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 geometric tools based on zero-area bands and cylindrical resolutions. This yields a canonical bulk-singular 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.