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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="chernsimons_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons theory</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> <p><strong><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/infinity-Wess-Zumino-Witten+theory+-+contents">∞-Wess-Zumino-Witten theory</a></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+cohomology">∞-Lie algebroid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a></p> </li> </ul> </li> </ul> <h2 id="definition">Definition</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">level (Chern-Simons theory)</a></li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p><strong>For <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></li> </ul> </li> <li> <p><strong>For <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></li> </ul> </li> <li> <p><strong>For discrete <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Crane-Yetter+model">Crane-Yetter model</a></li> </ul> </li> <li> <p><strong>For <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> targets</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/BF-theory">BF-theory</a> coupled to <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></li> </ul> </li> <li> <p><strong>For targets extending the <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></strong></p> <p>(such as the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+gravity">Chern-Simons gravity</a></p> </li> <li> <p>Chern-Simons-<a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-dimensional+Chern-Simons+theory">1-dimensional Chern-Simons theory</a></p> </li> <li> <p>for higher abelian targets</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+electric+background+charge+coupling">higher electric background charge coupling</a></p> </li> </ul> </li> <li> <p><strong>for <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> targets</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+sigma+model">Poisson sigma model</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+sigma+model">Courant sigma model</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-structure on the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the closed <a class="existingWikiWord" href="/nlab/show/string">string</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/1d+Chern-Simons+theory">1d Chern-Simons theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Wilson+loop">Wilson loop</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+Chern-Simons+theory">2d Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d+quantum+gravity">3d quantum gravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4d+Chern-Simons+theory">4d Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter+model">Yetter model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BF-theory">BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5d+Chern-Simons+theory">5d Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+Chern-Simons+theory">6d Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a></p> <ul> <li> <p>topological <a href="http://ncatlab.org/nlab/show/AdS-CFT#AdS7CFT6">AdS7/CFT6</a>-sector</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11d+Chern-Simons+theory">11d Chern-Simons theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Dijkgraaf-Witten+theory">∞-Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-Chern-Simons+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</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/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</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> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <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/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">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='#examples'>Examples</a></li> <ul> <li><a href='#AbelianTheory'>Abelian theory</a></li> <li><a href='#OnString2Connections'>Nonabelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> theory on String 2-connections</a></li> <li><a href='#TwoSpeciesCupProductTheoryOnG2Manifold'>Two-species cup-product theory on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> manifold</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#moduli_of_fields_abelian_case'>Moduli of fields (abelian case)</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#abelian_theory_2'>Abelian theory</a></li> <li><a href='#ReferencesNonabelianTheories'>Nonabelian theories</a></li> <li><a href='#on_manifolds'>On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds</a></li> <li><a href='#formulation_in_extended_tqft'>Formulation in extended TQFT</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The case of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 7.</p> <h2 id="examples">Examples</h2> <p>We discuss</p> <ol> <li> <p><a href="#AbelianTheory">Abelian 7d CS theory</a> of an abelian 3-form connection;</p> </li> <li> <p><a href="#OnString2Connections">7d p2 theory on String 2-connections</a></p> </li> <li> <p><a href="#TwoSpeciesCupProductTheoryOnG2Manifold">2-species cup-product theory on a G₂ manifold</a></p> </li> </ol> <h3 id="AbelianTheory">Abelian theory</h3> <p>A basic D=7 <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> is the abelian theory, whose <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> of the <a class="existingWikiWord" href="/nlab/show/cup+product+in+ordinary+differential+cohomology">cup product in ordinary differential cohomology</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>L</mi></mstyle> <mrow><mstyle mathvariant="bold"><mi>DD</mi></mstyle><mo>∪</mo><mstyle mathvariant="bold"><mi>DD</mi></mstyle></mrow></msub><mo lspace="verythinmathspace">:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mi>Δ</mi></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>×</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mover><mo>∪</mo><mo>^</mo></mover></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}} \colon \mathbf{B}^3 U(1)_{conn} \stackrel{\Delta}{\to} \mathbf{B}^3 U(1)_{conn} \times \mathbf{B}^3 U(1)_{conn} \stackrel{\widehat {\cup}}{\to} \mathbf{B}^7 U(1)_{conn} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of this to <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> 0 hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_7</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 7 is the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mn>7</mn></msub></mrow></msub><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mstyle mathvariant="bold"><mi>DD</mi></mstyle><mo>∪</mo><mstyle