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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="topological_physics">Topological physics</h4> <div class="hide"><div> <p><strong>Topological Physics</strong> – Phenomena in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> controlled by the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (often: the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>) of the <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a>.</p> <p>General theory:</p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/extended+topological+field+theory">extended</a>) <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+state+of+matter">topological state of matter</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gapped+Hamiltonian">gapped Hamiltonian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>, <a class="existingWikiWord" href="/nlab/show/symmetry+protected+trivial+order">symmetry protected trivial order</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+insulator">topological insulator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>, <a class="existingWikiWord" href="/nlab/show/quantum+spin+Hall+effect">quantum spin Hall effect</a></p> <p><a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>, <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a></p> </li> </ul> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/metamaterials">metamaterials</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+photonics">topological photonics</a> (<a class="existingWikiWord" href="/nlab/show/light+waves">light waves</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+phononics">topological phononics</a> (<span class="newWikiWord">sound waves<a href="/nlab/new/sound+waves">?</a></span>)</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#nontopological_qfts'>Non-topological QFTs</a></li> <li><a href='#Examples'>Examples</a></li> <li><a href='#homotopy_qfts'>Homotopy QFTs</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#OriginInPhysics'>Origin in physics</a></li> <li><a href='#global_1functorial_tqft'>Global (1-functorial) TQFT</a></li> <li><a href='#local_functorial_tqft'>Local (<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>-functorial) TQFT</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>topological quantum field theory</strong> is a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> which – as a <a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a> – is a functor on a flavor of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mi>S</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord_n^S</annotation></semantics></math>, where the <a class="existingWikiWord" href="/nlab/show/n-morphism">n-morphism</a>s are <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s without any non-topological further structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> – for instance no <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> structure – but possibly some “topological structure”, such as <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a> or similar.</p> <p>For more on the general idea and its development, see <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> and <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Often topological <em>quantum</em> field theories are just called <em>topological field theories</em> and accordingly the abbreviation TQFT is reduced to TFT. Strictly speaking this is a misnomer, which is however convenient and very common. It should be noted, however, that TQFTs may have classical counterparts which would better deserve to be called TFTs. But they are not usually.</p> </div> <h2 id="nontopological_qfts">Non-topological QFTs</h2> <p>In contrast to topological QFTs, non-topological quantum field theories in the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> description are <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>-functors 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>-categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mi>S</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^S_n</annotation></semantics></math> whose morphisms are manifolds with extra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-structure, for instance</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">S =</annotation></semantics></math> conformal structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">S =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> “euclidean QFT”</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">S =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> “relativistic QFT”</p> </li> </ul> <h2 id="Examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+Chern-Simons+theory">2d Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a></p> <ul> <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> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Turaev-Viro+model">Turaev-Viro model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4d+TQFT">4d TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topologically+twisted+D%3D4+super+Yang-Mills+theory">topologically twisted D=4 super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4d+Chern-Simons+theory">4d Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter+model">Yetter model</a></p> </li> </ul> </li> </ul> <h2 id="homotopy_qfts">Homotopy QFTs</h2> <p>These