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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="chernsimons_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons theory</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> <p><strong><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/infinity-Wess-Zumino-Witten+theory+-+contents">∞-Wess-Zumino-Witten theory</a></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">quantum field theory</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/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+cohomology">∞-Lie algebroid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a></p> </li> </ul> </li> </ul> <h2 id="definition">Definition</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">level (Chern-Simons theory)</a></li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p><strong>For <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></li> </ul> </li> <li> <p><strong>For <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></li> </ul> </li> <li> <p><strong>For discrete <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Crane-Yetter+model">Crane-Yetter model</a></li> </ul> </li> <li> <p><strong>For <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/BF-theory">BF-theory</a> coupled to <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></li> </ul> </li> <li> <p><strong>For targets extending the <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></strong></p> <p>(such as the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+gravity">Chern-Simons gravity</a></p> </li> <li> <p>Chern-Simons-<a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-dimensional+Chern-Simons+theory">1-dimensional Chern-Simons theory</a></p> </li> <li> <p>for higher abelian targets</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+electric+background+charge+coupling">higher electric background charge coupling</a></p> </li> </ul> </li> <li> <p><strong>for <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> targets</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+sigma+model">Poisson sigma model</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+sigma+model">Courant sigma model</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>for the <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>-structure on the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the closed <a class="existingWikiWord" href="/nlab/show/string">string</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/1d+Chern-Simons+theory">1d Chern-Simons theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Wilson+loop">Wilson loop</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+Chern-Simons+theory">2d Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d+quantum+gravity">3d quantum gravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4d+Chern-Simons+theory">4d Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter+model">Yetter model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BF-theory">BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5d+Chern-Simons+theory">5d Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+Chern-Simons+theory">6d Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a></p> <ul> <li> <p>topological <a href="http://ncatlab.org/nlab/show/AdS-CFT#AdS7CFT6">AdS7/CFT6</a>-sector</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11d+Chern-Simons+theory">11d Chern-Simons theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Dijkgraaf-Witten+theory">∞-Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-Chern-Simons+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</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> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </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/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</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/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> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><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">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/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='#properties'>Properties</a></li> <ul> <li><a href='#as_a_characteristic_class'>As a characteristic class</a></li> <li><a href='#LevelRenormalization'>Level renormalization</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesLevelRenormalization'>Level renormalization</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a> of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> gives rise to an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> which is <a class="existingWikiWord" href="/nlab/show/gauge+invariance">gauge invariant</a> only for certain discrete choices of the scale (a global prefactor) of the Lagrangian. This is called the <strong>level</strong> of the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a>. The fact that it takes values in a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> is called <strong>level quantization</strong> of the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a>.</p> <h2 id="properties">Properties</h2> <h3 id="as_a_characteristic_class">As a characteristic class</h3> <p>Formally, for <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> being some <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact 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>, the level is identified with a choice of <a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a>, hence an element</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>c</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [c] \in H^4(B G, \mathbb{Z}) </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Under the <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> every such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> corresponds to an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\langle -,-\rangle_c</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. (The point being here that while every scalar multiple of an invariant polynomial is itself again an invariant polynomial, only a lattice of these does corespond to a class in <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a>.)