Discrete Theta Angles, Symmetries and Anomalies

TL;DR

The paper develops a universal framework for discrete theta angles in gauge theories by coupling them to TQFTs and studying resulting symmetry extensions. It shows that discrete theta angles control two-group and higher-group structures, including mixed anomalies, in a wide range of theories from 4d $SU(N)/\mathbb{Z}_k$ and $SO(N)/Spin(N)/O(N)$ to 3d and 2d models. The analysis hinges on higher-form symmetries, the Pontryagin square, Bockstein maps, and cohomological data that govern symmetry enrichment and anomaly matching. These results clarify how SPT phases and discrete theta terms reorganize global and higher-form symmetries under gauging, with concrete implications for QCD-like theories and Chern-Simons/TQFT constructions. The findings provide a cohesive picture linking discrete theta angles, symmetry extension, and anomaly structure across dimensions, with potential applications to confinement, dualities, and topological phases of gauge theories.

Abstract

Gauge theories in various dimensions often admit discrete theta angles, that arise from gauging a global symmetry with an additional symmetry protected topological (SPT) phase. We discuss how the global symmetry and 't Hooft anomaly depends on the discrete theta angles by coupling the gauge theory to a topological quantum field theory (TQFT). We observe that gauging an Abelian subgroup symmetry, that participates in symmetry extension, with an additional SPT phase leads to a new theory with an emergent Abelian symmetry that also participates in a symmetry extension. The symmetry extension of the gauge theory is controlled by the discrete theta angle which comes from the SPT phase. We find that discrete theta angles can lead to two-group symmetry in 4d QCD with $SU(N),SU(N)/\mathbb{Z}_k$ or $SO(N)$ gauge groups as well as various 3d and 2d gauge theories.