mathvariant="bold"><mi>DD</mi></mstyle></mrow></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}}] \right) \;\colon\; [\Sigma_7, \mathbf{B}^3 U(1)_{conn}] \to U(1) \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> configuration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Σ</mi> <mn>7</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \Sigma_7 \to \mathbf{B}^3 U(1)_{conn} </annotation></semantics></math></div> <p>here is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a>. In the special case that the underlying <a class="existingWikiWord" href="/nlab/show/circle+3-group">circle 3-group</a> <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a> is trivializable and trivialized, this is equivalently a <a class="existingWikiWord" href="/nlab/show/differential+3-form">differential 3-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in \Omega^3(\Sigma_7)</annotation></semantics></math> and the above <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> takes this to the simple expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↦</mo><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mn>7</mn></msub></mrow></msub><mi>C</mi><mo>∧</mo><mi>d</mi><mi>C</mi><mo>)</mo></mrow><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C \mapsto \exp\left( 2 \pi i \int_{\Sigma_7} C \wedge d C \right) \in U(1) \,, </annotation></semantics></math></div> <p>where in the <a class="existingWikiWord" href="/nlab/show/exponent">exponent</a> we have the <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> over the <a class="existingWikiWord" href="/nlab/show/wedge+product">wedge product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a>. On general field configurations the action functional is the suitable globalization of this expression.</p> <p>In (<a href="#Witten97">Witten 97</a>), (<a href="#Witten98">Witten 98</a>) a slight refinement of this construction (a <a class="existingWikiWord" href="/nlab/show/quadratic+refinement">quadratic refinement</a> induced by an <a class="existingWikiWord" href="/nlab/show/integral+Wu+structure">integral Wu structure</a>) was argued to be the <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic dual</a> to the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of the abelian self-dual 2-form gauge field in the <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> on the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>. The issue of the quadratic refinement was discussed in more detail in (<a href="#HopkinsSinger">HopkinsSinger</a>). A refinement to <a class="existingWikiWord" href="/nlab/show/extended+Lagrangians">extended Lagrangians</a> as above is discussed in (<a href="#FSSII">FSSII</a>).</p> <p>By the argument in (<a href="#Witten98">Witten98</a>) the above relation holds when we interpret the fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mo>:</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\phi \colon : \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> after <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactification</a> on a 4-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> in the <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS7-CFT6</a> setup. By the discussion at <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> this field is in general not simply a 3-connection as above but receives corrections by a <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> and “flux quantization” which give it non-abelian components. This, and the resulting non-abelian generalization of the above extended Lagrangian is discussed in (<a href="#FSSI">FSSI</a>, <a href="#FSSII">FSSII</a>).</p> <p>The nonabelian 7d action functional this obtained contains the following two examples as summands.</p> <h3 id="OnString2Connections">Nonabelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> theory on String 2-connections</h3> <p>The <a class="existingWikiWord" href="/nlab/show/second+fractional+Pontryagin+class">second 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><mo stretchy="false">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mn>8</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>String</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [\tfrac{1}{6}p_2] \in H^8(B String, \mathbb{Z}) </annotation></semantics></math></div> <p>has a smooth and differential refinement (see at <em><a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></em>) to an <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tfrac{1}{2}\hat \mathbf{p}_2 \;\colon\; \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \,. </annotation></semantics></math></div> <p>where the domain is the <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli 2-stack</a> of <a class="existingWikiWord" href="/nlab/show/String+2-group">String 2-group</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">principal 2-connections</a> (see at <em><a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a></em> for more). This modulates the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle">Chern-Simons circle 7-bundle</a> with connection on <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>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}String_{conn}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of this to codimension 0 yields an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mn>7</mn></msub></mrow></msub><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \tfrac{1}{6}\hat \mathbf{p}_2] \right) \;\colon\; [\Sigma_7, \mathbf{B}String_{conn}] \to U(1) </annotation></semantics></math></div> <p>on string 2-connection fields. This is part of the quantum-corrected and flux-quantized extended action functional of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> in <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> by the analysis in (<a href="#FSSII">FSSII</a>).</p> <h3 id="TwoSpeciesCupProductTheoryOnG2Manifold">Two-species cup-product theory on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> manifold</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a> with characteristic <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>3</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega_3 \in \Omega^3(X) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><mo>⋆</mo><msub><mi>ω</mi> <mn>3</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega_4 = \star \omega_3\in \Omega^4(X) </annotation></semantics></math></div> <p>and 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 simply connected compact <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a> with <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math>, consider the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on the space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+1-forms">Lie algebra valued 1-forms</a> <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> given by the <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><msub><mi>ω</mi> <mn>4</mn></msub><mo>∧</mo><mi>CS</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A \mapsto \exp\left( 2 \pi i\int_{X} \omega_4 \wedge CS\left(A\right) \right) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS(A) \in \Omega^3(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. This, or some suitable globalization of this, has been considered as an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for 7-dimensional Chern-Simons-type theory in (<a href="#DonaldsonThomas">Donaldson-Thomas</a>) and (<a href="#BaulieuLosevNekrasov">Baulieu-Losev-Nekrasov</a>). This appears as an action functional in <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a> (<a href="#deBoerEtAl">deBoer et al</a>).