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space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </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/%3Fech+cohomology">?ech 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> <h4 id="higher_spin_geometry">Higher spin geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/fivebrane+geometry">fivebrane geometry</a></strong> …</p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></p> </li> </ul> <h2 id="spin_geometry">Spin geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/central+product+spin+group">central product spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor">spinor</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/charge+conjugation+matrix">charge conjugation matrix</a>, <a class="existingWikiWord" href="/nlab/show/Fierz+identity">Fierz identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+conjugate">Dirac conjugate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+spinor">Dirac spinor</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+spinor">Weyl spinor</a>, <a class="existingWikiWord" href="/nlab/show/Majorana+spinor">Majorana spinor</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+equation">Dirac equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/rotation+groups">rotation groups</a> in <a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low</a> <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/classification+of+simple+Lie+groups">Dynkin label</a></th><th><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">sp. orth. group</a></th><th><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></th><th><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%282%29">SO(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%282%29">Pin(2)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%283%29">SO(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%283%29">Pin(3)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%284%29">Pin(4)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%285%29">SO(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%285%29">Pin(5)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%286%29">SO(6)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%287%29">SO(7)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="semi-spin+group#SemiSpin8">SO(8)</a></td></tr> <tr><td style="text-align: left;">B4</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%289%29">SO(9)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B5</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D8</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2816%29">SemiSpin(16)</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D16</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2832%29">SO(32)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2832%29">Spin(32)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a></td></tr> </tbody></table> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%285%29.Spin%283%29">Spin(5).Spin(3)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a></p> </li> </ul> </div> <h2 id="string_geometry">String geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+2-group">string^c 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+structure">string^c structure</a></p> </li> </ul> <h2 id="fivebrane_geometry">Fivebrane geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></p> </li> </ul> <h2 id="ninebrane_geometry">Ninebrane geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</a></li> </ul> </div></div> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </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> <li><a href='#Properties'>Properties</a></li> <li><a href='#presentations'>Presentations</a></li> <ul> <li><a href='#ByLieIntegrationOfStringLie2Algebra'>By Lie integration of the string Lie 2-algebra</a></li> <li><a href='#PresentationByStrictTwoGroups'>By strict Lie <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>-group</a></li> <li><a href='#as_a_finitedimensional_weak_lie_2group'>As a finite-dimensional weak Lie 2-group</a></li> <li><a href='#as_an_automorphism_2group_of_fermionic_cft'>As an automorphism 2-group of fermionic CFT</a></li> <li><a href='#as_the_automorphisms_of_the_wesszuminowitten_gerbe_2connection'>As the automorphisms of the Wess-Zumino-Witten gerbe 2-connection</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>The <em>string 2-group</em> is a <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth 2-group</a>-refinement of the <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> called the <a class="existingWikiWord" href="/nlab/show/string+group">string group</a>. It is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> induced by the smooth/stacky version of the <a class="existingWikiWord" href="/nlab/show/first+fractional+Pontryagin+class">first fractional Pontryagin class</a>/<a class="existingWikiWord" href="/nlab/show/second+Chern+class">second Chern class</a>.</p> <h2 id="definition">Definition</h2> <p>A string 2-group extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String(G)</annotation></semantics></math> is defined for every <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple</a> <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> <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>, such as the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = Spin(n)</annotation></semantics></math> or the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = SU(n)</annotation></semantics></math> (for non-low <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>).