+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Quantum field theory +--{: .hide} [[!include functorial quantum field theory - contents]] =-- #### Physics +-- {: .hide} [[!include physicscontents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The _Turaev-Viro model_, or more precisely _Turaev-Viro-Barrett-Westbury model_, is a [[3d TQFT]] construction induced from a [[spherical category]]/[[fusion category]] $\mathcal{C}$. If $\mathcal{C}$ is moreover a [[modular tensor category]] then there exists also the [[Reshetikhin-Turaev construction]] [[3d TQFT]]. In this case the Turaev-Viro model is something like the "norm square" of the [[Reshetikhin-Turaev construction]] on $\mathcal{C}$. For $G$ a [[finite group]] and $\mathcal{C} = Vect_G$ the [[category]] of $G$-[[graded vector spaces]] the Turaev-Viro model describes the $G$-[[Dijkgraaf-Witten theory]], also the [[Levin-Wen model]]. See for instance the introduction of ([Kirillov-Balsam 10](#KirillovBalsam10)) for a review. ## Properties ### As an extended TQFT The Turaev-Viro model has been constructed as a 3-2-1 [[extended TQFT]] in ([Kirillov-Balsam 10](#KirillovBalsam10), [Balsam 10a](#Balsam10a), [Balsam 10b](#Balsam10b)). Related but different is the construction of fully extended [[3d TQFT]] from [[fusion categories]] via the [[cobordism theorem]], see at _[fusion category -- Relation to extended 3d TQFT](fusion%20category#RelationtoTQFT)_ for more on this. ## References The original articles * [[Vladimir Turaev]] and O. Ya. Viro, _State sum invariants of 3-manifolds and quantum 6jsymbols_, Topology 31 (1992), no. 4, 865–902, DOI 10.1016/0040-9383(92)90015-A. MR1191386 * [[Vladimir Turaev]], _Quantum invariants of knots and 3-manifolds_, de Gruyter Studies in Mathematics **18** Walter de Gruyter & Co., Berlin, 1994 ([doi:10.1515/9783110435221](https://doi.org/10.1515/9783110435221)) constructed 3-manifold invariants from quantum [[6j symbols]]. See also * [[Adrian Ocneanu]], _Chirality for operator algebras_, In _Subfactors (Kyuzeso, 1993), pp. 39-63. World Sci. Publ., River Edge, NJ, 1994 In * [[John Barrett]], [[Bruce Westbury]], _Invariants of piecewise-linear 3-manifolds_, Trans. Amer. Math. Soc. 348 (1996), no. 10, 3997–4022, ([doi:10.1090/S0002-9947-96-01660-1](https://doi.org/10.1090/S0002-9947-96-01660-1), MR1357878) * [[John Barrett]], [[Bruce Westbury]], _Spherical Categories_, Adv. Math. 143:357-375, 1999 ([arXiv:hep-th/9310164](http://arxiv.org/abs/hep-th/9310164), [doi:10.1006/aima.1998.1800](https://doi.org/10.1006/aima.1998.1800)) it was shown that this construction proceeds from any [[spherical category|spherical]] [[fusion category]]. Relation to the [[Crane-Yetter model]]: * {#BarrettGarciIslasMartins04} [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281)) Refinement of the construction to an [[extended TQFT]] is in * {#KirillovBalsam10} [[Alexander Kirillov, Jr.]], [[Benjamin Balsam]]: _Turaev-Viro invariants as an extended TQFT_ [[arXiv:1004.1533](http://arxiv.org/abs/1004.1533)] * {#Balsam10a} [[Benjamin Balsam]], _Turaev-Viro invariants as an extended TQFT II_ ([arXiv:1010.1222](http://arxiv.org/abs/1010.1222)) * {#Balsam10b} [[Benjamin Balsam]], _Turaev-Viro invariants as an extended TQFT II_ ([arXiv:1012.0560](http://arxiv.org/abs/1012.0560)) Discussion that relates the [[quantization of 3d Chern-Simons theory]] to the [[Reshetikhin-Turaev construction]] of a 3d-[[TQFT]] from the [[modular tensor category]] induced by $G$ is in * {#Andersen} [[Jørgen Andersen]], _A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications_ ([arXiv:1206.2785](arxiv.org/abs/1206.2785)) and references cited there. On [[string-net model]] realization related to the [[Levin-Wen model]]: * [[Alexander Kirillov, Jr.]]: _String-net model of Turaev-Viro invariants_ ([arXiv:1106.6033](http://arxiv.org/abs/1106.6033)) Discussion of the [[Ising model]] [[2d CFT]] as a boundary theory to a 3d [[TQFT]] based on the [[Turaev-Viro model]], similar to the [[CS-WZW correspondence]], and the phenomenon of [[Kramers-Wannier duality]], is in: * {#FreedTeleman18} [[Daniel Freed]], [[Constantin Teleman]], _Topological dualities in the Ising model_ ([arXiv:1806.00008](https://arxiv.org/abs/1806.00008)) Discussion of the connection with [[Dijkgraaf-Witten theory]] * Jerome Petit, 2006. _The invariant of Turaev-Viro from Group category_ ([math/0608614](https://arxiv.org/abs/math/0608614)) [[!redirects Turaev-Viro construction]] [[!redirects Turaev-Viro-Barrett-Westbury model]] [[!redirects Turaev-Viro-invariant]] [[!redirects Turaev-Viro-invariants]] [[!redirects Turaev-Viro invariant]] [[!redirects Turaev-Viro invariants]] [[!redirects TV invariant]] [[!redirects TV invariants]] [[!redirects TV-invariant]] [[!redirects TV-invariants]]