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Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d

by Po-Shen Hsin, Ho Tat Lam, Nathan Seiberg

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Submission summary

Authors (as registered SciPost users): Po-Shen Hsin · Nathan Seiberg
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-03-20
Date submitted: 2019-01-21 01:00
Submitted by: Hsin, Po-Shen
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on $\mathbb{Z}_N$ one-form symmetries. A 3d topological quantum field theory (TQFT) $\mathcal{T}$ with such a symmetry has $N$ special lines that generate it. The braiding of these lines and their spins are characterized by a single integer $p$ modulo $2N$. Surprisingly, if $\gcd(N,p)=1$ the TQFT factorizes $\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}$. Here $\mathcal{T}'$ is a decoupled TQFT, whose lines are neutral under the global symmetry and $\mathcal{A}^{N,p}$ is a minimal TQFT with the $\mathbb{Z}_N$ one-form symmetry of label $p$. The parameter $p$ labels the obstruction to gauging the $\mathbb{Z}_N$ one-form symmetry; i.e.\ it characterizes the 't Hooft anomaly of the global symmetry. When $p=0$ mod $2N$, the symmetry can be gauged. Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with gauge fields extended to the bulk. This understanding allows us to consider $SU(N)$ and $PSU(N)$ 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement -- probe quarks are confined. In the $PSU(N)$ theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent $\theta$-parameter, which leads to interfaces. Typically, the theory on the interface is not confining. Furthermore, the liberated probe quarks are anyons on the interface. The $PSU(N)$ theory is obtained by gauging the $\mathbb{Z}_N$ one-form symmetry of the $SU(N)$ theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the $PSU(N)$ theory.

Published as SciPost Phys. 6, 039 (2019)

Submission & Refereeing History

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Submission 1812.04716v1 on 21 January 2019

Reports on this Submission

Anonymous Report 1 on 2019-2-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1812.04716v1, delivered 2019-02-25, doi: 10.21468/SciPost.Report.845


1- Physics of 3d Abelian TQFT is studied in detail with clear explanation.
2- Important factorization theorem is shown for 3d Abelian TQFTs if they have the most 't Hooft anomalous $\mathbb{Z}_N$ one-form symmetry. In this process, they naturally explain the notion of minimal TQFT.
3- They introduce the "gauging" procedure for 't Hooft anomalous $\mathbb{Z}_N$ one-form symmetry of 3d TQFT using the minimal TQFTs.
4- These results are applied to conjecture the reasonable dynamics of 4d SU(N) and PSU(N) gauge theories with $\theta$ angles. Especially, it allows us to study interfaces of PSU(N) Yang-Mills with different discrete theta parameters.
5- The paper is self contained. The necessary information to read this paper is mostly written either in the main body or in the Appendix of this paper.


1- I do not come up with the weak point.


The motivation of this paper is to study the interface of 4d SU(N) and PSU(N) Yang-Mills theory with different discrete theta parameters. For SU(N) Yang-Mills theory at $\theta=\pi$, this is studied in detail in Ref.[2], and the authors made the statements more precise by introducing the notion of minimal TQFT in this paper. For PSU(N) Yang-Mills theory, the 4d bulk can be a nontrivial intrinsic topological order, and the situation becomes much more complicated. The authors tackle this problem by studying the 3d Abelian topological order in this paper.

The paper is well written. Physics of 3d Abelian TQFT is explained in an explicit physical language, so the readers are not required to be too familiar with mathematics, and the necessary backgrounds, like higher-form symmetries, Abelian anyson, etc. are summarized in the Appendix. Through this paper, the authors have clarified the rigorous fact about 4d Yang-Mills theory that can be said only by symmetry and 't Hooft anomaly, and this is a useful development of our understanding of nonperturbative QFT.

From these reasons, I suggest the publication of this paper.

Requested changes

1- No changes are needed.

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