Discrete theta angles, symmetries and anomalies

Po-Shen Hsin,Ho Tat Lam

doi:10.21468/scipostphys.10.2.032

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 4d4d QCD with SU(N),SU(N)/\mathbb{Z}_kSU(N),SU(N)/ℤk or SO(N)SO(N) gauge groups as well as various 3d3d and 2d2d gauge theories.

Highlights

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
In this note we study these universal aspects that depend on the topological quantum field theory (TQFT)
If the larger symmetry is a non-trivial extension of the gauged subgroup and its quotient, we observe that the resulting gauge theories have different extensions of global symmetries and ’t Hooft anomalies, that depend on the SPT phases i.e. the discrete theta angles for the gauged symmetry

Summary

Introduction

If the larger symmetry is a non-trivial extension of the gauged subgroup and its quotient (in other words, not a direct product), we observe that the resulting gauge theories have different extensions of global symmetries and ’t Hooft anomalies, that depend on the SPT phases i.e. the discrete theta angles for the gauged symmetry. When the discrete theta angle vanishes, our results agree with the general discussion in [6], where a mixed anomaly is observed in the resulting gauge theory due to the symmetry extension in the original theory.

Symmetry enrichment

Couple QFT to two-form gauge theory

Gauging k one-form symmetry in two-group

Bundle and classical action

Global symmetry

Continuous theta angle