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href='/nlab/show/diff/cobordism'>cobordism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extended+cobordism'>extended cobordism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category+of+cobordisms'>(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bordism+categories+following+Stolz-Teichner'>Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism+hypothesis'>cobordism hypothesis</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangle+hypothesis'>generalized tangle hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/On+the+Classification+of+Topological+Field+Theories'>classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/functorial+field+theory'>functorial field theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+functorial+field+theory'>unitary functorial field theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extended+functorial+field+theory'>extended functorial field theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conformal+field+theory'>CFT</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/vertex+operator+algebra'>vertex operator algebra</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+field+theory'>TQFT</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Reshetikhin-Turaev+construction'>Reshetikhin-Turaev model</a> / <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/HQFT'>HQFT</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/TCFT'>TCFT</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-model'>A-model</a>, <a class='existingWikiWord' href='/nlab/show/diff/B-model'>B-model</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gromov-Witten+invariants'>Gromov-Witten theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mirror+symmetry'>homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%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/diff/%282%7C1%29-dimensional+Euclidean+field+theory'>(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+model+for+elliptic+cohomology'>geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holographic+principle+of+higher+category+theory'>holographic principle of higher category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holographic+principle'>holographic principle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/AdS-CFT+correspondence'>AdS/CFT correspondence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quantization+via+the+A-model'>quantization via the A-model</a></p> </li> </ul> </li> </ul> </div> </div> </div> <h1 id='contents_2'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><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><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/diff/functorial+field+theory'>FQFT</a> construction of a 3d <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+field+theory'>TQFT</a> from the data of a <a class='existingWikiWord' href='/nlab/show/diff/modular+tensor+category'>modular tensor category</a> <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_1' xmlns='http://www.w3.org/1998/Math/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/diff/Turaev-Viro+model'>Turaev-Viro model</a> on <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>.</p> <p>In the case that <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_3' xmlns='http://www.w3.org/1998/Math/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/diff/positive+energy+representation'>positive energy representation</a>s of a <a class='existingWikiWord' href='/nlab/show/diff/loop+group'>loop group</a> <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_4' xmlns='http://www.w3.org/1998/Math/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/diff/Lie+group'>Lie group</a> <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_5' xmlns='http://www.w3.org/1998/Math/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/diff/quantization'>quantization</a> of <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a> over the group <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_6' xmlns='http://www.w3.org/1998/Math/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/diff/boundary+field+theory'>boundary field theory</a> of the <a class='existingWikiWord' href='/nlab/show/diff/D%3D4+TQFT'>4d TQFT</a> <a class='existingWikiWord' href='/nlab/show/diff/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/diff/Daniel+Freed'>Freed</a> “<a class='existingWikiWord' href='/nlab/show/diff/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 class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_7' xmlns='http://www.w3.org/1998/Math/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/diff/functorial+field+theory'>FQFT</a> of <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>, though a fully explicit proof of this via <a class='existingWikiWord' href='/nlab/show/diff/quantization'>quantization</a> is currently not in the literature.</p> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/quantization+of+D%3D3+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/diff/conformal+field+theory'>Fuchs-Runkel-Schweigert-construction</a> builds from the RT-construction explicitly the rational 2-dimensional <a class='existingWikiWord' href='/nlab/show/diff/conformal+field+theory'>2d CFT</a> boundary theory (see at <em><a class='existingWikiWord' href='/nlab/show/diff/holographic+principle'>holographic principle</a></em>).