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

We study a 3d lattice gauge theory with gauge group U(1)(N-1) ? S-N, which is obtained by gauging the S-N global symmetry of a pure 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 SU(N) root system. Even though the center symmetry of the semi-Abelian gauge theory is given by 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 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.

Automated summary

The study examines a 3D lattice gauge theory with semi-Abelian gauge group, calculating mass gaps and string tensions using a monopole-gas description. It finds that the string tensions deviate from the N-ality rule and provide more information about the representations of the gauge group. The presence of non-invertible topological lines from the original Abelian theory explains this deviation, which is broken by adding W-bosons leading to the emergence of the N-ality rule in the deep infrared regime.