A short proof of confinement in three-dimensional lattice gauge theories with a central $\mathrm{U}(1)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908801092812800 |
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| author | Chatterjee, Sourav |
| author_facet | Chatterjee, Sourav |
| contents | 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 $\lvert\langle W_\ell\rangle\rvert \le n\exp\{-c(1+nβ)^{-1}T\log(R+1)\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00436 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A short proof of confinement in three-dimensional lattice gauge theories with a central $\mathrm{U}(1)$ Chatterjee, Sourav Mathematical Physics High Energy Physics - Lattice Probability 70S15, 81T13, 81T25, 82B20 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 $\lvert\langle W_\ell\rangle\rvert \le n\exp\{-c(1+nβ)^{-1}T\log(R+1)\}$. |
| title | A short proof of confinement in three-dimensional lattice gauge theories with a central $\mathrm{U}(1)$ |
| topic | Mathematical Physics High Energy Physics - Lattice Probability 70S15, 81T13, 81T25, 82B20 |
| url | https://arxiv.org/abs/2602.00436 |