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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="functorial_quantum_field_theory">Functorial 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="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='#Classification'>Classification</a></li> <ul> <li><a href='#with_coefficients_in_algebras_in_chain_complexes'>With coefficients in (algebras in) chain complexes</a></li> <li><a href='#general_version'>General version</a></li> </ul> <li><a href='#ActionFunctionals'>Worldsheet and effective background theories</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 term <em>topological conformal field theory</em> (TCFT) is used for a linearization or <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of something that is like a <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> (CFT) up to <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>. It is a notion somewhere half-way between a (2-dimensional) <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> and a <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>.</p> <blockquote> <p>(Actually, the remnant of conformal structure here should be just an artefact of the way to parameterize the moduli space of surfaces. As the classification result by Lurie discussed below shows, TCFTs are really <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-TFTs.)</p> </blockquote> <p>This formalizes the physics notion of “the <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>”, a <em>topologically twisted</em> superconformal field theory, such as, notably, the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>. TCFTs are therefore a tool for formalizing <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a>.</p> <p>Recall that an ordinary <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> (CFT) is, in <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>-language, a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a> on a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>conf</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord_2^{conf}</annotation></semantics></math> whose objects are disjoint unions of intervals and circles, and whose <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s are <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>s with these 1d manifolds as incoming and outgoing punctures.</p> <p>Since Riemann surfaces form a well-understood <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>, one can turn this also into a <a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>, i.e. an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mrow><mi>conf</mi><mo>,</mo><mi>top</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Bord_{2}^{conf,top}</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/hom-space">hom-space</a>s are these <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>s of <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>s with given 1d manifolds as incoming and outgoing punctures.</p> <p>A “truly topological conformal field theory” would be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mrow><mi>conf</mi><mo>,</mo><mi>top</mi></mrow></msubsup><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> Bord_2^{conf,top} \to \infty Grpd </annotation></semantics></math></div> <p>or similar. But what is actually called a “topological conformal field theory” is the linearization or <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of this:</p> <p>in a TCFT, this <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of conformal <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s is replaced by a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> whose <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>s (when modeled by a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>) are just the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>s of the original <a class="existingWikiWord" href="/nlab/show/hom-space">hom-space</a>s.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mrow><mi>conf</mi><mo>,</mo><mi>dg</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Bord_2^{conf,dg}</annotation></semantics></math> for the resulting <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> of Riemann cobordisms. Then a TCFT is a an homotopy-symmetric monoidal <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msubsup><mi>Bord</mi> <mn>2</mn> <mrow><mi>conf</mi><mo>,</mo><mi>dg</mi></mrow></msubsup><mo>→</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> F : Bord_2^{conf,dg} \to Ch_\bullet </annotation></semantics></math></div> <p>to the symmetric monoidal <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> of chain complexes.</p> <p>This means in particular that when two <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> are homologous as chains in the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of Riemann surfaces, then the TCFT will send them to two equivalent morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\Sigma_1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\Sigma_2}</annotation></semantics></math> of chain complexes between the in- and the output states. The equivalence between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\Sigma_1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\Sigma_2}</annotation></semantics></math>, however, is not unique neither up to equivalence. Rather, it funtorially depends on the 1-chain realizing the homology equivalence between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> as 0-chains in the moduli space. In particular, two non-homologous 1-chains between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> will in general lead to non-equivalent equivalences between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\Sigma_1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\Sigma_2}</annotation></semantics></math>.</p> <h2 id="Definition">Definition</h2> <p>According to <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">ClassTFT</a> the original definition of the domain for TCFTs can be formulated as follows (without reference to any conformal or Riemann structure).