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## "Quantum invariants of knots and 3-manifolds", Vladimir Turaev, 1994

*Reviewed December 26, 2023*

*Citation:* Turaev, Vladimir G. Quantum invariants of knots and 3-manifolds. de Gruyter, 2010.

*Web:* https://www.maths.ed.ac.uk/~v1ranick/papers/turaev5.pdf

*Tags:* Mathematical, Foundational, Modular-tensor-categories, TQFT, Quantum-groups

This paper shows that every modular tensor category gives a topological quantum field theory. This is one
of the most essential theorems in the algebraic theory of topological quantum computation. The construction is based on the earlier paper

> Reshetikhin, Nicolai, and Vladimir G. Turaev. "Invariants of 3-manifolds via link polynomials and quantum groups." Inventiones mathematicae 103.1 (1991): 547-597.

which showed that modular Hopf algebras give TQFTs. This gives a solution to "Witten's conjecture on the existence of non-trivial 3 dimensional TQFTs".
The authors describe the paper as follows:
"To sum up, we start with a purely algebraic object (a modular category) and
build a topological theory of modules of states of surfaces and operator invariants of 3-cobordisms.
This construction reveals an algebraic background to 2-dimensional modular functors and 3-dimensional TQFT's. It is precisely because
there are non-trivial modular categories, that there exist non-trivial 3-dimensional TQFT's."

Additionally, this paper shows that a unitary modular tensor category gives rise to a unitary TQFT. Another fantastic feature of this paper is that they show that the map MTC => TQFT is injective. That is,
if you have a TQFT which comes from an MTC then it must come from a *unique* MTC.

Finally, this paper details the only construction of MTCs which was known at the time: quantum groups.
After giving this construction, they also present the Temperley-Lieb construction and show that it is equivalent
to the U_q(SL_2(C)) quantum group.