Stochastic quantisation of Yang-Mills

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.

32 pages, 1 figure. Fixed typos and updated references. To appear in Journal of Mathematical Physics for the proceedings of ICMP 2021

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Cited by in corpus (9)

• A stochastic analysis approach to lattice Yang--Mills at strong coupling
• Norm inflation for a non-linear heat equation with Gaussian initial conditions
• Gauge field marginal of an Abelian Higgs model
• Invariant measure and universality of the 2D Yang-Mills Langevin dynamic
• Well-posedness of a gauge-covariant wave equation with space-time white noise forcing
• Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs
• Lecture notes on Malliavin calculus in regularity structures
• Local well-posedness of subcritical non-linear heat equations with Gaussian initial data
• A scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills-Higgs theory