On time-reversal anomaly of 2+1d topological phases

TL;DR

This paper introduces a method to detect the time-reversal anomaly in 2+1d topological phases by calculating fractional anomalous momentum on a cross-cap background, linking boundary phases to bulk topological invariants.

Contribution

It provides a novel technique to identify the time-reversal anomaly and connect boundary topological phases with bulk topological invariants in 3+1d superconductors.

Findings

Method to compute fractional anomalous momentum on cross-cap backgrounds.

Identification of the parameter ν mod 16 for bulk topological superconductors.

Application to boundary phases with time-reversal symmetry.

Abstract

We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter $\nu$ mod 16 of the bulk 3+1d topological superconductor with $\mathsf{T}^2=(-1)^F$ on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.