Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality

We study a 3d lattice gauge theory with gauge group $\mathrm{U}(1)^{N-1}\rtimes \mathrm{S}_N$, which is obtained by gauging the $\mathrm{S}_N$ global symmetry of a pure $\mathrm{U}(1)^{N-1}$ gauge theory, and we call it the semi-Abelian gauge theory. We compute mass gaps and string tensions for both theories using the monopole-gas description. We find that the effective potential receives equal contributions at leading order from monopoles associated with the entire $\mathrm{SU}(N)$ root system. Even though the center symmetry of the semi-Abelian gauge theory is given by $\mathbb{Z}_N$, we observe that the string tensions do not obey the $N$-ality rule and carry more detailed information on the representations of the gauge group. We find that this refinement is due to the presence of non-invertible topological lines as a remnant of $\mathrm{U}(1)^{N-1}$ one-form symmetry in the original Abelian lattice theory. Upon adding charged particles corresponding to $W$-bosons, such non-invertible symmetries are explicitly broken so that the $N$-ality rule should emerge in the deep infrared regime.