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}_NℤN one-form symmetries. A 3d topological quantum field theory (TQFT) \mathcal{T}𝒯 with such a symmetry has NN special lines that generate it. The braiding of these lines and their spins are characterized by a single integer pp modulo 2N2N. Surprisingly, if \gcd(N,p)=1gcd(N,p)=1 the TQFT factorizes \mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}𝒯=𝒯′⊗𝒜N,p. Here \mathcal{T}'𝒯′ is a decoupled TQFT, whose lines are neutral under the global symmetry and \mathcal{A}^{N,p}𝒜N,p is a minimal TQFT with the \mathbb{Z}_NℤN one-form symmetry of label pp. The parameter pp labels the obstruction to gauging the \mathbb{Z}_NℤN one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When p=0p=0 mod 2N2N, 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)SU(N) and PSU(N)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)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 \thetaθ-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)PSU(N) theory is obtained by gauging the \mathbb{Z}_NℤN one-form symmetry of the SU(N)SU(N) theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the PSU(N)PSU(N) theory.