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

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.