# Discrete Theta Angles, Symmetries and Anomalies

### Submission summary

 As Contributors: Po-Shen Hsin · Ho Tat Lam Preprint link: scipost_202102_00001v1 Date accepted: 2021-02-04 Date submitted: 2021-02-01 05:01 Submitted by: Lam, Ho Tat Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### 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.

Published as SciPost Phys. 10, 032 (2021)

### List of changes

1. Footnote 4 is added for a brief discussion on the $Sp(N)$ gauge theory.

2. Footnote 6 is added for a brief explanation of the Bockstein homomorphism, and for pointing the readers to relevant references.

3. First paragraph of section 2.1 is added for an explanation of symmetry fractionalization.

### Submission & Refereeing History

Resubmission scipost_202102_00001v1 on 1 February 2021
Submission 2007.05915v2 on 20 September 2020