几乎平坦的最高权及其在紧致曲面上的Wilson圈中的应用
Almost flat highest weights and application to Wilson loops on compact surfaces
AI总结：
本文为亏格≥1紧致曲面上的Wilson圈导出期望与方差的新公式，证明N→∞时的定量收敛，并揭示大亏格极限下方差不消失但无高阶涨落。
摘要：
我们为亏格为1及以上的紧致可定向曲面上的任意可缩简单环路的Wilson圈的期望和方差推导了新的公式，该公式适用于结构群为$\mathrm{U}(N)$的二维Yang--Mills理论模型。这些公式用作者与M. Maïda最近引入的$\mathrm{U}(N)$对偶上的高斯测度来表示（参见文献\cite{LM3}）。基于这些公式，我们证明了当$N$趋于无穷大时期望和方差收敛的定量结果，细化了作者与A. Dahlqvist的结果（参见文献\cite{DL}）。最后，我们通过与紧致双曲曲面模空间上积分研究的类比，推导了Wilson圈期望和方差的大亏格极限。令人惊讶的是，方差在该极限下并不消失，但不存在任何阶的非平凡涨落。
英文摘要：
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 $\mathrm{U}(N)$. They are written in terms of a Gaussian measure on the dual of $\mathrm{U}(N)$ introduced recently by the author and M. Maïda \cite{LM3}. 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 the author and A. Dahlqvist \cite{DL}. 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.
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