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

Abstract

We introduce a space of distributional one-forms ${\Omega }_{\alpha }^1$ on the torus ${\mathbf{T}}^2$ for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an ${\Omega }_{\alpha }^1$-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 Lévy (2003). It holds furthermore that ${\Omega }_{\alpha }^1$ embeds into the Hölder-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.

arXiv comment37 pages, 4 figures. Minor revisions. To appear in Communications in Mathematical Physics

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