a line bundle gerbe is the unique 3-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H \in \Omega^3(X) </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mo>*</mo></msup><mi>H</mi><mo>=</mo><mi>d</mi><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi^* H = d B \,. </annotation></semantics></math></div> <p>The deRham class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[H]</annotation></semantics></math> of this 3-form is the image in real cohomology of the class in integral coholomology classifying the bundle gerbe.</p> <h4 id="for_principal_bundle_gerbes">For principal bundle gerbes</h4> <p>A connection on 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>-principal bundle gerbe is</p> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Lie</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Lie}(G)</annotation></semantics></math>-valued 2-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi mathvariant="normal">Lie</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B \in \Omega^2(Y,\mathrm{Lie}(G)) </annotation></semantics></math></div></li> <li> <p>together with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Lie</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Lie}(\mathrm{Aut}(G))</annotation></semantics></math>-valued 1-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</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>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi mathvariant="normal">Lie</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Aut</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \in \Omega^1(Y,\mathrm{Lie}(\mathrm{Aut}(G))) </annotation></semantics></math></div></li> <li> <p>and a certain twisted notion of connection on the <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>-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></p> </li> <li> <p>satisfying a couple of conditions that reduce to those described above in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = U(1)</annotation></semantics></math>.</p> </li> </ul> <p>For the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub><mo>+</mo><mi mathvariant="normal">ad</mi><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F_{A} + \mathrm{ad} B = 0</annotation></semantics></math>, these conditions are nothing but a tetrahedron law on a 2-functor from 2-paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>G</mi><mi mathvariant="normal">BiTor</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma(G\mathrm{BiTor})</annotation></semantics></math>. This is discussed in <a href="http://arxiv.org/abs/math.DG/0511710">math.DG/0511710</a>.</p> <p>For the more general case a choice for these conditions that harmonizes with the conditions found for (proper) gerbes with connection by Breen & Messing in <a href="http://arxiv.org/abs/math.AG/0106083">math.AG/0106083</a> has been given by Aschieri, Cantini & Jurčo in<br /><a href="http://arxiv.org/abs/hep-th/0312154">hep-th/0312154</a>.</p> <h3 id="surface_transport">Surface transport</h3> <p>From a line bundle gerbe with connection one obtains a notion of <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> along surfaces in a way that generalizes the procedure for locally trivialized fiber bundles with connection.</p> <p>Recall that in the case of fiber bundles, the holonomy associated to a based loop <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> is obtained by</p> <ul> <li> <p>choosing a triangulation of the loop (i.e., a decomposition into intervals) such that each vertex sits in a double intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">U_{ij}</annotation></semantics></math> and such that each edge sits in a patch <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math></p> </li> <li> <p>choosing for each edge a lift into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mo>⊔</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Y = \sqcup_i U_i</annotation></semantics></math></p> </li> <li> <p>choosing for each vertex a lift into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>=</mo><msub><mo>⊔</mo> <mi>ij</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">Y^{[2]} = \sqcup_{ij} U_i\cap U_j</annotation></semantics></math></p> </li> <li> <p>assigning to each edge lifted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> the transport computed from the connection 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></p> </li> <li> <p>assigning to each vertex lifted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j</annotation></semantics></math> the value of the transition function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">g_{ij}</annotation></semantics></math> at that point</p> </li> <li> <p>multiplying these data in the order given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> .</p> </li> </ul> <p>For bundle gerbes this generalizes to a procedure that assigns a triangulation to a closed surface, that lifts faces, edges, and vertices to single, double and triple intersections, respectively, and which assigns the exponentiated integrals of the 2-form over faces, of the connection 1-form over edges, and assigns the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>ijk</mi></msub></mrow><annotation encoding="application/x-tex">\mu_{ijk}</annotation></semantics></math> to vertices.</p> <p>For the abelian case (line bundle gerbes) this procedure has been first described in</p> <ul> <li>K. Gawedzki & N. Reis, <em>WZW branes and Gerbes</em> (<a href="http://arxiv.org/abs/hep-th/0205233">arXiv</a>)</li> </ul> <p>based on</p> <ul> <li>O. Alvarez, <em>Topological quantization and cohomology</em> Commun. Math. Phys. 100 (1985), 279-309.</li> </ul> <p>Further discussion can be found in</p> <ul> <li>A. Carey, S. Johnson & M. Murray, <em>Holonomy on D-branes</em>, (<a href="http://arxiv.org/abs/hep-th/0204199">arXiv</a>)</li> </ul> <p>Gawedzki and Reis showed this way that the Wess-Zumino term in the WZW-model is nothing but the surface holonomy of a (line bundle) gerbe.</p> <p>In terms of string physics this means that the string (the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-particle) couples to the Kalb-Ramond field – which hence has to be interpreted as the connection (“and curving”) of a gerbe – in a way that categorifies the coupling of the electromagnetically charged (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-)particle to a line bundle.</p> <p>The necessity to interpret the Kalb–Ramond field as a connection on a gerbe was originally discussed in</p> <ul> <li>D. Freed and E. Witten <em>Anomalies in string theory with D-branes</em>, Asian J. Math. 3 (1999), 819-851 (<a href="http://arxiv.org/abs/hep-th/9907189">arXiv</a>)</li> </ul> <p>Underlying the Gawedzki–Reis formula is a general mechanism of transition of transport <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functors, described in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Connections on non-abelian gerbes and their holonomy</em>, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (<a href="http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html">TAC</a>, <a href="http://arxiv.org/abs/0808.1923">arXiv:0808.1923</a>, <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SchrWalII+III">web</a>)</li> </ul> <p>and similarly in</p> <ul> <li>Joao Faria Martins, Roger Picken, <em>A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces</em> (<a href="http://arxiv.org/abs/0808.3964">arXiv</a>)</li> </ul> <p>This applies to more general situations than ordinary line bundle gerbes with connection.</p> <p>The generalization (“<a class="existingWikiWord" href="/nlab/show/Jandl+gerbes">Jandl gerbes</a>”) to unoriented surfaces (hence to type I strings)</p> <ul> <li>K. Waldorf, C. Schweigert & U. S., <em>Unoriented WZW Models and Holonomy of Bundle Gerbes</em> (<a href="http://arxiv.org/abs/hep-th/0512283">arXiv</a>)</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> </ul> <h2 id="further_references">Further references</h2> <p>(…)</p> <p>In relation to <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Byungdo+Park">Byungdo Park</a>, <em>Differential cohomology and gerbes: An introduction to higher differential geometry</em>, lecture notes (2023) [<a href="https://byungdo.github.io/seminars/IMSRS.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Park-DifferentialCohomology.pdf" title="pdf">pdf</a>]</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> and <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> in <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> (for <a class="existingWikiWord" href="/nlab/show/bundle+gerbes+with+connection">bundle gerbes with connection</a>) over <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ernesto+Lupercio">Ernesto Lupercio</a>, <a class="existingWikiWord" href="/nlab/show/Bernardo+Uribe">Bernardo Uribe</a>, <em>Holonomy for Gerbes over Orbifolds</em>, J. Geom.Phys. <strong>56</strong> (2006) 1534-1560 [<a href="https://arxiv.org/abs/math/0307114">arXiv:math/0307114</a>, <a href="https://doi.org/10.1016/j.geomphys.2005.08.006">doi:10.1016/j.geomphys.2005.08.006</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 8, 2024 at 11:50:05. 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