somehow lie between the previous two types. There is some simple extra structure in the form of a ‘characteristic map’ from the manifolds and bordisms to a ‘background space’ <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>. In many of the simplest examples, this is taken to be the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of a group, but this is not the only case that can be considered. The topic is explored more fully in <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+field+theory">cohomological field theory</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>, <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+phase+of+matter">topological phase of matter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></p> </li> </ul> <h2 id="References">References</h2> <p>See also the references at <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a>, <a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a> and <a class="existingWikiWord" href="/nlab/show/4d+TQFT">4d TQFT</a>.</p> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> for <a class="existingWikiWord" href="/nlab/show/topological+field+theories">topological field theories</a> via <a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+differential+cohomology">equivariant ordinary differential cohomology</a>:</p> <ul> <li>Joe Davighi, Ben Gripaios, <a class="existingWikiWord" href="/nlab/show/Oscar+Randal-Williams">Oscar Randal-Williams</a>, <em>Differential cohomology and topological actions in physics</em> (<a href="https://arxiv.org/abs/2011.05768">arXiv:2011.05768</a>)</li> </ul> <h3 id="OriginInPhysics">Origin in physics</h3> <p>The concept originates in the guise of <a class="existingWikiWord" href="/nlab/show/cohomological+quantum+field+theory">cohomological quantum field theory</a> motivated from TQFTs appearing in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> in</p> <ul> <li id="Witten88"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Topological quantum field theory</em>, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. (<a href="http://projecteuclid.org/euclid.cmp/1104161738">euclid:1104161738</a>)</p> </li> <li id="Witten91"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Introduction to cohomological field theory</em>, International Journal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (<a class="existingWikiWord" href="/nlab/files/WittenCQFT.pdf" title="pdf">pdf</a>)</p> </li> <li id="CordesMooreRamgoolam94"> <p>Stefan Cordes, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, Sanjaye Ramgoolam, <em>Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories</em>, Nucl. Phys. Proc. Suppl.41:184-244,1995 (<a href="http://arxiv.org/abs/hep-th/9411210">arXiv:hep-th/9411210</a>)</p> </li> </ul> <p>and in the discussion of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> (“Schwarz-type TQFT”) in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Quantum Field Theory and the Jones Polynomial</em>, Commun. Math. Phys. <strong>121(( (3) (1989) 351-399. [<a href="http://projecteuclid.org/euclid.cmp/1104178138">euclid:cmp/1104178138</a>, MR0990772]</strong></li> </ul> <p>Review:</p> <ul> <li>Danny Birmingham, <a class="existingWikiWord" href="/nlab/show/Matthias+Blau">Matthias Blau</a>, Mark Rakowski, <a class="existingWikiWord" href="/nlab/show/George+Thompson">George Thompson</a>, <em>Topological field theory</em>, Physics Reports <strong>209</strong> 4–5 (1991) 129-340 [<a href="https://doi.org/10.1016/0370-1573(91)90117-5">doi:10.1016/0370-1573(91)90117-5</a>]</li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Markus+Banagl">Markus Banagl</a>, <em>Positive topological quantum field theories</em>, Quantum Topology <strong>6</strong> 4 (2015) 609-706 [<a href="http://arxiv.org/abs/1303.4276">arxiv/1303.4276</a>, <a href="https://doi.org/10.4171/qt/71">doi:10.4171/qt/71</a>]</li> </ul> <h3 id="global_1functorial_tqft">Global (1-functorial) TQFT</h3> <p>The <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>-<a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> for global (i.e. <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">1-functorial</a>) TQFTs are due to:</p> <ul> <li id="Atiyah89"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>Topological quantum field theories</em>, Publications Mathématiques de l’IHÉS <strong>86</strong> (1989) 175-186 [<a href="http://www.numdam.org/item?id=PMIHES_1988__68__175_0">numdam:PMIHES_1988__68__175_0</a>]</li> </ul> <p>Exposition of the conceptual ingredients:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Quantum Quandaries: a Category-Theoretic Perspective</em> (<a href="http://arxiv.org/abs/quant-ph/0404040">arXiv:quant-ph/0404040</a>)</li> </ul> <p>More technical lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Lectures on topological quantum field theory</em>, in: <em>Integrable Systems, Quantum Groups, and Quantum Field Theories</em>, NATO ASI Series <strong>409</strong> (1992) [<a href="https://doi.org/10.1007/978-94-011-1980-1_5">doi:10.1007/978-94-011-1980-1_5</a>, <a href="http://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Freed-TQFTLectures.pdf" title="pdf">pdf</a>]</p> </li> <li id="Quinn95"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Quinn">Frank Quinn</a>, <em>Lectures on axiomatic topological quantum field theory</em>, in <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Karen+Uhlenbeck">Karen Uhlenbeck</a> (eds.) <em>Geometry and Quantum Field Theory</em> <strong>1</strong> (1995) [<a href="https://doi.org/10.1090/pcms/001">doi:10.1090/pcms/001</a>]</p> </li> <li id="Walker06"> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Walker">Kevin Walker</a>, <em>TQFTs</em>, 2006 (<a href="http://canyon23.net/math/tc.