The Dehn Twist Action for Quantum Representations of Mapping Class Groups

TL;DR

This paper investigates the Dehn twist actions on conformal blocks within non-semisimple modular categories, determining their order and implications for mapping class group representations, with applications to Johnson and Torelli groups.

Contribution

It extends the understanding of Dehn twist orders in non-semisimple settings and relates these to ribbon twists and group actions, generalizing previous results.

Findings

Dehn twist order equals the ribbon twist order for non-separating curves.

Explicit expression for the order in separating cases using monoidal powers.

Johnson kernels act trivially iff the ribbon twist and double braiding are trivial.

Abstract

We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For Dehn twists about non-separating simple closed curves, we prove that this order is the order of the ribbon twist, thereby generalizing a result that De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel obtained for the small quantum group. In the separating case, we express the order using the order of the ribbon twist on monoidal powers of the canonical end. As an application, we prove that the Johnson kernels of the mapping class groups act trivially if and only if for the canonical end the ribbon twist and double braiding with itself are trivial. We give a similar result for the visibility of the Torelli groups.