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对与超重整量子场论相关的朗之万动力学做出严格的理解。理解这些发展对构成量子力学数学基础的测量理论的影响是相当有趣的。这种量子规范理论的一个重要例子是杨-米尔斯测量。二维Yang-Mills测度的Langevin动力学最近在[CCHS20]中被证明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)。这个项目的目的是理解当N趋于无穷时这些解的行为。在[Lev17]中，当N趋于无穷大时，二维Yang-Mills测量收敛于确定性主场，并且预计大N的朗之万动态在该主场周围波动。最近关于更简单的线性西格玛模型的大N极限的研究[SSZZ20]进一步证明了这一点。这个项目将涉及正则结构的组合[Hai14]，它允许人们在有限N下控制动态的短期行为，自由概率及其与随机矩阵的联系，这允许人们理解收敛到主场。此外，在[Che19]中介绍的测量固定程序中建立一个统一的N版本将是有兴趣的，这将有助于研究大N动态的长期行为。应用于这个问题的技术很可能会推广到进一步的模型，比如杨-米尔斯-希格斯模型，对于这个模型，物质场的存在还不为人所知。参考文献：A. Chandra， I. Chevyrev， M. Hairer， H. Shen。二维杨-米尔斯测量的朗之万动力学。ArXiv e-print（2020）。[j] .中国农业大学学报：自然科学版。Yang-Mills在二维环面上作为随机分布进行测量。通讯。数学。物理学报，372 (2019),no。3, 1027 - 1058。[Lev17] T. lsamvy。平面上的主域。中国农业科学学报第388期（2017），9 +201页。规则结构的理论。发明。数学。198(2014)，第1期。[20]沈慧，史密斯，朱仁，朱晓霞。随机量化的O(N)线性Sigma模型的大N极限。ArXiv e-print（2020）。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