Anomalies of discrete symmetries in various dimensions and group cohomology

TL;DR

This paper investigates 't Hooft anomalies for discrete symmetries across various dimensions, revealing their origins in gauge theories with topological terms and their relation to symmetry protected topological phases.

Contribution

It provides a comprehensive analysis of how anomalies arise in gauge theories with topological terms and their connection to higher-dimensional topological theories and SPT phases.

Findings

Anomalies can originate from nontrivial extensions of symmetry groups.

Anomaly inflow from (d+1)-dimensional topological theories can cancel certain anomalies.

Some anomalies are too severe to be canceled by inflow, indicating intrinsic obstructions.

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.