Discrete Theta Angles, Symmetries and Anomalies

TL;DR

This paper explores how discrete theta angles influence symmetries and anomalies in gauge theories across various dimensions, revealing their role in emergent symmetries and topological phases.

Contribution

It demonstrates the impact of discrete theta angles on symmetry extensions and anomalies by coupling gauge theories to topological quantum field theories, introducing new emergent symmetries.

Findings

Discrete theta angles affect global symmetries and anomalies.

Gauging Abelian subgroups with SPT phases leads to emergent Abelian symmetries.

Discrete theta angles can induce two-group symmetries in 4d and lower-dimensional 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.