Langevin dynamic for the 2D Yang-Mills measure

TL;DR

This paper constructs a Markov process for the stochastic Yang-Mills heat flow in two dimensions using regularity structures, defining a suitable state space of distributional connections and gauge invariance properties.

Contribution

It introduces a novel state space for the 2D stochastic Yang-Mills measure and develops a basis-free regularity structures framework for vector-valued noise.

Findings

Defined a natural state space for the stochastic Yang-Mills heat flow

Constructed a gauge-invariant Markov process in this space

Developed a basis-free regularity structures framework for vector-valued noise

Abstract

We define a natural state space and Markov process associated to the stochastic Yang-Mills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric. To construct the Markov process we show that the stochastic Yang-Mills heat flow takes values in our space of connections and use the "DeTurck trick" of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations. Our main tool for solving for the Yang-Mills heat flow is the theory of regularity structures and along the way we also develop a "basis-free" framework for applying the theory of regularity structures in the context of vector-valued noise - this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.