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We study 3d and 4d systems with a one-form global symmetry, explore theirconsequences, 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 generateit. The braiding of these lines and their spins are characterized by a singleinteger $p$ modulo $2N$. Surprisingly, if $\gcd(N,p)=1$ the TQFT factorizes$\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}$. Here $\mathcal{T}'$ is adecoupled 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 symmetryof label $p$. The parameter $p$ labels the obstruction to gauging the$\mathbb{Z}_N$ one-form symmetry; i.e.\ it characterizes the 't Hooft anomalyof 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 withgauge 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 isassociated with confinement and oblique confinement -- probe quarks areconfined. In the $PSU(N)$ theory the low-energy theory can include a discretegauge theory. We will study the behavior of the theory with a space-dependent$\theta$-parameter, which leads to interfaces. Typically, the theory on theinterface is not confining. Furthermore, the liberated probe quarks are anyonson 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 in3d TQFTs allows us to describe the interface in the $PSU(N)$ theory.

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