</p> <p>To refine this to an <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> and then fully globalize the action functional we can ask for a <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric prequantization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\omega_4</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/n-plectic+structure">3-plectic structure</a>, by a <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum 3-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat \mathbf{G}_2</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></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi mathvariant="normal">conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ω</mi> <mn>4</mn></msub></mrow></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>4</mn></msubsup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}^3 U(1)_{\mathrm{conn}} \\ & {}^{\mathllap{\hat \mathbf{G}_2}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega_4}{\to}& \Omega^4_{cl} } \,, </annotation></semantics></math></div> <p>where <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><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^3 U(1)_{conn} \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> is the smooth <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundles with connection</a>.</p> <p>If moreover we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \hat \mathbf{c} \;:\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} </annotation></semantics></math></div> <p>for the universal differential characteristic map which is 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><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle-,-\rangle</annotation></semantics></math> (as discussed at <em><a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a></em>), hence the <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> for ordinary <a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d</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>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>, then an <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> for the above <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> is given by the <a class="existingWikiWord" href="/nlab/show/cup+product+in+ordinary+differential+cohomology">cup product in ordinary differential cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mn>7</mn></msub></mrow></msub><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mn>7</mn></msub><mo>,</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>4</mn></msub><mover><mo>∪</mo><mo stretchy="false">^</mo></mover><mover><mstyle mathvariant="bold"><mi>S</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mover><mstyle mathvariant="bold"><mi>G</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>,</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>×</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mover><mo>∪</mo><mo stretchy="false">^</mo></mover></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \hat {\mathbf{G}}_4 \hat \cup \hat \mathbf{S}] \right) \;\colon\; X \times \mathbf{B}G_{conn} \stackrel{(\hat \mathbf{G}_2, \hat \mathbf{c})}{\to} \mathbf{B}^3 U(1)_{conn} \times \mathbf{B}^3 U(1)_{conn} \stackrel{\hat \cup}{\to} \mathbf{B}^7 U(1)_{conn} \,. </annotation></semantics></math></div> <p>(This is an cup product extended Lagrangian of the kind considered in (<a href="#FSSIII">FSSIII</a>).)</p> <p>Notice that the prequantization lift to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> is entirely demanded by the interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\omega_4</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> in interpretations of this setup in <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a>.</p> <p>Moreover, the above considerations do not really need <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to be a <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a> to go through, a manifold with <a class="existingWikiWord" href="/nlab/show/weak+G%E2%82%82+holonomy">weak G₂ holonomy</a> is just as well, hence equipped with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi \in \Omega^3(X)</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><mi>λ</mi><mo>⋆</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex"> \omega_4 = \lambda \star \phi </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ϕ</mi><mo>=</mo><msub><mi>ω</mi> <mn>4</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \phi = \omega_4 \,. </annotation></semantics></math></div> <p>This arises from <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactifications">Freund-Rubin compactifications</a> with <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> (<a href="#BilalDerendingerSfetsos">Bilal-Derendinger-Sfetsos</a>).</p> <h2 id="properties">Properties</h2> <h3 id="moduli_of_fields_abelian_case">Moduli of fields (abelian case)</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles+with+connection">line n-bundles with connection</a> on <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>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></strong></p> <table><thead><tr><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><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau n-fold</a></th><th><a class="existingWikiWord" href="/nlab/show/line+n-bundle">line n-bundle</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/flat+infinity-connection">flat</a>/degree-0 n-bundles</th><th><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> of <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation moduli</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/modular+functor">modular functor</a>/<a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> in <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>/<a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>/<a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+3-fold">Calabi-Yau 3-fold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+3-bundle">line 3-bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+cohomology">CY3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D1+Chern-Simons+theory">D=1 Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D2+Chern-Simons+theory">D=2 Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D3+Chern-Simons+theory">D=3 Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D4+Chern-Simons+theory">D=4 Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D5+Chern-Simons+theory">D=5 Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D6+Chern-Simons+theory">D=6 Chern-Simons theory</a></p> </li> <li> <p><strong>D=7 Chern-Simons theory</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D11+Chern-Simons+theory">D=11 Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-models">AKSZ sigma-models</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite-dimensional+Chern-Simons+theory">infinite-dimensional Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a>, <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hitchin+functional">Hitchin functional</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="abelian_theory_2">Abelian theory</h3> <p>The abelian 7d <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> of a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a> was considered in</p> <ul> <li id="Witten97"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Five-Brane Effective Action In M-Theory</em> J. Geom. Phys.22:103-133,1997 (<a href="http://arxiv.org/abs/hep-th/9610234">arXiv:hep-th/9610234</a>)</p> </li> <li id="Witten98"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>AdS/CFT Correspondence And Topological Field Theory</em> JHEP 9812:012,1998 (<a href="http://arxiv.org/abs/hep-th/9812012">arXiv:hep-th/9812012</a>)</p> </li> </ul> <p>and argued to be the <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic dual</a> to the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of an abelian 2-form connection on a <em>single</em> <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> in its <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> on the worldvolume.</p> <p>The precise formulation of this functional in terms of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> and <a class="existingWikiWord" href="/nlab/show/integral+Wu+structure">integral Wu structure</a> was given in</p> <ul id="HopkinsSinger"> <li><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em><a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Quadratic Functions in Geometry, Topology, and M-Theory</a></em></li> </ul> <h3 id="ReferencesNonabelianTheories">Nonabelian theories</h3> <p>In</p> <ul id="FSSI"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">7d Chern-Simons theory and the 5-brane</a></em></li> </ul> <ul id="FSSII"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+moduli+3-stack+of+the+C-field">The moduli 3-stack of the C-field</a></em></li> </ul> <p>the 7d Chern-Simons action obtained by <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+reduction">compactifying</a> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> including the quantum corrections of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> on a 4-sphere (the <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS7/CFT6</a> setup) is considered and refined to an <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a>. It contains the <a href="#DonaldsonThomas">Donaldson-Thomas</a>-functional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>X</mi></msub><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∧</mo><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\int_X CS(A) \wedge G_4</annotation></semantics></math> as one summand and the <a href="#Witten97">Witten 97</a>-functional as another.</p> <p>Further discussion of <a class="existingWikiWord" href="/nlab/show/extended+Lagrangians">extended Lagrangians</a> for 7d CS theories is in</p> <ul id="FSSIII"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Extended+higher+cup-product+Chern-Simons+theories">Extended higher cup-product Chern-Simons theories</a></em></li> </ul> <h3 id="on_manifolds">On <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds</h3> <p>The Chern-Simons type action functionals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><msub><mo>∫</mo> <mi>X</mi></msub><mi>CS</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">A \mapsto \int_X CS(A) \wedge \omega_4</annotation></semantics></math> on a 7d <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ω</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega_3)</annotation></semantics></math> was first considered in</p> <ul id="DonaldsonThomas"> <li><a class="existingWikiWord" href="/nlab/show/S.+Donaldson">S. Donaldson</a>, R. Thomas, <em>Gauge theory in higher dimensions</em> (<a href="http://www2.imperial.ac.uk/~rpwt/skd.pdf">pdf</a>)</li> </ul> <p>and around (3.23) of</p> <ul id="BaulieuLosevNekrasov"> <li>L. Baulieu, A. Losev, <a class="existingWikiWord" href="/nlab/show/Nikita+Nekrasov">Nikita Nekrasov</a>, <em>Chern-Simons and Twisted Supersymmetry in Higher Dimensions</em>, Nucl.Phys. B522 (1998) 82-104 (<a href="http://arxiv.org/abs/hep-th/9707174">arXiv:hep-th/9707174</a>)</li> </ul> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jan+de+Boer">Jan de Boer</a>, <a class="existingWikiWord" href="/nlab/show/Paul+de+Medeiros">Paul de Medeiros</a>, Sheer El-Showk, Annamaria Sinkovics, <em>Open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Strings</em> (<a href="http://arxiv.org/abs/hep-th/0611080">arXiv:hep-th/0611080</a>)</li> </ul> <p>this is put into the context of <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a> (see around equation (2) in the introduction).</p> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/weak+G%E2%82%82-holonomy">weak G₂-holonomy</a> is in</p> <ul id="BilalDerendingerSfetsos"> <li>A. Bilal, J.-P. Derendinger, K. Sfetsos, <em>(Weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy from Self-duality, Flux and Supersymmetry</em>, Nucl.Phys. B628 (2002) 112-132 (<a href="http://arxiv.org/abs/hep-th/0111274">arXiv:hep-th/0111274</a>)</li> </ul> <h3 id="formulation_in_extended_tqft">Formulation in extended TQFT</h3> <p>Formulation in <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/4-3-2+8-7-6">4-3-2 8-7-6</a></em>, talk at <em><a href="https://people.maths.ox.ac.uk/tillmann/ASPECTS.html">ASPECTS of Topology</a></em> Dec 2012</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 10:48:52. 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