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"State sum invariants of 3-manifolds and quantum 6j-symbols", Vladimir Turaev, Oleg Viro, 1992

Reviewed August 31, 2023

Citation: Turaev, Vladimir G., and Oleg Ya Viro. "State sum invariants of 3-manifolds and quantum 6j-symbols." Topology 31.4 (1992): 865-902.

Web: http://www.pdmi.ras.ru/~olegviro/6j.pdf

Tags: Mathematical, Foundational, TQFT

This is the paper which introduces the Turaev-Viro TQFT. The TV TQFT takes as input a spherical fusion category and is defined on unorientable manifolds, which seems to make it more general than the Reshetikhin-Turaev TQFT which requires an MTC and is only defined for orientable manifolds, and is sensitive to orientation change. The idea is that, actually, the RT TQFT is more general. The TV is exactly the RT invariant evaluated the Drinfeld center of the input category, and hence it is like a RT TQFT which is only defined for doubled topological phases.

One nice thing about this invariant is that it is much more elementary, and is simpler to define. It is a very explicit state sum over 6j symbols, defined on an arbitrary triangulation. The only subtle part of the theory is checking that the invariant does not depend on the choice of triangulation, but this is not very hard.

This is also the paper which popularized the term "6j" symbol for the associativity of fusion categories, because in this context there is a very literal connection to Wigner's much earlier concept of 6j symbol. Note that I am unsure whether or not Wigner 6j symbols were actually introduced by Wigner. My understanding is that he introduced a certain class of 3j symbols, which were generalized to 6j by Moshinsky:

> Moshinsky, Marcos. "Bases for the irreducible representations of the unitary groups and some applications." Journal of Mathematical Physics 4.9 (1963): 1128-1139.