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Invertible braided
tensor categories
Adrien Brochier, David Jordan, Pavel Safronov and Noah Snyder |
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Algebraic & Geometric Topology 21 (2021) 2107–2140
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Abstract |
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We prove that a finite braided tensor category is invertible in the Morita –category 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 –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 – and –algebras in an arbitrary symmetric monoidal –category, and we conjecture a similar characterization of invertible –algebras for any . Finally, we propose the Picard group of as a generalization of the Witt group of nondegenerate braided fusion categories, and pose a number of open questions about it. |
Keywords
braided tensor category, topological field theory, higher
categories
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Mathematical Subject Classification
Primary: 18M20, 57R56
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References |
Publication
Received: 8 June 2020
Revised: 13 July 2020
Accepted: 27 July 2020
Published: 18 August 2021
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Authors |
| Adrien Brochier | |
| Institut de Mathématiques de
Jussieu-Paris Rive Gauche Université de Paris Sorbonne Université, CNRS Paris France |
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| David Jordan | |
| School of Mathematics University of Edinburgh Edinburgh United Kingdom |
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| Pavel Safronov | |
| School of Mathematics University of Edinburgh Edinburgh United Kingdom |
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| Noah Snyder | |
| Mathematics Department Indiana University Bloomington, IN United States |
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