Comments on one-form global symmetries and their gauging in 3d and 4d

We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on Z_N ℤ N one-form symmetries. A 3d topological quantum field theory (TQFT) T 𝒯 with such a symmetry has N N special lines that generate it. The braiding of these lines and their spins are characterized by a single integer p p modulo 2N 2 N . Surprisingly, if (N,p)=1 gcd ( N , p ) = 1 the TQFT factorizes T=T'⊗ AN,p 𝒯 = 𝒯 ′ ⊗ 𝒜 N , p . Here T' 𝒯 ′ is a decoupled TQFT, whose lines are neutral under the global symmetry and AN,p 𝒜 N , p is a minimal TQFT with the Z_N ℤ N one-form symmetry of label p p . The parameter p p labels the obstruction to gauging the Z_N ℤ N one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When p=0 p = 0 mod