</p> <p>Since <a class="existingWikiWord" href="/nlab/show/string+structures">string structures</a> arise predominantly as higher analogs of <a class="existingWikiWord" href="/nlab/show/spin+structures">spin structures</a>, the default choice is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">G = Spin</annotation></semantics></math> and in that case one usually just writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo>=</mo><mi>String</mi><mo stretchy="false">(</mo><mi>Spin</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String = String(Spin)</annotation></semantics></math>, for short.</p> <p>Recall first that the <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is one step in the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>.</p> <div class="num_defn" id="AbstractDefinitionDiscrete"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">B Spin(n) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> for its <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>B</mi> <mn>4</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \frac{1}{2}p_1 : B Spin(n) \to B^4 \mathbb{Z} </annotation></semantics></math></div> <p>for a representative of the <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> called the first fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a>. Its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">⟨</mo><mn>7</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">B String(n) := B O(n)\langle 7 \rangle</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>B</mi><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mn>4</mn></msup><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ B String(n) &\to& * \\ \downarrow && \downarrow \\ B Spin(n) &\stackrel{\frac{1}{2} p_1}{\to}& B^4 \mathbb{Z} } \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Ω</mi><mi>B</mi><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> String(n) := \Omega B String(n) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>-object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> called the <strong><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></strong>.</p> </div> <p>Write now</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>Π</mi><mo>⊣</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo>⊣</mo><mi>coDisc</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mover><mover><mover><munder><mo>←</mo><mi>coDisc</mi></munder><mover><mo>→</mo><mi>Γ</mi></mover></mover><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>→</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : Smooth\infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd \simeq Top </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <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, regarded as a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <div class="num_prop" id="SmoothFractionalPontryaginClass"> <h6 id="proposition">Proposition</h6> <p>There is a lift through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2} p_1</annotation></semantics></math> to the <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#SmoothFirstFracPontryaginClass">smooth first fractional Pontryagin class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin(n) \to \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> (see <a href="http://ncatlab.org/nlab/show/smooth+infinity-groupoid#LieGroups">Smooth ∞-groupoids – Cohesive ∞-groups – Lie groups</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^3 U(1)</annotation></semantics></math> is the three-fold delooping of the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> (see <a href="http://ncatlab.org/nlab/show/smooth+infinity-groupoid#CircleLienGroup">Smooth ∞-groupoids – Cohesive ∞-groups – Circle Lie n-group</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\mathbf{p}_1</annotation></semantics></math> is the image under <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the canonical <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">cocycle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>:</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu = \langle -,[-,-]\rangle : \mathfrak{so}(n) \to b^2 \mathbb{R} \,. </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>.</p> </li> </ul> </div> <p>This is shown in (<a href="#FSS">FSS</a>).</p> <div class="num_defn" id="AbstractDefinitionSmooth"> <h6 id="definition_3">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}String(n)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the smooth first fractional Pontryagin class</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. Its <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> String(n) := \Omega \mathbf{B}String(n) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">smooth ∞-group</a> called <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <strong>smooth string 2-group</strong>.