Almost flat highest weights and application to Wilson loops on compact surfaces

• 成果类型: Article; Early Access
• 署名作者: Lemoine, Thibaut
• 署名单位: Universite PSL; College de France
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01388-3
• 发表日期: 2025
• 关键词: weil-petersson volumes; heat kernel measure; yang-mills; MODULI SPACES; limit; EQUATIONS; geodesics

摘要

We derive new formulas for the expectation and variance of Wilson loops for any contractible simple loop on a compact orientable surface of genus 1 and higher, in the model of two-dimensional Yang-Mills theory with structure group U(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm U}(N)$$\end{document}. They are written in terms of a Gaussian measure on the dual of U(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm U}(N)$$\end{document} introduced recently by the author and M. Ma & iuml;da [26]. From these formulas, we prove a quantitative result on the convergence of the expectation and variance as N tends to infinity, refining a result of [10]. We finally derive the large g limit of the Wilson loop expectation and variance, by analogy with the study of integrals on moduli spaces of compact hyperbolic surfaces. Surprisingly, the variance does not vanish in this regime, but there are no nontrivial fluctuations of any order.