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

TL;DR

The paper proves confinement for a three-dimensional Wilson lattice gauge theory with a gauge group that contains a central $\mathrm{U}(1)$ subgroup, delivering a bound $|\langle W_{\ell} \rangle| \le n e^{-C(1+n\beta)^{-1} T \log(R+1)}$ for rectangular loops with $R\le T$. The approach adapts a Mermin–Wagner-type complex rotation to introduce auxiliary $\mathrm{U}(1)$ edge variables, decomposes the Wilson loop into a phase and a bounded matrix factor, and conditions to reduce to a two-dimensional XY-type model where a powerful anti-concentration lemma yields the required decay. This yields a logarithmic quark-antiquark potential, establishing confinement in this class, though it does not reach the area-law strength known for Villain-type theories. The results highlight a pathway toward stronger area-law statements via comparison inequalities, remaining, however, technically demanding beyond the scope of this short lattice proof.

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 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 $\mathrm{U}(1)$. Indeed, combining a comparison inequality of Fr{ö}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 $\lvert\langle W_\ell\rangle\rvert \le n\exp\{-c(1+nβ)^{-1}T\log(R+1)\}$.

Theorems & Definitions (8)

• Theorem 3.1
• Lemma 4.1
• proof
• Corollary 4.2
• proof
• Lemma 4.3
• proof
• proof : Proof of Theorem \ref{['confine3d']}