</p> <p><strong>Definition</strong> The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> of non-compact 2-dimensional cobordism is defined as follows:</p> <ul> <li> <p>The objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> are oriented 0-manifolds.</p> </li> <li> <p>Given a pair of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">X, Y \in Bord^{nc}_2</annotation></semantics></math> , a 1-morphism from <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is an oriented bordism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">B : X \to Y</annotation></semantics></math>.</p> </li> <li> <p>Given a pair of 1-morphsims <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>,</mo><mi>B</mi><mo>′</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">B,B' : X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math>, a 2-morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">B'</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> is an oriented bordism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\Sigma: B \to B'</annotation></semantics></math> (which is trivial along <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>) with the following property: every connected component of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> has nonempty intersection with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">B'</annotation></semantics></math>.</p> </li> <li> <p>Higher morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> are given by (orientation preserving) diffeomorphisms, isotopies between diffeomorphisms, and so forth.</p> </li> </ul> <p>Then, the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> becomes</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C2%29-category">symmetric monoidal (∞,2)-category</a>. Then symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-functor">(∞,2)-functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>:</mo><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> Z : Bord^{nc}_2 \to C </annotation></semantics></math></div> <p>are equivalent to <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+objects">Calabi-Yau objects</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> in <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>: the functor <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">Z</annotation></semantics></math> sends the point to <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>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">ClassTFT, theorem 4.2.11</a>. One can “unfold” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> and the theorem above, obtaining a statement in terms of <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-categories</a>. Actually it was the unfolded version to be proven first, (<a href="#Costello04">Costello 04</a>).</p> <p>in the particular case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C=Ch_\bullet</annotation></semantics></math>. We state it below in the general version given by <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a> in <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">ClassTFT</a>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{OC}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of open-closed strings, described as follows:</p> <ol> <li> <p>objects are oriented 1-manifolds with boundary;</p> </li> <li> <p>morphisms are oriented bordisms between 1-manifolds such that each connected component has non-vanishing intersection with the codomain 1-manifold;</p> </li> <li> <p>the higher morphisms are given by orientation preserving <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>s, isotopies between these, and so forth.</p> </li> </ol> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> for the full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> on disjoint unions of intervals (open strings sector).</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">ClassTFT, above theorem 4.2.13</a>.</p> <h2 id="Classification">Classification</h2> <h3 id="with_coefficients_in_algebras_in_chain_complexes">With coefficients in (algebras in) chain complexes</h3> <p>The original statement of the classification result for TCFTs concerned symmetric homotopy-monoidal functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mrow><mi>conf</mi><mo>,</mo><mi>dg</mi></mrow></msubsup><mo>→</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Bord_2^{conf,dg} \to Ch_\bullet</annotation></semantics></math>:</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>(<a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Costello</a>, following <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Kontsevich</a>)</p> <ol> <li> <p>The category of open TCFTs with set <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> of D-branes is equivalent to that of <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+categories">Calabi-Yau categories</a> with set <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> of objects.</p> </li> <li> <p>The homology of the chain complex of closed states of the universal extension of an open TCFT to an open-closed TCFT is the <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+category">Calabi-Yau category</a>.</p> </li> </ol> </div> <p>In (<a href="#Costello04">Costello 04</a>) this is proven using information about cell decompositions of the moduli space of punctured Riemann surfaces, thus effectively presenting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mrow><mi>conf</mi><mo>,</mo><mi>dg</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Bord_2^{conf,dg}</annotation></semantics></math> by generators-and-relations, The then theorem amounts to noticing that representations of these generators and relations define the operations in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category with pairing operation.