Turaev-Viro model


Quantum field theory


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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 𝒞\mathcal{C}.

If 𝒞\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 𝒞\mathcal{C}.

For GG a finite group and 𝒞=Vect G\mathcal{C} = Vect_G the category of GG-graded vector spaces the Turaev-Viro model describes the GG-Dijkgraaf-Witten theory, also the Levin-Wen model.

See for instance the introduction of (Kirillov-Balsam 10) for a review.


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.


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


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

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 GG 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 December 27, 2016 11:16:49 by Manuel Bärenz? (