Stochastic quantisation of Yang–Mills

We review two works arXiv:2006.04987 and arXiv:2201.03487 which study the stochastic quantisation equations of Yang-Mills on two and three dimensional Euclidean space with finite volume. The main result of these works is that one can renormalise 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 ∼ extends and such that the renormalised 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.