A short proof of confinement in three-dimensional lattice gauge theories with a central U (1)

Pure lattice gauge theories in three dimensions are widely expected to confine. A rigorous proof of confinement for three-dimensional U (1) lattice gauge theory with Villain action was given by Göpfert and Mack. Beyond the abelian case, rigorous confinement results are comparatively scarce; one general mechanism applies when the gauge group has a central copy of U (1). Indeed, combining a comparison inequality of Fröhlich with earlier work of Glimm and Jaffe yields confinement with a logarithmically growing quark–antiquark potential for this class of theories. The purpose of this note is to give a short, self-contained proof of this classical result for three-dimensional Wilson lattice gauge theory: when G ⊆ U( n ) contains the full circle of scalar matrices { zI : | z | = 1}, rectangular Wilson loops obey an explicit upper bound of the form |⟨ W ℓ ⟩| ≤ n exp{− c (1 + nβ ) −1 T log( R + 1)}.