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

TL;DR

The paper develops a framework to diagnose the time-reversal anomaly of 2+1d fermionic topological phases by computing fractional anomalous momentum on a cross-cap background, linking the boundary invariant ν (mod 16) to a bulk 3+1d topological superconductor. It translates the anomaly into a cross-cap momentum p and uses crosscap states on $T^2$ to extract p, thereby determining ν for various theories. Through detailed analyses of concrete examples (semion-fermion and T-Pfaffian) and constructions arising from gapping free fermions, the authors demonstrate that ν can be read off from the crosscap data and crosscheck with the known Z16 classification. The work clarifies how non-orientable backgrounds encode time-reversal anomalies and provides a practical method to match boundary TQFTs with their bulk SPT origins, with implications for identifying gapped boundaries and domain-wall realizations.

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 $ν$ 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.