Invertible braided tensor categories

Abstract

We prove that a finite braided tensor category $\mathcal{𝒜}$ is invertible in the Morita $4$–category $\mathbf{BrTens}$ 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 $\mathbf{BrTens}$ as a generalization of the Witt group of nondegenerate braided fusion categories, and pose a number of open questions about it.