Large N limit of the stochastic Yang-Mills model

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