</p> <h2 id='references'>References</h2> <p><span> Original<del class='diffmod'> articles</del><ins class='diffmod'> articles:</ins><del class='diffdel'> include</del></span></p> <ul> <li id='ReshetikhinTuraev91'><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Nicolai+Reshetikhin'>N. Reshetikhin</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><p><a class='existingWikiWord' href='/nlab/show/diff/Nicolai+Reshetikhin'>Nikolai Reshetikhin</a>, <a class='existingWikiWord' href='/nlab/show/diff/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></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Turaev'>V. Turaev</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><em>Invariants of 3-manifolds via link polynomials and quantum groups</em></del><del class='diffdel'>. Invent. Math. 103 (1991), no. 3, 547–597. (</del><del class='diffdel'><a href='http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf'>pdf</a></del><del class='diffdel'>)</del></li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/modular+tensor+category'>modular tensor categories</a>)</p> </blockquote> </li></ins> </ul> <p><span><del class='diffmod'> A</del><ins class='diffmod'> Textbook</ins><del class='diffmod'> standard</del><ins class='diffmod'> accounts:</ins><del class='diffdel'> textbook</del><del class='diffdel'> account</del><del class='diffdel'> is</del></span></p> <ul> <li><span><del class='diffdel'> B.</del><del class='diffdel'> Bakalov</del><del class='diffdel'> &</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Alexandre+Kirillov'>Alexandre Kirillov</a></del><ins class='diffmod'><p><a class='existingWikiWord' href='/nlab/show/diff/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></ins><span><del class='diffdel'> ,</del></span><del class='diffdel'><em>Lectures on tensor categories and modular functors</em></del><del class='diffdel'> AMS, University Lecture Series, (2000) (</del><del class='diffdel'><a href='http://www.math.sunysb.edu/~kirillov/tensor/tensor.html'>web</a></del><del class='diffdel'>).</del></li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bojko+Bakalov'>Bojko Bakalov</a>, <a class='existingWikiWord' href='/nlab/show/diff/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></ins> </ul> <p><span><del class='diffmod'> (See</del><ins class='diffmod'> Review:</ins><del class='diffdel'> the</del><del class='diffdel'> dedicated</del><del class='diffdel'> page</del></span><del class='diffdel'><em><a class='existingWikiWord' href='/nlab/show/diff/Help+me%21+I%27m+trying+to+understand+Bakalov+and+Kirillov'>Help me! I'm trying to understand Bakalov and Kirillov</a></em></del><del class='diffdel'> for help with understanding the computations in this book.)</del></p> <del class='diffmod'><p>See also</p></del><ins class='diffmod'><ul> <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Reshetikhin%E2%80%93Turaev_invariant'>Reshetikhin-Turaev invariant</a></em></li> </ul></ins> <del class='diffmod'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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>)</li> </ul></del><ins class='diffmod'><p>See also:</p></ins> <del class='diffmod'><p>Discussion that relates the <a class='existingWikiWord' href='/nlab/show/diff/geometric+quantization'>geometric quantization</a> of <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a> to the Reshetikhin-Turaev construction of a 3d-<a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+field+theory'>TQFT</a> from the <a class='existingWikiWord' href='/nlab/show/diff/modular+tensor+category'>modular tensor category</a> induced by <math class='maruku-mathml' display='inline' id='mathml_e06109506c1fa5636be0c8a8da7ee9e0246299ba_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is in</p></del><ins class='diffmod'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/J%C3%B8rgen+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/diff/J%C3%B8rgen+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/diff/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></ins> <ins class='diffins'><p>Discussion that relates the <a class='existingWikiWord' href='/nlab/show/diff/geometric+quantization'>geometric quantization</a> of <math class='maruku-mathml' display='inline' id='mathml_f689bf203ba41bbc021ce3f786504ea1b41650a3_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a> to the Reshetikhin-Turaev construction of a 3d-<a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+field+theory'>TQFT</a> from the <a class='existingWikiWord' href='/nlab/show/diff/modular+tensor+category'>modular tensor category</a> induced by <math class='maruku-mathml' display='inline' id='mathml_f689bf203ba41bbc021ce3f786504ea1b41650a3_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is in</p></ins><ins class='diffins'> </ins><ul id='Andersen'> <li><a class='existingWikiWord' href='/nlab/show/diff/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><span><del class='diffmod'> The</del><ins class='diffmod'> Relation</ins><del class='diffdel'> relation</del> to the</span><a class='existingWikiWord' href='/nlab/show/diff/Crane-Yetter+model'>Crane-Yetter model</a><span><del class='diffmod'> </del><ins class='diffmod'> :</ins><del class='diffdel'> was</del><del class='diffdel'> discussed</del><del class='diffdel'> in</del></span></p> <ul> <li id='BarrettGarciIslasMartins04'><a class='existingWikiWord' href='/nlab/show/diff/John+Barrett'>John Barrett</a>, J. Garcia-Islas, <a class='existingWikiWord' href='/nlab/show/diff/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> <p><del class='diffmod'> </del><ins class='diffmod'> </ins></p><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on May 23, 2021 at 05:17:32. 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