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"Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups I", Simon Lentner, et al., 2023

Reviewed September 4, 2023

Citation: Lentner, Simon, et al. Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups I. Vol. 44. Springer Nature, 2023.

Web: https://arxiv.org/abs/2003.06527

Tags: Mathematical, Expository, Modular-tensor-categories, Non-finite/semisimple

This paper discussed the action induced projective representations of mapping class groups induced by non semisimple modular tensor categories. The big idea is that the associated functors to the theory are no longer exact, but they are left exact. Hence, you get derived functors. These give interesting cohomology groups. In the special case of factorizable ribbon Hopf algebras acting on the torus, one recovers Hochschild cohomology. he nice thing is that your action of the mapping class group goes all the way up, inducing actions on every cohomology group.

This paper is a nice piece of motivation for the fact that you can loosen up your restrictions on the category and still get interesting theories. This motivates the use of categories, because if categories weren't used then one would have to re-prove everything in this new setting. Another nice thing about this paper is that it is published as a "Springer Brief", and so there is lots of background given. The first half of the paper or so is an introduction to the theory of mapping class groups. Of course, this is still not the best place to learn about mapping class groups. That remains the primer by Farb and Margalit,

> Farb, Benson, and Dan Margalit. A primer on mapping class groups (pms-49). Vol. 41. Princeton university press, 2011.

Non-semisimple categories also do give rise to TQFTs and Levin-Wen type models, as described in

> Geer, Nathan, et al. "Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs." Annals of Physics 442 (2022): 168937.

In short, letting your category be non-semisimple is a natural direction to go! The original article on non semisimplicity in quantum algebra comes back in 2001, in the same paper where extended TQFTs were introduced:

> Kerler, Thomas, and Volodymyr V. Lyubashenko. Non-semisimple topological quantum field theories for 3-manifolds with corners. Vol. 1765. Springer Science & Business Media, 2001.