</p> <h3 id="general_version">General version</h3> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a>. Then symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>:</mo><mi>𝒪</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> Z : \mathcal{O} \to C </annotation></semantics></math></div> <p>are equivalent to <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+algebra">Calabi-Yau algebra</a> <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">objects</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> in <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>: the functor <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">Z</annotation></semantics></math> sends the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> to <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>.</p> </div> <p>This is the result of spring <a href="http://arxiv.org/abs/math/0412149">Cos04</a> reformulated and generalized according to <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">ClassTFT, theorem 4.2.14</a>.</p> <p>This is a special case of the general <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem.</p> <p>The idea of the proof is that a topological open string theory, i.e., a symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>:</mo><mi>𝒪</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Z : \mathcal{O} \to C</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> to an open-closed topological string theory, i.e., to a symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>:</mo><mi>𝒪𝒞</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Z : \mathcal{OC} \to C</annotation></semantics></math>, which is the unfolded version of a symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-functor">(∞,2)-functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mi>nc</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^{nc}_2</annotation></semantics></math> to a symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math>.</p> <h2 id="ActionFunctionals">Worldsheet and effective background theories</h2> <p>One imagines generally that one obtains TCFTs, in their formal definition given above, from worldsheet <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>s as familiar from the physics literature (such as on the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>) by performing the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> and finding from it a collection of <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s on <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of bosonic field configurations.</p> <p>It seems there is at this point no literature giving a direct construction along these lines, but there is the following:</p> <p>In <a href="http://arxiv.org/abs/math/0605647">Cos06</a> is constructed from the geometric input datum of a generalized <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+space">Calabi-Yau space</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><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Q)</annotation></semantics></math> and it is shown that</p> <ol> <li> <p>there is a collection of differential forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>g</mi><mo>,</mo><mi>h</mi></mrow></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_{g,h}(\cdots)</annotation></semantics></math>on the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℳ</mi> <mi>g</mi> <mrow><mi>h</mi><mo>,</mo><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{M}_{g}^{h,n}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>s such that these define a 2d TCFT;</p> <p>(In the discussion leading up to Lemma 4.5.1 there. The proof that this yields a TCFT is theorem 4.5.4.)</p> </li> <li> <p>the partition function of the string perturbation series for the above TCFT is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mfrac linethickness="0"><mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo>></mo><mn>0</mn></mrow></mrow><mrow><mrow><mn>2</mn><mi>g</mi><mo>−</mo><mn>2</mn><mo>+</mo><mi>h</mi><mo>+</mo><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></mrow></mfrac></munder><msup><mi>λ</mi> <mrow><mn>2</mn><mi>g</mi><mo>−</mo><mn>2</mn><mo>+</mo><mi>h</mi></mrow></msup><msup><mi>N</mi> <mi>h</mi></msup><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><msub><mo>∫</mo> <mrow><msubsup><mi>ℳ</mi> <mi>g</mi> <mrow><mi>h</mi><mo>,</mo><mi>n</mi></mrow></msubsup></mrow></msub><msub><mi>K</mi> <mrow><mi>g</mi><mo>,</mo><mi>h</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>a</mi> <mo>⊗</mo></msup><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \sum_{{g,n \geq 0, h \gt 0} \atop {2g-2+h+\frac{n}{2}}} \lambda^{2g-2+h}N^h \frac{1}{n!} \int_{\mathcal{M}_g^{h,n}} K_{g,h}(a^\otimes n) </annotation></semantics></math></div> <p>which is shown to be the partition function of a background <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> coming from 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>a</mi><mo>↦</mo><mi>S</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mi>X</mi></msub><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>Q</mi><mi>a</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>a</mi> <mn>3</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a \mapsto S(a) = \int_X \frac{1}{2} a Q a + \frac{1}{3}a^3 \,. </annotation></semantics></math></div></li> </ol> <p>So this constructs a 2d TCFT and shows that its <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> background quantum field theory is a <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>. While the action functionl on the worldsheet itself, whose <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> should give the <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> considered above, is not explicitly considered here, this does formalizes at least some aspects of an observation that was earlier made in (<a href="#Witten92">Witten 92</a>) where it was observed that Chern-Simons theory is the effective background string theory of 2d TFTs obtained from action functionals of the A-model and the B-model.</p> <p>Similarly the effective background QFT of the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> topological string can be identified. This is known as <em><a class="existingWikiWord" href="/nlab/show/Kodeira-Spencer+gravity">Kodeira-Spencer gravity</a></em> or as <em><a class="existingWikiWord" href="/nlab/show/BCOV+theory">BCOV theory</a></em>.