Abstract
We study a 3d lattice gauge theory with gauge group U(1)N−1 ⋊ SN, which is obtained by gauging the SN 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 ℤN, 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.
Highlights
The characterization of string tensions is an especially important point of difference between Abelianizing and non-Abelianizing confining gauge theories
We study a 3d lattice gauge theory with gauge group U(1)N−1 SN, which is obtained by gauging the SN global symmetry of a pure U(1)N−1 gauge theory, and we call it the semi-Abelian gauge theory
We find that the effective potential receives equal contributions at leading order from monopoles associated with the entire SU(N ) root system
Summary
To give the Villain formulation, we consider a link field A valued in RN−1 and a plaquette field np valued in the root lattice Γr ⊂ RN−1 of SU(N ). Considered gauging the discrete subgroup 2πΓr of the RN−1 1-form center symmetry group, which acts according to. As already noted above, there is a U(1)N−1 1-form center symmetry (2.6), where the group is U(1)N−1 rather than RN−1 thanks to the 1-form gauge structure (2.4). Under O(N − 1) transformations Π that preserve the root lattice Γr. Note that the existence of the non-Abelian global symmetry (2.8) is somewhat unusual for a pure gauge theory. Pure gauge theories without matter fields, either Abelian or non-Abelian, do not possess non-Abelian global symmetries. With w in the weight lattice Γw of SU(N ) Note that it is invariance under the 1-form gauge transformations (2.4) that requires the electric charge to be a weight.
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