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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) for a review.
The Turaev-Viro model has been constructed as a 3-2-1 extended TQFT in (Kirillov-Balsam 10, Balsam 10a, Balsam 10b).
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 for more on this.
The original article
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, vol. 18 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994
constructed 3-manifold invariants from quantum 6j symbols. See also
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. MR1357878 (97f:57017)
John Barrett, Bruce Westbury, Spherical Categories (arXiv:hep-th/9310164)
it was shown that this construction proceeds from any spherical fusion category.
Refinement of the construction to an extended TQFT is in
Alexander Kirillov Jr., Benjamin Balsam, Turaev-Viro invariants as an extended TQFT (arXiv:1004.1533)
Benjamin Balsam, Turaev-Viro invariants as an extended TQFT II (arXiv:1010.1222)
Benjamin Balsam, Turaev-Viro invariants as an extended TQFT II (arXiv: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
and references cited there.
A relation to the Levin-Wen model is discussed in