Mapping class group representations from non-semisimple TQFTs

Abstract

In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category C. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467-516, arXiv:hep-th/9405167]. Finally, we show that, when C is the category of finite-dimensional representations of the small quantum group of sl2, the action of all Dehn twists for surfaces without marked points has infinite order.