The Hochschild Complex of a Finite Tensor Category

TL;DR

This paper explores the relationship between topological field theory and homological algebra by constructing a homotopy coherent action of the mapping class group on the Hochschild complex of a modular category, advancing understanding of conformal blocks.

Contribution

It introduces a homotopy coherent projective action of the torus mapping class group on the Hochschild complex, linking topological and algebraic structures in non-semisimple modular categories.

Findings

Constructed a homotopy coherent action of SL(2,Z) on the Hochschild complex.

Connected Hochschild complex to differential graded conformal blocks.

Described a differential graded Verlinde algebra.

Abstract

Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the non-semisimple case. In the present paper, we elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus on the Hochschild complex of a modular category. This is a further step towards understanding the Hochschild complex of a modular category as a differential graded conformal block for the torus. Moreover, we describe a differential graded version of the Verlinde algebra.