</p> </div> <h2 id="Properties">Properties</h2> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>:</mo><mo>=</mo><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><mo>:</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mi>Π</mi></mover><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> \vert - \vert := \vert\Pi(-)\vert : Smooth \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\vert - \vert}{\to} Top </annotation></semantics></math></div> <p>for the <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#Homotopy">intrinsic geometric realization</a> in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The smooth string 2-group, def. <a class="maruku-ref" href="#AbstractDefinitionSmooth"></a>, indeed maps under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\vert-\vert</annotation></semantics></math> to the topological string group:</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≃</mo><mi>B</mi><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \vert \mathbf{B}String(n) \vert \simeq B String(n) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^3 U(1)</annotation></semantics></math> is presented by a simplicial presheaf that is degreewise presented by a paracompact smooth manifold (a finite product of the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> with itself), it follows from the general properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> discussed at <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> preserves the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\mathbf{p}_1</annotation></semantics></math>.</p> </div> <h2 id="presentations">Presentations</h2> <p>Several explicit presentations of the string Lie 2-group are known.</p> <h3 id="ByLieIntegrationOfStringLie2Algebra">By Lie integration of the string Lie 2-algebra</h3> <p>We discuss a presentation of the smooth string 2-group by <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the skeletal version of the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a>.</p> <p>Recall the identification of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>s <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> with their dual <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>s <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>.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo><mo>:</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \mu := \langle - ,[-,-]\rangle : \mathfrak{so}(n) \to b^2 \mathbb{R} </annotation></semantics></math></div> <p>for the canonical degree-3 <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a>, normalized such that the 3-form</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>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↩</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>←</mo><mi>μ</mi></mover><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(Spin(n)) \hookleftarrow CE(\mathfrak{so}(n)) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R}) </annotation></semantics></math></div> <p>represents the image in <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of a generators of the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H^3(G,\mathbb{Z}) \simeq \mathbb{Z}</annotation></semantics></math>.</p> <p>Define the <strong><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>μ</mi></msub></mrow><annotation encoding="application/x-tex"> \mathfrak{string}(n) := \mathfrak{so}(n)_\mu </annotation></semantics></math></div> <p>to be given by 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><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>⊕</mo><mo stretchy="false">⟨</mo><mi>b</mi><mo stretchy="false">⟩</mo><mo>,</mo><msub><mi>d</mi> <mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{string}(n)) := \wedge^\bullet ( \mathfrak{so}(n)^* \oplus \langle b\rangle , d_{\mathfrak{string}}) </annotation></semantics></math></div> <p>which is that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(n)</annotation></semantics></math> with a single generator <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> in degree 3 adjoined and the differential given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></msub><msub><mo stretchy="false">|</mo> <mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo>;</mo></mrow><annotation encoding="application/x-tex"> d_{\mathfrak{string}}|_{\mathfrak{so}(n)^*} = d_{\mathfrak{so}(n)}; </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></msub><mo>:</mo><mi>b</mi><mo>↦</mo><mi>μ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{\mathfrak{string}} : b \mapsto \mu \,. </annotation></semantics></math></div></div> <div class="num_prop" id="HomotopyFiberOfLInftyAlgebraCocycle"> <h6 id="proposition_3">Proposition</h6> <p>We have a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi><mi>b</mi><mi>ℝ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>μ</mi></mover></mtd> <mtd><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathfrak{string}(n) &\to& e b \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R} } \,. </annotation></semantics></math></div></div> <p>See <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> for more discussion.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The <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>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{string}(n)</annotation></semantics></math> yields a presentation of the smooth String 2-group, def. <a class="maruku-ref" href="#AbstractDefinitionSmooth"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{cosk}_3 \exp(\mathfrak{string}(n)) \simeq \mathbf{B} String(n) \,. </annotation></semantics></math></div></div> <p>This is essentially the model considered in (<a href="#Henriques">Henriques</a>), discussed here in the context of <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> as described in (<a href="#FSS">FSS</a>).