Large N limit of the stochastic Yang-Mills model

奇异随机偏微分方程的最新发展使我们能够严格意义上的朗之万动力学与超重整化量子场论。理解这些发展对规范理论的影响是相当有趣的，规范理论构成了量子力学的数学基础。这种量子规范理论的一个重要例子是杨-米尔斯测度。最近在[CCHS 20]中证明了2D杨-米尔斯测度的朗之万动力学对任何结构群都有短时解。典型的结构群是酉群U（N）和正交群O（N）。这个项目的目的是了解这些解决方案的行为，N趋于无穷大。2D杨米尔斯措施被证明收敛到一个确定性的主字段N趋于无穷大在[Lev 17]，预计朗之万动态大N波动围绕这个主字段。在最近的工作[SSZZ 20]中，关于更简单的线性σ模型的大N极限，有进一步的证据。该项目将涉及正则性结构的组合[Hai 14]，它允许人们控制有限N处动态的短期行为，以及自由概率及其与随机矩阵的连接，它允许人们理解收敛到主场。此外，建立[Che 19]中介绍的规范固定程序的N版本的统一也是有意义的，这将有助于研究大N动态的长时间行为。应用于这个问题的技术很可能会推广到其他模型，例如杨-米尔斯-希格斯模型，其中还不知道是否存在Mater场。参考文献：[CCHS 20] A.钱德拉岛谢弗列夫湾Hairer，H.沈二维杨-米尔斯测度的朗之万动力学。ArXiv电子版（2020年）。arXiv：2006.04987[Che19] I.谢弗列夫二维环面上的Yang-Mills测度为随机分布。Comm. Math. Phys. 372（2019），no. 3，1027- 1058。[Lev17] T.莱维飞机上的主磁场Astérisque No. 388（2017），ix+201 pp. [Hai14] M.海尔规则结构理论。发明。Math.198（2014），no.2，269- 504] H. Shen，S.史密斯河Zhu，X.竹通过随机量化的O（N）线性Sigma模型的大N极限。ArXiv电子版（2020年）。arXiv：2005.09279

Recent developments in singular stochastic PDEs have allowed us to make rigorous sense of the Langevin dynamic associated to super-renormalisable quantum field theories. It is of considerable interest to understand the implications of these developments to gauge theories, which form the mathematical basis of quantum mechanics. An important example of such a quantum gauge theory is the Yang-Mills measure.The Langevin dynamic of the 2D Yang-Mills measure was recently shown in [CCHS20] to have short-time solutions for any structure group. Typical structures groups of interest are the unitary groups U(N) and orthogonal groups O(N). The aim of this project is to understand the behaviour of these solutions as N tends to infinity.The 2D Yang-Mills measure was shown to converge to a deterministic Master Field as N tends to infinity in [Lev17], and it is expected that the Langevin dynamic for large N fluctuates around this Master Field. There is further evidence of this in the recent work [SSZZ20] on the large N limit of the simpler linear sigma-models. This project will involve a combination of regularity structures [Hai14], which allows one to control short-time behaviour of the dynamic at finite N, and free probability and its connection to random matrices, which allows one to understand the convergence to the Master Field. It will furthermore be of interest to establish a uniform in N version of the gauge-fixing procedure introduced in [Che19], which would be useful in studying the long-time behaviour of the large N dynamic. It is likely that the techniques applied to this problem will generalise to further models, such as the Yang-Mills-Higgs model, for which the existence of a Mater Field is not yet known.References:[CCHS20] A. Chandra, I. Chevyrev, M. Hairer, H. Shen. Langevin dynamic for the 2D Yang-Mills measure. ArXiv e-print (2020). arXiv:2006.04987[Che19] I. Chevyrev. Yang-Mills measure on the two-dimensional torus as a random distribution. Comm. Math. Phys. 372 (2019), no. 3, 1027--1058.[Lev17] T. Lévy. The master field on the plane. Astérisque No. 388 (2017), ix+201 pp.[Hai14] M. Hairer. A theory of regularity structures. Invent. Math. 198 (2014), no. 2, 269--504[SSZZ20] H. Shen, S. Smith, R. Zhu, X. Zhu. Large N Limit of the O(N) Linear Sigma Model via Stochastic Quantization. ArXiv e-print (2020). arXiv:2005.09279