A short proof of confinement in three-dimensional lattice gauge theories with a central $\mathrm{U}(1)$

TL;DR

This paper provides a concise proof of confinement in three-dimensional lattice gauge theories with a central U(1) subgroup, extending known results and offering explicit bounds on Wilson loop expectations.

Contribution

It offers a short, self-contained proof of confinement for 3D Wilson lattice gauge theories with groups containing U(1), generalizing previous results and providing explicit Wilson loop bounds.

Findings

Confinement is proven for a class of 3D lattice gauge theories with U(1) central subgroup.

Explicit upper bounds on Wilson loop expectations are derived.

The proof simplifies and extends classical confinement results in lattice gauge theories.

Abstract

Pure lattice gauge theories in three dimensions are widely expected to confine. A rigorous proof of confinement for three-dimensional $\mathrm{U}(1)$ lattice gauge theory with Villain action was given by G\"opfert 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 $\mathrm{U}(1)$. Indeed, combining a comparison inequality of Fr{\"o}hlich 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 $G\subseteq \mathrm{U}(n)$ contains the full circle of scalar matrices $\{zI:\ |z|=1\}$, rectangular Wilson loops obey an explicit upper bound of the form $\lvert\langle W_\ell\rangle\rvert \le n\exp\{-c(1+n\beta)^{-1}T\log(R+1)\}$.