Mathematics

In this talk I will discuss the problem of stochastic quantisation of Parisi-Wu in the context of Yang-Mills (YM) theories. I will present several works which make sense of the YM Langevin dynamic on the 2D and 3D torus in a way that respects the underlying gauge symmetries. I will particularly focus on a recent result that the 2-dimensional Euclidean YM measure is invariant for its associated dynamic. I will describe some steps in the proof of this result, which is based on lattice approximations and Bourgain’s globalisation argument. A novelty of the approach is in the use of geometric arguments to show uniqueness of the limiting continuum dynamic. I will also present some consequences, including universality and regularity properties of the 2D YM measure, and discuss the challenges in extending the result to 3 dimensions. Based on joint works with Ajay Chandra, Martin Hairer, and Hao Shen.