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

TL;DR

This paper proves a conjectured formula for the time-reversal anomaly in 2+1d fermionic topological phases, linking the anomaly to the modular S matrix and $ ext{T}^2$ eigenvalues, extending previous bosonic case analyses.

Contribution

It provides an explicit proof of the anomaly formula, generalizing the understanding of time-reversal symmetry in fermionic topological quantum field theories.

Findings

Explicit formula for time-reversal anomaly proven

Crosscap state expressed via modular S matrix and $ ext{T}^2$ eigenvalues

Extension of bosonic case analysis to fermionic theories

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.