</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We observe the image under <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of 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>-algebra pullback diagram from prop. <a class="maruku-ref" href="#HomotopyFiberOfLInftyAlgebraCocycle"></a> is a pullback diagram 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><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp_{smooth}^{op}, sSet]_{proj}</annotation></semantics></math> that presents the defining <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>. Before applying the <a class="existingWikiWord" href="/nlab/show/coskeleton">coskeleton</a> operation we have immediately</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi><mi>b</mi><mi>ℝ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>μ</mi></mover></mtd> <mtd><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>e</mi><mi>b</mi><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>μ</mi></mover></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \exp(-) \; :\; \left( \array{ \mathfrak{string}(n) &\to& e b \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R} } \right) \;\mapsto \; \left( \array{ \exp(\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R}) \\ \downarrow && \downarrow \\ \exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& \exp(b^2 \mathbb{R}) } \right) </annotation></semantics></math></div> <p>such that on the right we still have a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram.</p> <p>We discuss the descent o this pullback diagram along the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(\mathfrak{so}(n)) \to \mathbf{cosk}_3 \exp(\mathfrak{so}(n))</annotation></semantics></math>.</p> <p>Notice from <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> the weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub><mo>:</mo><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℝ</mi> <mi>c</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int_{\Delta^\bullet} : \exp(b^n \mathbb{R}) \simeq \mathbf{B}^{n+1}\mathbb{R}_c \,. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be the set of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>→</mo><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[4] \to \exp(b^2 \mathbb{R})</annotation></semantics></math> that fit into a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><msub><mi>ℝ</mi> <mi>c</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[4] &\to& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}} \\ && \mathbf{B}^3 \mathbb{R}_c \\ \downarrow && \downarrow \\ \Delta[4] &\to& \mathbf{B}^3 (\mathbb{Z} \to \mathbb{R})_c } </annotation></semantics></math></div> <p>(closed 3-forms on 3-balls whose integral is an integer).</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>I</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>I</mi><mo>×</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mrow></munder><mstyle mathvariant="bold"><mrow><msub><mi>cosk</mi> <mn>3</mn></msub></mrow></mstyle><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \exp(b^2 \mathbb{R}/\mathbb{Z}) := \mathbf{cosk}_3 \left( (I \times \Delta[4])\coprod_{I \times \partial \Delta[4]} \mathbf{cosk_3} \exp(b^2 \mathbb{R}) \right) </annotation></semantics></math></div> <p>for the result of filling all these by 4-cells. Similarly define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>e</mi><mi>b</mi><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(e b \mathbb{R}/\mathbb{Z})</annotation></semantics></math>.</p> <p>Then applying the <a class="existingWikiWord" href="/nlab/show/coskeleton">coskeleton</a> functor to the above pullback diagram and using the projection (<a href="#FSS">FSS</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \exp(\mathfrak{so}(n)) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{cosk}_3\exp(\mathfrak{so}(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \exp(b^2 \mathbb{R}/\mathbb{Z}) } </annotation></semantics></math></div> <p>we get the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><mi>e</mi><mi>b</mi><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</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><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mn>3</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔰𝔬</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>exp</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mn>2</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{cosk}_3 (\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R}/\mathbb{Z}) \\ \downarrow && \downarrow \\ \mathbf{cosk}_3 \exp(\mathfrak{so}) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \exp(b^2 \mathbb{R}/\mathbb{Z}) } \,. </annotation></semantics></math></div> <p>This is again a pullback diagram of a fibration resolution of the point inclusion, hence presents the homotopy fiber in question.</p> </div> <h3 id="PresentationByStrictTwoGroups">By strict Lie <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>-group</h3> <p>A realization of the string 2-group as a <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>s was given in (<a href="#BCSS">BCSS</a>).</p> <p>This is one of three different (there should be more), weakly equivalent such <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> models that are discussed in the (to date unpublished)</p> <ul> <li><a href="http://www.math.uni-hamburg.de/home/schreiber/nactwist.pdf#page=91">nactwist, section 5.2.3</a></li> </ul> <p>(This particular section, and its results, are joint work of <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> and <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>).