</p> <p>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> is furthermore connected and simply connected, then the <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a> of Chern-Simons theory is the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CS</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">CS_c</annotation></semantics></math> of this specific invariant polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\langle -,-\rangle_c</annotation></semantics></math>.</p> <p>Generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>B</mi><mi>G</mi><mo>→</mo><msup><mi>B</mi> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \;\colon\; B G \to B^3 U(1) \simeq K(\mathbb{Z},4)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/universal+characteristic+map">universal characteristic map</a> the <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a> is a <a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential refinement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> of a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> through <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+infinity-groupoids">geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoids">smooth infinity-groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">|</mo><mi>Π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mi>c</mi></mover></mtd> <mtd><msup><mi>B</mi> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}G_{conn} &\stackrel{\hat \mathbf{c}}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow && \downarrow &&& \mathbf{H} \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1) \\ && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}} \\ B G &\stackrel{c}{\to}& B^3 U(1) &&& L_{whe} Top } \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> is the “differentially refined level” of the theory. Notice that in terms of this the statement that the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> is <a class="existingWikiWord" href="/nlab/show/gauge+invariance">gauge invariant</a> for “given level” is the statement that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_3</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 3, the <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of the <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> fo <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_3</annotation></semantics></math> is a map of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth stacks</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mn>3</mn></msub></mrow></msub><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>3</mn></msub><mo>,</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>3</mn></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp\left( 2 \pi i \int_{\Sigma_3} [\Sigma_3, \hat \mathbf{c}] \right) \;\colon\; [\Sigma_3, \mathbf{B}G_{conn}] \to U(1) \,. </annotation></semantics></math></div> <p>Analogous reasoning and termionology applies to <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> and generally to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a>.</p> <h3 id="LevelRenormalization">Level renormalization</h3> <p>Upon <a class="existingWikiWord" href="/nlab/show/quantization+of+3d+Chern-Simons+theory">quantization</a>, the <a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">Chern-Simons level</a> receives a <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> which shifts it by the <a class="existingWikiWord" href="/nlab/show/dual+Coxeter+number">dual Coxeter number</a> of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> (see the references <a href="#ReferencesLevelRenormalization">below</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/su%282%29-anyon">su(2)-anyon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+of+3d+Chern-Simons+theory">quantization of 3d Chern-Simons theory</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/extended+prequantum+field+theory">extended prequantum field theory</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq k \leq n</annotation></semantics></math></th><th>(off-shell) <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum (n-k)-bundle</a></th><th>traditional terminology</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential</a> <a class="existingWikiWord" href="/nlab/show/universal+characteristic+map">universal characteristic map</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">level</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum (n-1)-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinity-Wess-Zumino-Witten+model">WZW bundle (n-2)-gerbe</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum (n-k)-bundle</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum 1-bundle</a></td><td style="text-align: left;">(off-shell) <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantum+0-bundle">prequantum 0-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>For traditional accounts see at <em><a href="#Chern-Simons+theory">Chern-Simons theory - References</a></em>.</p> <p>Introductory discussion is in the section <em><a href="geometry%20of%20physics#PhysicsMotivationAndSurvey">Physics in Higher Geometry: Motivation and Survey</a></em> at</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em>.</li> </ul> <h3 id="ReferencesLevelRenormalization">Level renormalization</h3> <p>On the <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> of the Chern-Simons level (shift by the <a class="existingWikiWord" href="/nlab/show/dual+Coxeter+number">dual Coxeter number</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Luis+Alvarez-Gaum%C3%A9">Luis Alvarez-Gaumé</a>, J. M. F. Labastida, A. V. Ramallo, <em>A note on perturbative Chern-Simons theory</em>, Nuclear Physics B <strong>334</strong> 1 (1990) 103-124 (<a href="https://doi.org/10.1016/0550-3213(90)90658-Z">doi:10.1016/0550-3213(90)90658-Z</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+A.+Shifman">Mikhail A. Shifman</a>, <em>Four-dimension aspect of the perturbative renormalization in three-dimensional Chern-Simons theory</em>, Nuclear Physics B <strong>352</strong> 1 (1991) 87-112 (<a href="https://doi.org/10.1016/0550-3213(91)90130-P">doi:10.1016/0550-3213(91)90130-P</a>)</p> </li> <li> <p>M. Asorey, F. Falceto, J. L. Lopez, G. Luzon, <em>Universalty and Ultraviolet Regularizations of Chern-Simons Theory</em>, Nucl.Phys. B <strong>429</strong> (1994) 344-374 (<a href="https://arxiv.org/abs/hep-th/9403117">arXiv:hep-th/9403117</a>, <a href="https://doi.org/10.1016/0550-3213(94)00331-9">doi:10.1016/0550-3213(94)00331-9</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 30, 2022 at 10:18:08. 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