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"Construction of the Reshetikhin-Turaev TQFT from conformal field theory", Jørgen Ellegaard Andersen, Kenji Ueno, 2014

Reviewed December 28, 2023

Citation: Andersen, Jørgen Ellegaard, and Kenji Ueno. Construction of the Reshetikhin-Turaev TQFT from conformal field theory. No. arXiv: 1110.5027. preparation, 2011.

Web: https://arxiv.org/abs/1110.5027

Tags: Foundational, Mathematical, Physical

This paper settles a long story of the equivalence between two different quantization of Chern-Simons theory, at least in the case where the gauge group is SU(n).

There are two standard approaches to the quantization of Chern-Simons theory. There is a version of geometric quantization of character varieties:

> Axelrod, Scott Elliot. Geometric quantization of Chern-Simons gauge theory. Princeton University, 1991.

and there is the skein-theoretic quantization using quantum groups:

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

These are both mathematical realizations of Witten's programme, but it was not immediately clear that they were mathematically equivalent. In fact, their mathematical equivalence is a very deep fact. One way of describing their equivalence is by looking at their associated modular tensor categories. In the Reshetikhin-Turaev picture, the MTC is the semisimplification of the category of finite dimensional representations of the quantum group of the gauge Lie algebra g. The geometric picture gives the category of level k integrable highest weight modules over the affine Lie algebra \hat{g}. These categories have long been known to be equivalent:

> Finkelberg, Michael. "An equivalence of fusion categories." Geometric & Functional Analysis GAFA 6 (1996): 249-267.

It's good to remember, though, that this paper does have an error and its method does not work in all cases:

> Finkelberg, Michael. "Erratum to: An equivalence of fusion categories." Geometric and Functional Analysis 23 (2013): 810-811.

According to Victor Ostrik in a math overflow post, this is a minor issue and modern work assures us of the validity of this equivalence in all cases.

However, the equivalence of these modular tensor categories is not immediately enough to deduce the two theories are the same. One first needs to prove that the geometric quantization is uniquely determined by its genus zero data, which by standard machinery is determined by the modular tensor category. This is a very deep result:

> Andersen, Jørgen Ellegaard, and Kenji Ueno. "Modular functors are determined by their genus zero data." Quantum Topology 3.3 (2012): 255-291.

The result was proved using machinery established in two earlier papers:

> Ellegaard Andersen, Jorgen, and Kenji Ueno. "Abelian Conformal Field theories and Determinant Bundles." arXiv Mathematics e-prints (2003): math-0304135.
> Andersen, Jørgen Ellegaard, and Kenji Ueno. "Geometric construction of modular functors from conformal field theory." Journal of Knot theory and its Ramifications 16.02 (2007): 127-202.

Still, there are some details to be checked. Even once you know that the two modular tensor categories are isomorphic and Chern-Simons quantizations are all determined by their underlying MTC, its nice to track through the explicit equivalances to get a proper isomorphism between the theories and check that everything works correctly and as intended. This is exactly what this paper does.