The Yang--Mills measure on compact surfaces as a universal scaling limit of lattice gauge models

Abstract

In this article, we study the 2 dimensional Yang--Mills measure on compact surfaces from a unified continuum and discrete perspective. We construct the Yang--Mills measure as a random distributional 1 form on surfaces of arbitrary genus equipped with an arbitrary smooth area form, using the analytic concept of pseudo-coordinates. Our approach yields a canonical noise-flat decomposition of the measure, reflecting the topology of the surface. We prove a universality theorem stating that the continuum Yang--Mills measure arises as the scaling limit of a wide class of lattice gauge theories -- including Wilson, Manton, and Villain actions -- on any compact surface. We study the convergence in natural spaces of distributions with anisotropic regularity. As further consequences, we obtain a new intrinsic construction of the Yang--Mills measure, independent of the previous constructions in the literature, and prove the convergence of correlation functions and Segal amplitudes on all compact surfaces.