Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and planar master field

Abstract

This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang-Mills measure on all orientable compact surfaces of genus larger or equal to 1, with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [16], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3,32] and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms the liberation process of Voiculescu [55]. In contrast to [16], our main tool is a fine use of Makeenko-Migdal equations, proving their uniqueness under suitable assumptions, generalising the arguments of [17,29].