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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='#construction'>Construction</a></li> <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 <strong>fivebrane 6-group</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fivebrane(n)</annotation></semantics></math> is a smooth version of the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> that appears in the second step of 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> <p>It is a lift of this through the <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#GeometricRealization">geometric realization</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo></mrow><annotation encoding="application/x-tex">\Pi : </annotation></semantics></math> <span class="newWikiWord">?LieGrpd<a href="/nlab/new/%3FLieGrpd">?</a></span> <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/%E2%88%9EGrpd">∞Grpd</a>.</p> <p>One step below the fivebrane 6-group in the Whitehead tower is the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-group">string Lie 2-group</a>.</p> <p>For the time being see the discussions at</p> <p><a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#SmoothWhitehead">smooth Whitehead tower</a></p> <p>and the Motivation section at</p> <p><a class="existingWikiWord" href="/nlab/show/infinity-Chern-Weil+theory">infinity-Chern-Weil theory</a></p> <p>for more background.</p> <h2 id="definition">Definition</h2> <p>In the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <span class="newWikiWord">?LieGrpd<a href="/nlab/new/%3FLieGrpd">?</a></span> we have a smooth refinement of the second fractional <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \frac{1}{6} \mathbf{p}_2 : \mathbf{B} String(n) \to \mathbf{B}^7 \mathbb{R}/\mathbb{Z} </annotation></semantics></math></div> <p>defined on the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-group">string Lie 2-group</a>. Strictly speaking, we need <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>6</mn></mrow><annotation encoding="application/x-tex">n\gt 6</annotation></semantics></math>, since for 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>, <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">String(n)</annotation></semantics></math> is not 6-connected. See <a class="existingWikiWord" href="/nlab/show/orthogonal+group#HomotopyGroups">orthogonal group</a> for a table of the relevant homotopy groups.</p> <p>The <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}Fivebrane(n)</annotation></semantics></math> of the <strong>fivebrane 6-group</strong> is the <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> classified by this in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, that is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}String(n) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z} } \,. </annotation></semantics></math></div> <h2 id="construction">Construction</h2> <p>Along the lines of the description at <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> and <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, in a canonical <a class="existingWikiWord" href="/nlab/show/models+for+infinity-stack+%28infinity%2C1%29-toposes">model</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{6}\mathbf{p}_2</annotation></semantics></math> is given by a morphism out of a <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Q</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Q</annotation></semantics></math> of <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> that is built in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≤</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">k \leq 7</annotation></semantics></math> from smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplices">simplices</a> in the <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>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math>. This morphism assigns to a 7-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>7</mn></msubsup><mo>→</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : \Delta^7_{Diff} \to Spin(n)</annotation></semantics></math> the integral</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><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>7</mn></msubsup></mrow></msub><msup><mi>ϕ</mi> <mo>*</mo></msup><msub><mi>μ</mi> <mn>7</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \int_{\Delta^7_{Diff}} \phi^* \mu_7 \;\;\in \mathbb{R}/\mathbb{Z} </annotation></semantics></math></div> <p>of the degree 7 <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">\mu_7</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> <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> which is normalized such that its pullback to <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">String(n)</annotation></semantics></math> (..explain…) is the deRham image of the generator in <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> there.</p> <p>More in detail, a resolution of <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> is given by the <a class="existingWikiWord" href="/nlab/show/coskeleton">coskeleton</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>7</mn></msub><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msub><mi>Q</mi> <mn>7</mn></msub><mo>⊂</mo><mi>hom</mi><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>7</mn></msubsup><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mn>8</mn><mo>⋅</mo><mn>7</mn><mo>⋅</mo><mn>6</mn><mo>⋅</mo><mn>5</mn><mo>⋅</mo><mn>4</mn></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Q</mi> <mn>4</mn></msub><mo>⊂</mo><mi>hom</mi><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>4</mn></msubsup><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>20</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Q</mi> <mn>3</mn></msub><mo>⊂</mo><mi>hom</mi><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>3</mn></msubsup><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>4</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>hom</mi><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>2</mn></msubsup><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>hom</mi><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>Diff</mi> <mn>1</mn></msubsup><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{cosk}_7 \left( \array{ Q_7 \subset hom(\Delta^7_{Diff}, G) \times (U(1))^{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4} \\ \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow \downarrow \downarrow \\ \vdots \\ \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow \\ Q_4 \subset hom(\Delta^4_{Diff}, G) \times (U(1))^{20} \\ \downarrow \downarrow \downarrow\downarrow \downarrow \\ Q_3 \subset hom(\Delta^3_{Diff}, G) \times (U(1))^4 \\ \downarrow \downarrow \downarrow\downarrow \\ hom(\Delta^2_{Diff}, G) \times U(1) \\ \downarrow \downarrow \downarrow \\ hom(\Delta^1_{Diff}, G) \\ \downarrow \downarrow \\ * } \right) </annotation></semantics></math></div> <p>where the subobjects are those consisting of 3-simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with 2-faces labeled in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> such that the integral of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mu_3</annotation></semantics></math> over the 3-simplex in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}/\mathbb{Z}</annotation></semantics></math> is the signed product of these labels.</p> <p>(…)</p> <h2 id="related_concepts">Related concepts</h2> <div> <table><thead><tr><th></th><th><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></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th><th>16</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ninebrane+group">ninebrane group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">higher versions</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></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/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/stable+orthogonal+group">stable orthogonal group</a></td><td style="text-align: left;"><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><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(O)</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a></td><td style="text-align: left;"><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><mi>𝕊</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\mathbb{S})</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>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>24</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{24}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>240</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{240}</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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_6</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><msub><mi>ℤ</mi> <mn>504</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{504}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>480</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{480} \oplus \mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \oplus \mathbb{Z}_2 </annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/image+of+J-homomorphism">image of J-homomorphism</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(\pi_n(J))</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>24</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{24}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>240</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{240}</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>504</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{504}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>480</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{480}</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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The topological <a class="existingWikiWord" href="/nlab/show/fivebrane+group">fivebrane group</a> with its interpretation in <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a> was discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Fivebrane structures</em> Reviews in mathematical physics, 10 (2009) 1197 (<a href="http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SSSII">web</a>)</li> </ul> <p>and the smooth fivebrane 6-group was indicated. The latter is discussed in more detail in section 4.1 of</p> <ul> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jesse+Wolfson">Jesse Wolfson</a> says he has shown the existence of a presentation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/smooth+infinity-group">smooth 6-group</a> by a locally Kan and degreewise finite-dimensional simplicial smooth manifold.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 27, 2022 at 02:58:10. See the <a href="/nlab/history/fivebrane+6-group" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/fivebrane+6-group" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/192/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/fivebrane+6-group/12" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/fivebrane+6-group" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/fivebrane+6-group" accesskey="S" class="navlink" id="history" rel="nofollow">History (12 revisions)</a> <a href="/nlab/show/fivebrane+6-group/cite" style="color: black">Cite</a> <a href="/nlab/print/fivebrane+6-group" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/fivebrane+6-group" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>