Invertible braided tensor categories

Brochier, Adrien; Jordan, David; Safronov, Pavel; Snyder, Noah

doi:10.2140/agt.2021.21.2107

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Invertible braided tensor categories

Adrien Brochier, David Jordan, Pavel Safronov and Noah Snyder

Algebraic & Geometric Topology 21 (2021) 2107–2140

DOI: 10.2140/agt.2021.21.2107

Abstract

We prove that a finite braided tensor category $𝒜$ is invertible in the Morita $4$ –category $BrTens$ of braided tensor categories if and only if it is nondegenerate. This includes the case of semisimple modular tensor categories, but also nonsemisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible $4$ –dimensional framed topological field theories, which we regard as a nonsemisimple framed version of the Crane–Yetter–Kauffman invariants, after the Freed–Teleman and Walker constructions in the semisimple case. More generally, we characterize invertibility for $E_{1}$ – and $E_{2}$ –algebras in an arbitrary symmetric monoidal $\infty$ –category, and we conjecture a similar characterization of invertible $E_{n}$ –algebras for any $n$ . Finally, we propose the Picard group of $BrTens$ as a generalization of the Witt group of nondegenerate braided fusion categories, and pose a number of open questions about it.

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Keywords

braided tensor category, topological field theory, higher categories

Mathematical Subject Classification

Primary: 18M20, 57R56

References

Publication

Received: 8 June 2020

Revised: 13 July 2020

Accepted: 27 July 2020

Published: 18 August 2021

Authors

Adrien Brochier
Institut de Mathématiques de Jussieu-Paris Rive Gauche Université de Paris Sorbonne Université, CNRS Paris France

David Jordan
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Pavel Safronov
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Noah Snyder
Mathematics Department Indiana University Bloomington, IN United States

Volume 21, issue 4 (2021)