Yang-Mills measure on the two-dimensional torus as a random distribution

TL;DR

This paper constructs a distributional framework for Yang-Mills fields on a 2D torus, establishing the existence of a random distribution matching Wilson loop observables and exhibiting expected regularity properties.

Contribution

It introduces a novel space of distributional one-forms on the torus and proves the existence of a random variable with Wilson loop observables matching the Yang-Mills measure.

Findings

Existence of an $ ext{Ω}^1_ ext{α}$-valued random variable $A$ with correct Wilson loop law

Embedding of $ ext{Ω}^1_ ext{α}$ into $ ext{C}^{ ext{α}-1}$ space for regularity

Method based on Landau gauge applied to lattice approximations

Abstract

We introduce a space of distributional one-forms $\Omega^1_\alpha$ on the torus $\mathbf{T}^2$ for which holonomies along axis paths are well-defined and induce H\"older continuous functions on line segments. We show that there exists an $\Omega^1_\alpha$-valued random variable $A$ for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang-Mills measure in the sense of L\'evy (2003). It holds furthermore that $\Omega^1_\alpha$ embeds into the H\"older-Besov space $\mathcal{C}^{\alpha-1}$ for all $\alpha\in(0,1)$, so that $A$ has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.