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Khovanov">Mikhail Khovanov</a> (notes by <a class="existingWikiWord" href="/nlab/show/You+Qi">You Qi</a>), §2 in: <em>Introduction to categorification</em>, lecture notes, Columbia University (2010, 2020) [<a href="https://www.math.columbia.edu/~khovanov/cat2020/">web</a>, <a href="https://you-qi2121.github.io/mypage/categorificationnotes.html">web</a>, full:<a href="https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0">pdf</a>]</p> <blockquote> <p>(with an eye towards <a class="existingWikiWord" href="/nlab/show/link+homology">link homology</a>)</p> </blockquote> </li> </ul> <p>An introduction specifically to <a class="existingWikiWord" href="/nlab/show/2d+TQFTs">2d TQFTs</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joachim+Kock">Joachim Kock</a>, <em>Frobenius algebras and 2D topological quantum field theories</em>, No. 59 of LMSST, Cambridge University Press, 2003., (full information <a href="http://mat.uab.es/~kock/TQFT.html">here</a>).</li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></em>.</p> <p>Relation to <a class="existingWikiWord" href="/nlab/show/cutting+and+pasting+of+manifolds">cut-and-paste-ivariants</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Carmen+Rovi">Carmen Rovi</a>, Matthew Schoenbauer, <em>Relating Cut and Paste Invariants and TQFTs</em>, The Quarterly Journal of Mathematics <strong>73</strong> 2 (2022) 579–607 [<a href="https://arxiv.org/abs/1803.02939">arXiv:1803.02939</a>, <a href="https://doi.org/10.1093/qmath/haab044">doi:10.1093/qmath/haab044</a>]</li> </ul> <p>See also:</p> <ul> <li id="Torzewska22"> <p><a class="existingWikiWord" href="/nlab/show/Fiona+Torzewska">Fiona Torzewska</a>, <em>Topological quantum field theories and homotopy cobordisms</em> [<a href="https://arxiv.org/abs/2208.14504">arXiv:2208.14504</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fiona+Torzewska">Fiona Torzewska</a>, <em>Topological Quantum Field Theories and Homotopy Cobordisms</em>, <a href="CQTS##TorzewskaDec2023">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Dec 2023) [slides:<a class="existingWikiWord" href="/nlab/files/Torzewska-TQFTandHomCob.pdf" title="pdf">pdf</a>, video:<a href="https://youtu.be/7rtJ61EPL-M">YT</a>]</p> </li> </ul> <h3 id="local_functorial_tqft">Local (<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>-functorial) TQFT</h3> <p>The local <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> formulation (i.e. <a class="existingWikiWord" href="/nlab/show/n-functor">n-functorial</a>) together with the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> was suggested in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/James+Dolan">James Dolan</a>, <em>Higher dimensional algebra and Topological Quantum Field Theory</em> J.Math.Phys. 36 (1995) 6073-6105 (<a href="http://arxiv.org/abs/q-alg/9503002">arXiv:q-alg/9503002</a>)</li> </ul> <p>and formalized and proven in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a>; <em>TQFT and the Cobordism Hypothesis</em> (<a href="http://www.ma.utexas.edu/video/dafr/lurie/">video</a>, <a href="http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html">notes</a>)</li> </ul> <p>This also shows how <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> fits in, which formalizes the original proposal of 2d <a class="existingWikiWord" href="/nlab/show/cohomological+quantum+field+theory">cohomological quantum field theory</a>.</p> <p>Lecture notes:</p> <ul> <li id="Teleman14"><a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Five lectures on topological field theory</em>, in <em>Geometry and Quantization of Moduli Spaces</em>, CRM Advanced Courses in Mathematics, Birkhäuser (2016) [<a href="https://doi.org/10.1007/978-3-319-33578-0_3">doi:10.1007/978-3-319-33578-0_3</a>, <a href="http://math.berkeley.edu/~teleman/math/barclect.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Teleman-LecturesOnTFT.pdf" title="pdf">pdf</a>]</li> </ul> <p>A discussion amplifying the aspects of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <em>Topological field theory, higher categories, and their applications</em>, survey for ICM 2010, (<a href="http://arxiv.org/abs/1004.2307">arxiv/1004.2307</a>)</li> </ul> <p>See also</p> <ul> <li>Mark Feshbach, <a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, <em>A higher category of cobordisms and topological quantum field theory</em> [<a href="http://arxiv.org/abs/1108.3349">arxiv/1108.3349</a>]</li> </ul> <p>Indication of local <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> in the context of <a class="existingWikiWord" href="/nlab/show/infinity-Dijkgraaf-Witten+theory">infinity-Dijkgraaf-Witten theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em><a class="existingWikiWord" href="/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups">Topological Quantum Field Theories from Compact Lie Groups</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 4, 2024 at 15:15:33. 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