Mapping Class Group Representations From Non-Semisimple TQFTs

TL;DR

This paper explores how non-semisimple TQFTs produce projective representations of surface mapping class groups, linking algebraic data to these representations and demonstrating infinite order actions in specific quantum group cases.

Contribution

It establishes the equivalence of projective representations from non-semisimple TQFTs with Lyubashenko's, and analyzes the order of Dehn twist actions in quantum group representations.

Findings

Projective representations match Lyubashenko's constructions.

Dehn twists have infinite order in certain quantum group cases.

Algebraic data explicitly describes mapping class group actions.

Abstract

In [arXiv:1912.02063], 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 $\mathcal{C}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of [arXiv:1912.02063] are equivalent to those obtained by Lyubashenko via generators and relations in [arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{C}$ is the category of finite-dimensional representations of the small quantum group of $\mathfrak{sl}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.