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Large N behaviour of the two-dimensional Yang–Mills partition function
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2021-06-30 , DOI: 10.1017/s0963548321000262 Thibaut Lemoine
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2021-06-30 , DOI: 10.1017/s0963548321000262 Thibaut Lemoine
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. B 400 (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. B 303 (1) 95–98).
中文翻译:
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)。
更新日期:2021-06-30
中文翻译:
二维 Yang-Mills 配分函数的大 N 行为
我们计算大