physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
The Turaev-Viro 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 is a boundary field theory of the 4d TQFT Yetter model (Barrett&Garci-Islas&Martins 04, theorem 2) Related discussion is in Freed “4-3-2 8-7-6”.
The original article is
V. G. 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 (94d:57044
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, J. Garcia-Islas, João Faria Martins, Observables in the Turaev-Viro and Crane-Yetter models, J. Math. Phys. 48:093508, 2007 (arXiv:math/0411281)
Refinement of the construction to an extended TQFT is in
Discussion that relates the geometric quantization of $G$-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