# Contents

## Idea

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.

## Properties

### As a boundary of the Yetter model

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 Freed4-3-2 8-7-6”.

## References

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

• Alexander Kirillov Jr, String-net model of Turaev-Viro invariants (arXiv:1106.6033)

Revised on October 10, 2013 21:05:49 by Urs Schreiber (89.204.130.176)