Contents

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) 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, 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.

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)

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, MR1357878)

• John Barrett, Bruce Westbury, Spherical Categories, Adv. Math. 143:357-375, 1999 (arXiv:hep-th/9310164, doi:10.1006/aima.1998.1800)

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

Relation to the Crane-Yetter model:

• 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

• 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

• Jørgen Andersen, A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications (arXiv:1206.2785)

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)

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:

• Daniel Freed, Constantin Teleman, Topological dualities in the Ising model (arXiv:1806.00008)

Discussion of the connection with Dijkgraaf-Witten theory

• Jerome Petit, 2006. The invariant of Turaev-Viro from Group category (math/0608614)