More on time-reversal anomaly of 2+1d topological phases

Abstract

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in 2+1 dimensional fermionic topological quantum field theories. The crucial step is to determine the crosscap state in terms of the modular S matrix and $\mathsf{T}^2$ eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.

9 pages; v2:heuristic explanation added, title in the journal version changed to "Derivation of the Time-Reversal Anomaly for (2+1)-Dimensional Topological Phases"