The Hochschild Complex of a Finite Tensor Category

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.

arXiv comment26 pages, 5 figures; small changes, to appear in Alg. Geom. Top

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Cited by in corpus (2)

• Homotopy Coherent Mapping Class Group Actions and Excision for Hochschild Complexes of Modular Categories
• Remarks on the derived center of small quantum groups