Anomalies of discrete symmetries in three dimensions and group cohomology

TL;DR

This paper investigates 't Hooft anomalies for finite discrete internal symmetries in three dimensions, framing them within group cohomology and anomaly inflow. It constructs explicit 3d bosonic theories with nontrivial 't Hooft anomalies for G=Z_n×Z_n and analyzes how central extensions realize these anomalies. It then shows ABJ-type phenomena in 3d where gauging a factor of G1×G2 necessarily breaks the other, with concrete Z3×Z3 examples and embeddings into U(3) to illustrate obstructions. The work clarifies how discrete anomalies constrain gauging, relate to parity-like and SPT boundary physics, and inform possible gapped phases with topological order beyond the group-cohomology SPT classification.

Abstract

We study 't Hooft anomalies for a global discrete internal symmetry G. We construct examples of bosonic field theories in three dimensions with a non-vanishing 't Hooft anomaly for a discrete global symmetry. We also construct field theories in three dimensions with a global discrete internal symmetry G_1 x G_2 such that gauging G_1 necessarily breaks G_2 and vice versa. This is analogous to the Adler-Bell-Jackiw axial anomaly in four dimensions and parity anomaly in three dimensions.