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Discrete Theta Angles, Symmetries and Anomalies

by Po-Shen Hsin, Ho Tat Lam

Submission summary

As Contributors: Po-Shen Hsin · Ho Tat Lam
Arxiv Link: (pdf)
Date submitted: 2020-09-20 20:19
Submitted by: Lam, Ho Tat
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2007.05915v2 on 20 September 2020

Reports on this Submission

Anonymous Report 1 on 2020-10-7 Invited Report


The article contains new and interesting results, especially on two-group symmetries and mixed anomalies.


1. The style of writing is rather terse.
2. The article was written such that the target readers are experts in the field of anomalies. Certain concepts, such as the symmetry fractionalization, in this article could be better explained or reviewed.


This article studied gauge theories with different discrete theta angles by gauging a global symmetry with different symmetry-protected topological (SPT) phases. The authors observed that when the symmetries in question are of higher-form with different degrees, gauging a subgroup symmetry that does not have a mixed anomaly with the remaining symmetry could lead to theories with two-group symmetries. As a result, two-group symmetries studied in this article, in general, depend on the discrete theta angles of the gauge theory. The authors also studied various mixed anomalies. For example, one of the interesting results is in section 4.5 where it was shown that, in the SO(N) QCD, there is a mixed anomaly between the magnetic $\mathbb{Z}_2$ one-form symmetry, the flavor symmetry and the charge conjugation symmetry. In the referee's opinion, the article deserves publication after minor improvements on the presentation.

Requested changes

1. The authors should explain or motivate in physical terms the following concepts: the Bockenstein homomorphism and the symmetry fractionalization. This would make the article more accessible to a wider audience.
2. In the article, the authors omitted the discussion on the groups $Sp(N)$ and $Sp(N)/\mathbb{Z}_2$. For the completeness of the paper, the authors should mention how their techniques can be applied to such cases.

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: perfect
  • grammar: perfect

Author Ho Tat Lam on 2020-11-06
(in reply to Report 1 on 2020-10-07)

We thank the referee for comments and suggestions! Please see the reply below.

  1. We will add paragraphs elaborating on the two concepts and point the readers to relevant references.

  2. Unlike the $SU(N)$, $Spin(N)$ gauge group discussed in the paper, $Sp(N)$ gauge group only has a $\mathbb{Z}_2$ center which does not have any nontrivial proper subgroup. Hence $Sp(N)$ or $Sp(N)/\mathbb{Z}_2$ gauge theories with matters do not have two-group symmetries with nontrivial Postnikov class. Therefore, we didn't discuss them in the paper. We will add a footnote explaining this.

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