</p> <p>We have the following pattern of routes through <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>StrLie</mi><mi>ω</mi><mi>Grpd</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>StrLie</mi><mi>ω</mi><mi>Grpd</mi></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mi>LieCrsdCmplx</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msub><mi>Π</mi> <mi>n</mi></msub><mi>S</mi><mi>CE</mi></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algebras</mi></mtd> <mtd></mtd> <mtd><mo>←</mo></mtd> <mtd></mtd> <mtd><mi>Str</mi><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algebras</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>DiffCrsdCmplx</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ StrLie \omega Grpd &&&& StrLie \omega Grpd &\stackrel{\simeq}{\leftarrow}& LieCrsdCmplx \\ \uparrow^{\Pi_n S CE} &&&& \uparrow && \uparrow^{\exp(-)} \\ L_\infty Algebras && \leftarrow&& Str L_\infty Algebras &\to& DiffCrsdCmplx } </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>StrLie</mi><mi>ω</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">StrLie \omega Grpd</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/strict+omega-groupoid">strict omega-groupoid</a>s internal to <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>s, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LieCrsCmplx</mi></mrow><annotation encoding="application/x-tex">LieCrsCmplx</annotation></semantics></math> is accordingly smooth <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a>es , <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algebra</mi></mrow><annotation encoding="application/x-tex">L_\infty Algebra</annotation></semantics></math> is all <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</a>s and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Str</mi><msub><mi>L</mi> <mn>∞</mn></msub><mi>Algebra</mi></mrow><annotation encoding="application/x-tex">Str L_\infty Algebra</annotation></semantics></math> is <em>strict</em> <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>-algebras. The vertical morphism on the right is term-wise ordinary <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>. The other vertical morphisms take an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</a>, form the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> of flat <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Lie+algebroid+valued+differential+forms">∞-Lie algebroid differential form</a>s, and then take <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_n(-)</annotation></semantics></math> of that.</p> <p>For the String-case this yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>fl</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>𝔰𝔬</mi> <mrow><msub><mi>μ</mi> <mn>3</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mo>≃</mo></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>Mick</mi></msub></mtd> <mtd><mover><mo>↦</mo><mo>≃</mo></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>BCSS</mi></msub></mtd> <mtd><mo>←</mo><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>Ω</mi><mo stretchy="false">^</mo></mover><mi>Spin</mi><mo>→</mo><mi>P</mi><mi>Spin</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msub><mi>𝔰𝔬</mi> <mrow><msub><mi>μ</mi> <mn>3</mn></msub></mrow></msub></mtd> <mtd></mtd> <mtd><mover><mo>↦</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>Ω</mi><mo stretchy="false">^</mo></mover><mi>𝔰𝔬</mi><mo>→</mo><mi>P</mi><mi>𝔰𝔬</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{so}_{\mu_3})) &\stackrel{\simeq}{\mapsto}& \mathbf{B} String_{Mick} &\stackrel{\simeq}{\mapsto}& \mathbf{B} String_{BCSS} &\leftarrow|& (\hat \Omega Spin \to P Spin) \\ \uparrow &&&\nearrow& \uparrow && \uparrow \\ \mathfrak{so}_{\mu_3} &&\stackrel{\simeq}{\mapsto}&& \mathfrak{string} &\mapsto& (\hat \Omega \mathfrak{so} \to P \mathfrak{so}) } \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔰𝔬</mi> <mrow><msub><mi>μ</mi> <mn>3</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{so}_{\mu_3}</annotation></semantics></math> denotes the weak, skeletal <a class="existingWikiWord" href="/nlab/show/String+Lie+2-algebra">String Lie 2-algebra</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{string}</annotation></semantics></math> its equivalent strict version given by BCSS</p> </li> <li> <p>the diagonal morphism is the construction in BCSS.</p> </li> <li> <p>the strict 2-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>fl</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>𝔤</mi> <mrow><msub><mi>μ</mi> <mn>3</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{g}_{\mu_3}))</annotation></semantics></math> has, notice, as morphism smooth paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> that are composed by concatenation</p> </li> <li> <p>the 2-groupoid <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>String</mi> <mi>Mick</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}String_{Mick}</annotation></semantics></math> is a version of the String Lie 2-group that manifestly uses the <span class="newWikiWord">Mickelsson cocycle<a href="/nlab/new/Mickelsson+cocycle">?</a></span> (morphism are paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> that are composed using the group product)</p> </li> <li> <p>the 2-groupoid <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>String</mi> <mi>BCSS</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}String_{BCSS}</annotation></semantics></math> is the version given in BCSS (morhisms again are paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> that are composed using the group product).