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arXiv · 2602.00436

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

Sourav Chatterjee
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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 $\lvert\langle W_\ell\rangle\rvert \le n\exp\{-c(1+n\beta)^{-1}T\log(R+1)\}$.

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Sourav Chatterjee. 2026-01-31. A short proof of confinement in three-dimensional lattice gauge theories with a central $\mathrm{U}(1)$. https://arxiv.org/abs/2602.00436

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