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 $\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}$. 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 $θ$-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.

Figures (3)

• Figure 1: The interfaces for different profiles of $\theta$ that interpolate from $\theta=\theta_0$ to $\theta=\theta_0+2\pi k$. The dashed lines are the profile of the $\theta$ parameter and the solid lines are the locations of the interfaces. In (a), there are $k$ domain walls located at the transitions when $\theta$ crosses an odd multiple of $\pi$. The theory on each domain wall is ${\cal T}_1$, which we argue is $SU(N)_1$Gaiotto:2014kfa. When the $\theta$ variation is more rapid, as in (b), there is only one interface and the theory on it is ${\cal T}_k$. One option for that theory is $SU(N)_k$, but we will argue that other options are also possible. Finally, as in (c), $\theta$ can be discontinuous. In this case the theory on the interface $\cal T$ is not determined uniquely by the microscopic dynamics. But it is constrained by anomaly considerations.
• Figure 2: Braiding the line operators supported on the curves $\gamma$ and $\gamma'$.
• Figure 3: If a boundary line $W (\gamma)$ is at the fixed point of the identification using $\widehat{a} = a^{K}$, it can form a junction by emanating a bulk line $\exp(iK\int_{\gamma_\perp} A)$.