Stochastic quantization of Yang–Mills

Abstract

We review two works [Chandra et al., Publ. Math. l’IHÉS (published online, 2022) and Chandra et al., arXiv:2201.03487 (2022)] that study the stochastic quantization equations of Yang–Mills on two- and three-dimensional Euclidean space with finite volume. The main result of these works is that one can renormalize the 2D and 3D stochastic Yang–Mills heat flow so that the dynamic becomes gauge covariant in law. Furthermore, there is a state space of distributional 1-forms S to which gauge equivalence approximately extends and such that the renormalized stochastic Yang–Mills heat flow projects to a Markov process on the quotient space of gauge orbits S/∼. In this Review, we give unified statements of the main results of these works, highlight differences in the methods, and point out a number of open problems.