Large N limit of the stochastic Yang-Mills model

随机 Yang-Mills 模型的大 N 极限

• 批准号: 2771450
• 金额: --
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2771450
• 关键词: Large; limit; stochastic; Yang; Mills

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

最近的奇异随机偏微分方程使我们能够严格理解与超重整化量子场论相关的朗之万动力学，了解这些发展对规范理论的影响具有相当大的发展意义，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[CCHS20]中显示了二维杨米尔斯测度的朗之万动力学对于任何典型结构都有短时解。感兴趣的群是酉群 U(N) 和正交群 O(N)。该项目的目的是了解这些解在 N 趋于无穷大时的行为。2D Yang-Mills 测度被证明收敛于 a。 [Lev trend17] 中的确定性主场为 N 到无穷大，并且预计大 N 的 Langevin 动态会围绕该主场波动。最近的工作 [SSZZ20] 进一步证明了这一点。更简单的线性 sigma 模型的大 N 限制 该项目将涉及正则结构 [Hai14] 的组合，它允许人们控制有限 N 下动态的短时行为，以及自由概率及其与随机矩阵的连接，这使得人们能够理解与主场的收敛性，此外，建立[Che19]中引入的规范固定程序的N版本的统一也将是有意义的，这对于研究该模型的长期行为非常有用。应用于此问题的技术很可能会推广到进一步的模型，例如 Yang-Mills-Higgs 模型，其中 Mater 场的存在尚不清楚。参考文献：[CCHS20] A. Chandra、I. Chevyrev、M. Hairer、H. Shen。2D Yang-Mills 测量动态（2020）。 arXiv：2006.04987[Che19] I. Chevyrev 测量二维环面作为随机分布。 Comm. Phys. 372 (2019)，第 3 期，1027--1058。飞机上的主场。388 (2017)， ix+201 pp.[Hai14] M. Hairer. 数学 198 (2014), no. 2, 269--504[SSZZ20] H. Shen, S. Smith, R. Zhu, X. Zhu。通过随机量化的 O(N) 线性西格玛模型的大 N 极限 (2020)。 arXiv:2005.09279