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Invertible braided tensor categories

Adrien Brochier, David Jordan, Pavel Safronov and Noah Snyder

Algebraic & Geometric Topology 21 (2021) 2107–2140

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 E1– and E2–algebras in an arbitrary symmetric monoidal –category, and we conjecture a similar characterization of invertible En–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.

braided tensor category, topological field theory, higher categories
Mathematical Subject Classification
Primary: 18M20, 57R56
Received: 8 June 2020
Revised: 13 July 2020
Accepted: 27 July 2020
Published: 18 August 2021
Adrien Brochier
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Université de Paris
Sorbonne Université, CNRS
David Jordan
School of Mathematics
University of Edinburgh
United Kingdom
Pavel Safronov
School of Mathematics
University of Edinburgh
United Kingdom
Noah Snyder
Mathematics Department
Indiana University
Bloomington, IN
United States