# 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 Crane-Yetter model

The Turaev-Viro model is a boundary field theory of the 4d TQFT Crane-Yetter model (Barrett&Garci-Islas&Martins 04, theorem 2) Related discussion is in Freed4-3-2 8-7-6”.

### As an extended TQFT

The Turaev-Viro model has been constructed as a 3-2-1 extended TQFT in (Kirillov-Balsam 10, Balsam 10a, Balsam 10b).

Related but differnt 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.

## References

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

• Adrian Ocneanu, Chirality for operator algebras, In Subfactors (Kyuzeso, 1993), pp. 39-63. World Sci. Publ., River Edge, NJ, 1994

In

it was shown that this construction proceeds from any spherical fusion category.

Relation to the Crane-Yetter model was discussed in

Refinement of the construction to an extended TQFT is in

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

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

Revised on November 18, 2014 15:20:18 by Manuel Bärenz? (128.243.59.6)