</p> </li> </ul> <h3 id="as_a_finitedimensional_weak_lie_2group">As a finite-dimensional weak Lie 2-group</h3> <p>(<a href="#Schommer-Pries">Schommer-Pries</a>)</p> <h3 id="as_an_automorphism_2group_of_fermionic_cft">As an automorphism 2-group of fermionic CFT</h3> <p>The string 2-group also appears as a certain <a class="existingWikiWord" href="/nlab/show/automorphism+2-group">automorphism 2-group</a> inside the <a class="existingWikiWord" href="/nlab/show/3-category+of+fermionic+conformal+nets">3-category of fermionic conformal nets</a> (<a href="#DouglasHenriques">Douglas-Henriques</a>)</p> <h3 id="as_the_automorphisms_of_the_wesszuminowitten_gerbe_2connection">As the automorphisms of the Wess-Zumino-Witten gerbe 2-connection</h3> <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> a compact simply connected simple <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, there is the “<a class="existingWikiWord" href="/nlab/show/WZW+gerbe">WZW gerbe</a>”, hence the <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</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> whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 3-form is the <a class="existingWikiWord" href="/nlab/show/left+invariant+differential+form">left invariant</a> extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>θ</mi><mo>∧</mo><mo stretchy="false">[</mo><mi>θ</mi><mo>∧</mo><mi>θ</mi><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \theta \wedge [\theta \wedge \theta]\rangle</annotation></semantics></math> of the canonical Lie algebra 3-cocycle to the group</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>WZW</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2U(1) \,. </annotation></semantics></math></div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>The string 2-group is the <a class="existingWikiWord" href="/nlab/show/smooth+infinity-group">smooth 2-group</a> of <a class="existingWikiWord" href="/nlab/show/automorphism+infinity-group">automorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{L}_{WZW}</annotation></semantics></math> which cover the left <a class="existingWikiWord" href="/nlab/show/action">action</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> on itself (hence the “<a class="existingWikiWord" href="/nlab/show/Heisenberg+2-group">Heisenberg 2-group</a>” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{L}_{WZW}</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/prequantum+2-bundle">prequantum 2-bundle</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ℒ</mi> <mi>WZW</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>String</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G) \,, </annotation></semantics></math></div></div> <p>This is due to (<a href="#FiorenzaRogersSchreiber13">Fiorenza-Rogers-Schreiber 13, section 2.6.1</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+algebroid">string Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Platonic+2-group">Platonic 2-group</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <strong>string 2-group</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin%5Ec">spin^c</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+string+structure">twisted string structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/string+cobordism">string cobordism</a></p> <p><a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality+2-group">T-duality 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string%5Ec+2-group">string^c 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> <h2 id="References">References</h2> <p>A <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> presentation of a topological realization of the string 2-group is implicit in</p> <ul> <li id="StolzTeichner"><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>What is an elliptic object?</em> (<a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.5463&rep=rep1&type=pdf">pdf</a>)</li> </ul> <p>A realization of the string 2-group in <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>s by <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the skeletal version of the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> is in</p> <ul> <li id="Henriques"><a class="existingWikiWord" href="/nlab/show/Andre+Henriques">Andre Henriques</a>, <em>Integrating <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>-algebras</em>, Compositio Mathematica, Volume 144, Issue 4 July 2008 , pp. 1017-1045 (<a href="http://arxiv.org/abs/math/0603563">arXiv:math/0603563</a>, <a href="https://doi.org/10.1112/S0010437X07003405">doi:10.1112/S0010437X07003405</a>)</li> </ul> <p>A realization of the string 2-group in <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a>s internal to <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>s by <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of a <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> incarnation of the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a> in in</p> <ul> <li id="BCSS"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Alissa+Crans">Alissa Crans</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em>From loop groups to 2-groups</em>, Homology Homotopy Appl. Volume 9, Number 2 (2007), 101-135. (<a href="https://arxiv.org/abs/math/0504123">arXiv:math/0504123</a>, <a href="https://projecteuclid.org/euclid.hha/1201127333">euclid:hha/1201127333</a>)</li> </ul> <p>A realization of the string 2-group as a <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> in finite-dimensional <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s in in</p> <ul> <li id="Schommer-Pries11"><a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, <em>Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group</em> Geometry & Topology <strong>15</strong> (2011) 609-676 [<a href="http://arxiv.org/abs/0911.2483">arXiv:0911.2483</a>, <a href="https://doi.org/10.2140/gt.2011.15.609">doi:10.2140/gt.2011.15.609</a>]</li> </ul> <p>A discussion as an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> object