Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

We introduce a space of distributional 1-forms Ωα1 on the torus T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an Ωα1-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 (Mem Am Math Soc 166(790), 2003). It holds furthermore that Ωα1 embeds into the Hölder–Besov space Cα-1 for all α∈ (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.