Modular functors from non-semisimple 3d TFTs

TL;DR

This paper constructs explicit modular functors from non-semisimple 3d topological field theories using a categorical approach, recovering known results and addressing anomalies in the process.

Contribution

It provides an explicit description of modular functors from non-semisimple modular tensor categories via 3d TFTs, extending Lyubashenko's work and resolving gluing anomalies.

Findings

Explicit construction of modular functors from non-semisimple categories

Recovery of Lyubashenko's result via categorical methods

Resolution of the gluing anomaly in the modular functor context

Abstract

Given a not necessarily semisimple modular tensor category C, we use the corresponding 3d TFT defined in [arXiv:1912.02063] to explicitly describe a modular functor as a symmetric monoidal 2-functor from a 2-category of oriented bordisms to a 2-category of finite linear categories. This recovers a result by Lyubashenko [arXiv:hep-th/9405168] obtained via generators and relations. Pulling back the modular functor for C to a 2-category of bordisms with orientation reversing involution cancels the gluing anomaly, and further pulling back to the original bordism category along a doubling functor leads to the modular functor for the Drinfeld centre Z(C).