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

Abstract

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 $\mathcal{T}={\mathcal{T}}^{\prime }\otimes {\mathcal{A}}^{N,p}$. Here ${\mathcal{T}}^{\prime }$ 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.

56 pages, 3 figures, 5 tables

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