Anomalies of discrete symmetries in various dimensions and group cohomology

TL;DR

The paper systematically analyzes 't Hooft anomalies for discrete global symmetries in bosonic theories across 2, 3, and 4 dimensions, showing that anomalies often arise when the total symmetry is a nontrivial extension and that some can be canceled by inflow from DW theories, tying these anomalies to surfaces of SPT phases. It provides a concrete, obstruction-based framework using Lyndon–Hochschild–Serre spectral sequences to determine when a given boundary theory admits a DW-type inflow, and it works out explicit 2d, 3d, and 4d examples including cubic and quintic anomalies. It highlights the role of Deligne–Beilinson cocycles in formulating topological actions and analyzes both DW and Chern–Simons theories, showing cases where inflow suffices and cases where anomalies are non-cancelable by DW inflow alone. Overall, the work reveals a richer landscape of discrete anomalies beyond free-fermion theories, including nonabelian extensions and higher-degree invariants, with implications for SPT/SET phases and topological gauge theories.

Abstract

We study 't Hooft anomalies for discrete global symmetries in bosonic theories in 2, 3 and 4 dimensions. We show that such anomalies may arise in gauge theories with topological terms in the action, if the total symmetry group is a nontrivial extension of the global symmetry by the gauge symmetry. Sometimes the 't Hooft anomaly for a d-dimensional theory with a global symmetry G can be canceled by anomaly inflow from a (d+1)-dimensional topological gauge theory with gauge group G. Such d-dimensional theories can live on the surfaces of Symmetry Protected Topological Phases. We also give examples of theories with more severe 't Hooft anomalies which cannot be canceled in this way.