in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> and the realization of the smooth first fractional Pontryagin class is in</p> <ul> <li id="FSS"><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>Cech cocycles for differential characteristic classes</em> (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#FSS">web</a>)</li> </ul> <p>and in section 4.1 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>A 2-group model which has a smoothening of the <em>topological</em> <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> in lowest degree has been given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Sachse">Christoph Sachse</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Wockel">Christoph Wockel</a>: <em>A Smooth Model for the String Group</em>, International Mathematics Research Notices, <strong>2013</strong> 16 (2013) 3678-3721 [<a href="http://arxiv.org/abs/1104.4288">arXiv:1104.4288</a>, <a href="https://doi.org/10.1093/imrn/rns154">doi:10.1093/imrn/rns154</a>]</li> </ul> <p>A construction explicitly in terms of the “<a class="existingWikiWord" href="/nlab/show/basic+bundle+gerbe">basic</a>” <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</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>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>A Construction of String 2-Group Models using a Transgression-Regression Technique</em>, Contemp. Math. <strong>584</strong> (2012) 99-115 [<a href="http://arxiv.org/abs/1201.5052">arXiv:1201.5052</a>, <a href="https://doi.org/10.1090/conm/584/11588">doi:10.1090/conm/584/11588</a>]</li> </ul> <p>Via fermionic nets/<a class="existingWikiWord" href="/nlab/show/2-Clifford+algebra">2-Clifford algebra</a>:</p> <ul> <li id="DouglasHenriques"><a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <em>Geometric string structures</em> (<a class="existingWikiWord" href="/nlab/files/Tring.pdf" title="TringWP">TringWP</a>)</li> </ul> <p>The realization of the string 2-group as the <a class="existingWikiWord" href="/nlab/show/Heisenberg+2-group">Heisenberg 2-group</a> of the <a class="existingWikiWord" href="/nlab/show/WZW+gerbe">WZW gerbe</a> is due to</p> <ul> <li id="FiorenzaRogersSchreiber13"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Higher+geometric+prequantum+theory">Higher geometric prequantum theory</a></em>, 2013 (<a href="http://arxiv.org/abs/1304.0236">arXiv:1304.0236</a>)</li> </ul> <p>A model of the string 2-group using the smooth <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a> (instead of the based <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>) is dicussed in</p> <ul> <li id="MurrayRobertsWockel17"><a class="existingWikiWord" href="/nlab/show/Michael+Murray">Michael Murray</a>, <a class="existingWikiWord" href="/nlab/show/David+Roberts">David Roberts</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Wockel">Christoph Wockel</a>, <em>Quasi-periodic paths and a string 2-group model from the free loop group</em> (<a href="https://arxiv.org/abs/1702.01514">arXiv:1702.01514</a>)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/matrix+factorizations">matrix factorizations</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>:</p> <ul> <li id="FreedTeleman14"><a class="existingWikiWord" href="/nlab/show/Daniel+S.+Freed">Daniel S. Freed</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Dirac families for loop groups as matrix factorizations</em>, <a href="http://arxiv.org/abs/1409.6051">arxiv/1409.6051</a></li> </ul> <p>Further on <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>-<a class="existingWikiWord" href="/nlab/show/higher+central+extensions">extensions</a> by the <a class="existingWikiWord" href="/nlab/show/circle+2-group">circle 2-group</a>:</p> <p>of <a class="existingWikiWord" href="/nlab/show/tori">tori</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, <em>Categorical Tori</em>, SIGMA 14 (2018), 014, 18 (<a href="https://arxiv.org/abs/1406.7046">arXiv:1406.7046</a>)</li> </ul> <p>of <a class="existingWikiWord" href="/nlab/show/finite+subgroups+of+SU%282%29">finite subgroups of SU(2)</a> (to <a class="existingWikiWord" href="/nlab/show/Platonic+2-groups">Platonic 2-groups</a>):</p> <ul> <li id="EpaGanter16"><a class="existingWikiWord" href="/nlab/show/Narthana+Epa">Narthana Epa</a>, <a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, <em>Platonic and alternating 2-groups</em>, Higher Structures 1(1):122-146, 2017 (<a href="http://arxiv.org/abs/1605.09192">arXiv:1605.09192</a>, <a href="https://journals.mq.edu.au/index.php/higher_structures/article/view/30">hs:30</a>)</li> </ul> <p>Discussion of a general definition of smooth <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi mathvariant="normal">String</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">A\rightarrow \mathrm{String}(H)\rightarrow H</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/compact">compact</a> <a class="existingWikiWord" href="/nlab/show/simply+connected">simply connected</a> <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>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, with <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> not necessarily chosen to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math> but only of the same <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <em>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>-Bundles and Smooth String Group Models</em>, <a href="http://arxiv.org/abs/2008.12263">arXiv:2008.12263</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 2, 2024 at 15:28:47. 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