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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#as_a_boundary_of_the_craneyetter_model'>As a boundary of the Crane-Yetter model</a></li> <li><a href='#relation_to_chernsimons_theory'>Relation to Chern-Simons theory</a></li> <li><a href='#relation_to_conformal_field_theory'>Relation to conformal field theory</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The Reshetikhin-Turaev construction is the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> construction of a 3d <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> from the data of a <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> <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">\mathcal{C}</annotation></semantics></math>. It is something like the “square root” of the <a class="existingWikiWord" href="/nlab/show/Turaev-Viro+model">Turaev-Viro model</a> on <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">\mathcal{C}</annotation></semantics></math>.</p> <p>In the case that <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> is a category of <a class="existingWikiWord" href="/nlab/show/positive+energy+representation">positive energy representation</a>s of a <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\Omega G</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, then this algebraically defined QFT is thought to be the result of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> over the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="as_a_boundary_of_the_craneyetter_model">As a boundary of the Crane-Yetter model</h3> <p>The Reshetikhin-Turaev model is a <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a> of the <a class="existingWikiWord" href="/nlab/show/4d+TQFT">4d TQFT</a> <a class="existingWikiWord" href="/nlab/show/Crane-Yetter+model">Crane-Yetter model</a> (<a href="#BarrettGarciIslasMartins04">Barrett&Garci-Islas&Martins 04, theorem 2</a>) Related discussion is in <a class="existingWikiWord" href="/nlab/show/Freed">Freed</a> “<a class="existingWikiWord" href="/nlab/show/4-3-2+8-7-6">4-3-2 8-7-6</a>”.</p> <h3 id="relation_to_chernsimons_theory">Relation to Chern-Simons theory</h3> <p>The RT-construction for group <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> is expected to be the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>, though a fully explicit proof of this via <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> is currently not in the literature.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/quantization+of+Chern-Simons+theory">quantization of Chern-Simons theory</a></em> for more on this.</p> <h3 id="relation_to_conformal_field_theory">Relation to conformal field theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/CFT">Fuchs-Runkel-Schweigert-construction</a> builds from the RT-construction explicitly the rational 2-dimensional <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> boundary theory (see at <em><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></em>).</p> <h2 id="references">References</h2> <p>Original articles:</p> <ul> <li id="ReshetikhinTuraev91"> <p><a class="existingWikiWord" href="/nlab/show/Nikolai+Reshetikhin">Nikolai Reshetikhin</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Invariants of 3-manifolds via link polynomials and quantum groups</em>. Invent. Math. 103 (1991), no. 3, 547–597. (<a href="https://doi.org/10.1007/BF01239527">doi:10.1007/BF01239527</a>, <a href="http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Modular categories and 3-manifold invariants</em>, International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) (<a href="https://doi.org/10.1142/S0217979292000876">doi:10.1142/S0217979292000876</a>)</p> <blockquote> <p>(introducing <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>)</p> </blockquote> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Quantum invariants of knots and 3-manifolds</em>, de Gruyter Studies in Mathematics <strong>18</strong> Walter de Gruyter & Co., Berlin, 1994 (<a href="https://doi.org/10.1515/9783110435221">doi:10.1515/9783110435221</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bojko+Bakalov">Bojko Bakalov</a>, <a class="existingWikiWord" href="/nlab/show/Alexandre+Kirillov">Alexandre Kirillov</a>, <em>Lectures on tensor categories and modular functors</em> AMS, University Lecture Series, (2000) (<a href="https://bookstore.ams.org/ulect-21">ISBN:978-1-4704-2168-7</a>, <a href="http://www.math.sunysb.edu/~kirillov/tensor/tensor.html">webpage</a>).</p> </li> </ul> <p>Review:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Reshetikhin%E2%80%93Turaev_invariant">Reshetikhin-Turaev invariant</a></em></li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%B8rgen+Ellegaard+Andersen">Jørgen Ellegaard Andersen</a>, William Petersen, <em>Construction of Modular Functors from Modular Tensor Categories</em> (<a href="https://arxiv.org/abs/1602.04907">arXiv:1602.04907</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%B8rgen+Ellegaard+Andersen">Jørgen Ellegaard Andersen</a>, <em>Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory</em> (<a href="https://arxiv.org/abs/1110.5027">arXiv:1110.5027</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Walker">Kevin Walker</a>, <em>On Witten’s 3-Manifold Invariants</em> (<a href="http://canyon23.net/math/1991TQFTNotes.pdf">old version</a> <a href="http://canyon23.net/math/tc.pdf">draft of new version</a>)</p> </li> </ul> <p>Discussion that relates the <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> to the Reshetikhin-Turaev construction of a 3d-<a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> from the <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> induced by <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> is in</p> <ul id="Andersen"> <li><a class="existingWikiWord" href="/nlab/show/J%C3%B8rgen+Andersen">Jørgen Andersen</a>, <em>A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications</em> (<a href="http://arxiv.org/abs/1206.2785">arXiv:1206.2785</a>)</li> </ul> <p>and references cited there.</p> <ul> <li>Alain Bruguières, Alexis Virelizier, <em>Hopf diagrams and quantum invariants</em>, <a href="http://arxiv.org/abs/math/0505119">math.QA/0505119</a>; <em>Categorical centers and Reshetikhin-Turaev invariants</em>, <a href="http://arxiv.org/abs/0812.2426">arxiv/0812.2426</a></li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/Crane-Yetter+model">Crane-Yetter model</a>:</p> <ul> <li id="BarrettGarciIslasMartins04"><a class="existingWikiWord" href="/nlab/show/John+Barrett">John Barrett</a>, J. Garcia-Islas, <a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>, <em>Observables in the Turaev-Viro and Crane-Yetter models</em>, J. Math. Phys. 48:093508, 2007 (<a href="http://arxiv.org/abs/math/0411281">arXiv:math/0411281</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 17, 2025 at 09:40:16. 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