The Hochschild complex of a finite tensor category

Schweigert, Christoph; Woike, Lukas

doi:10.2140/agt.2021.21.3689

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The Hochschild complex of a finite tensor category

Christoph Schweigert and Lukas Woike

Algebraic & Geometric Topology 21 (2021) 3689–3734

DOI: 10.2140/agt.2021.21.3689

Abstract

Modular functors, ie consistent systems of projective representations of mapping class groups of surfaces, were constructed for nonsemisimple modular categories 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 nonsemisimple case. 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 $SL (2, ℤ)$ 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.

Keywords

modular functor, Hochschild complex, finite tensor category, mapping class group

Mathematical Subject Classification

Primary: 13D03, 18M15, 57K16

References

Publication

Received: 4 October 2020

Revised: 7 January 2021

Accepted: 30 January 2021

Published: 28 December 2021

Authors

Christoph Schweigert
Fachbereich Mathematik Universität Hamburg Hamburg Germany

Lukas Woike
Institut for Matematiske Fag Københavns Universitet København Denmark

Volume 21, issue 7 (2021)