Stochastic quantization of Yang–Mills

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.