Stochastic quantisation of Yang-Mills

TL;DR

This paper reviews recent advances in the stochastic quantisation of Yang-Mills theories in two and three dimensions, focusing on renormalisation, gauge covariance, and the construction of a Markov process on gauge orbits.

Contribution

It unifies and summarizes key results on the renormalisation and gauge covariance of stochastic Yang-Mills heat flow in low dimensions, highlighting differences in methods and open problems.

Findings

Renormalisation of stochastic Yang-Mills heat flow in 2D and 3D

Gauge covariance of the dynamic in law

Construction of a Markov process on gauge orbit space

Abstract

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 $\mathcal{S}$ to which gauge equivalence $\sim$ extends and such that the renormalised stochastic Yang-Mills heat flow projects to a Markov process on the quotient space of gauge orbits $\mathcal{S}/{\sim}$. 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.