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"Modular categories as representations of the 3-dimensional bordism 2-category", Bruce Bartlett, et. al, 2015

Reviewed October 10, 2023

Citation: Bartlett, Bruce, et al. "Modular categories as representations of the 3-dimensional bordism 2-category." arXiv preprint arXiv:1509.06811 (2015).

Web: https://arxiv.org/abs/1509.06811

Tags: Foundational, Modular-tensor-categories, TQFT

This is the paper that finally establishes a bijection between appropriately defined 1-extended TQFTs and MTCs. Interestingly enough, you need to specify just a little bit of extra data for your MTC - namely, a square root of its global dimension. Different choices of square root result in non-equivalent TQFTs.

While such a bijection was accepted in the community to be true, before this paper it had not been conclusively shown. The correct definition of TQFT in terms of higher category theory is very subtle, and the details of the bijection are very complicated. The Reshetikhin-Turaev construction takes in an MTC and spits out a TQFT. From a TQFT one shows that its "circle sector" is an MTC, which was not known in literature before this paper. The circle sector construction is very straightforward, and is illustrated beautifully in Section 3. Objects in the MTC are generated by circles. Morphisms are generated by the pair of pants bordism, with an implicit appeal to the classification of surfaces. Braiding is the obvious braiding of cylinders.

It should be stressed that this is a deep theorem, and in particular it takes lots of space and formalism to prove. This work, for instance is the last in a four-part series building up to the proof. Only one of the previous three parts is available on the internet:

>Bartlett, Bruce, et al. "Extended 3-dimensional bordism as the theory of modular objects." arXiv preprint arXiv:1411.0945 (2014).

Another point to stress is that fully extended (2+1)-dimensional TQFTs will correspond to 2-categories instead of MTCs, and so the full categorical formalism underlying TQFT is not MTCs. This more complete but still conjectural story is given by the cobordism hypothesis.