</p> <p>(See also at <em><a class="existingWikiWord" href="/nlab/show/world+sheets+for+world+sheets">world sheets for world sheets</a></em> for a similar mechanism and see at <em><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a></em> for related “physical” strings.)</p> <p>So via the detour over the effective background field theory, this sort of shows that the physicist’s A-model and B-model are indeed captured by the abstract <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> definition of TCFT as given above.</p> <h2 id="related_concepts">Related concepts</h2> <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/Landau-Ginzburg+model">Landau-Ginzburg model</a>, <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HCFT">HCFT</a></p> </li> </ul> <h2 id="references">References</h2> <p>The concept is essentially a formalization of what used to be called <a class="existingWikiWord" href="/nlab/show/cohomological+field+theory">cohomological field theory</a> in</p> <ul> <li id="Witten91"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Introduction to cohomological field theory</em>, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (<a class="existingWikiWord" href="/nlab/files/WittenCQFT.pdf" title="pdf">pdf</a>)</li> </ul> <p>The definition was given independently by</p> <ul> <li id="Getzler92"><a class="existingWikiWord" href="/nlab/show/Ezra+Getzler">Ezra Getzler</a>, <em>Batalin-Vilkovisky algebras and two-dimensional topological field theories</em> , Comm. Math. Phys. 159(2), 265–285 (1994) (<a href="http://arxiv.org/abs/hep-th/9212043">arXiv:hep-th/9212043</a>)</li> </ul> <p>and</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Topological field theory</em> , (1999), Notes of lectures at Stanford university. (<a href="http://www.cgtp.duke.edu/ITP99/segal/">web</a>). See in particular <a href="http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf">lecture 5</a> (“topological field theory with cochain values”).</li> </ul> <p>The classification of TCFTs by <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+categories">Calabi-Yau categories</a> was discussed in</p> <ul> <li id="Costello04"> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>Topological conformal field theories and Calabi-Yau categories</em>, Advances in Mathematics, Volume 210, Issue 1, (2007), (<a href="http://arxiv.org/abs/math/0412149">arXiv:math/0412149</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>The Gromov-Witten potential associated to a TCFT</em> (<a href="http://arxiv.org/abs/math/0509264">arXiv:math/0509264</a>)</p> </li> </ul> <p>following conjectures by <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, e.g.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <em>Homological algebra of mirror symmetry</em> , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139, Basel, 1995, Birkhäuser.</li> </ul> <p>This classification is a precursor of the full <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem. This, and the reformulation of the original TCFT constructions in full generality is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a></em></li> </ul> <p>Here are notes from a seminar on these definitions and results:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a> and <a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a> <em>TCFT seminar</em> (<a href="http://math.berkeley.edu/~cpries/Hot-Topics-07.pdf">pdf notes</a>)</li> </ul> <p>Discussion of the construction of TCFTs from differential forms on moduli space and the way this induces by “<a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a>” effective background Chern-Simons theories is in</p> <ul> <li id="Costello06"><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>Topological conformal field theories and gauge theories</em> (<a href="http://arxiv.org/abs/math/0605647">arXiv:math/0605647</a>)</li> </ul> <p>formalizing at least aspects of the observations in</p> <ul> <li id="Witten92"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Chern-Simons Gauge Theory As A String Theory</em> (<a href="http://arxiv.org/abs/hep-th/9207094">arXiv:hep-th/9207094</a>)</p> </li> <li> <p>P.A. Grassi, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Policastro">Giuseppe Policastro</a>, <em>Super-Chern-Simons Theory as Superstring Theory</em> (<a href="http://arxiv.org/abs/hep-th/0412272">arXiv:hep-th/0412272</a>)</p> </li> </ul> <p>On how the <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> of the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> yields <a class="existingWikiWord" href="/nlab/show/Kodeira-Spencer+gravity">Kodeira-Spencer gravity</a>/<a class="existingWikiWord" href="/nlab/show/BCOV+theory">BCOV theory</a>:</p> <ul> <li id="BCOV93"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Bershadsky">Michael Bershadsky</a>, <a class="existingWikiWord" href="/nlab/show/Sergio+Cecotti">Sergio Cecotti</a>, <a class="existingWikiWord" href="/nlab/show/Hirosi+Ooguri">Hirosi Ooguri</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes</em>, Commun. Math. Phys. <strong>165</strong> (1994) 311-428 [<a href="http://arxiv.org/abs/hep-th/9309140">arXiv:hep-th/9309140</a>, <a href="https://doi.org/10.1007/BF02099774">doi:10.1007/BF02099774</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <a class="existingWikiWord" href="/nlab/show/Si+Li">Si Li</a>, <em>Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model</em> (<a href="http://arxiv.org/abs/1201.4501">arXiv:1201.4501</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Si+Li">Si Li</a>, <em>BCOV theory on the elliptic curve and higher genus mirror symmetry</em> (<a href="http://arxiv.org/abs/1112.4063">arXiv:1112.4063</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Si+Li">Si Li</a>, <em>Variation of Hodge structures, Frobenius manifolds and Gauge theory</em> (<a href="http://arxiv.org/abs/1303.2782">arXiv:1303.2782</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 21, 2023 at 07:17:24. 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