We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$ . Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B303(1) 95–98).

中文翻译：

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二维 Yang-Mills 配分函数的大 N 行为

我们计算大ñ欧几里得杨-米尔斯测度在可定向紧致曲面上的配分函数极限$g\geqslant 1$和不可定向的紧凑表面与属$g\geqslant 2$, 结构群为酉群${\mathrm U}(N)$或特殊的酉群${\mathrm{SU}}(N)$. 我们的证明基于渐近表示理论：更具体地说，我们控制不可约表示的维数和卡西米尔数${\mathrm U}(N)$和${\mathrm{SU}}(N)$什么时候ñ趋于无穷大。我们的主要技术工具，包括“几乎平坦的”杨氏图，使得 Gross 和 Taylor (1993,核物理。乙400(1-3) 181-208) 在 QCD 的设置中，在某些情况下，我们恢复了 Douglas (1995,量子场论和弦论 (Cargèse, 1993)，卷。328 个北约高级科学研究所系列 B：物理学, Plenum, New York, pp. 119–135) 和 Rusakov (1993,物理。莱特。乙303(1) 95-98)。