Yang\u2013Mills Measure and the Master Field on the Sphere

Antoine Dahlqvist,James R Norris

doi:10.1007/s00220-020-03773-6

Abstract

We study the Yang–Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that, given its values on simple loops, the master field is characterized on all loops of finite length by a system of differential equations, known as the Makeenko–Migdal equations. We obtain a number of further properties of the master field. On specializing to families of simple loops, our results identify the high-dimensional limit, in non-commutative distribution, of the Brownian bridge in the group of unitary matrices starting and ending at the identity.

Highlights

The Yang–Mills measure, associated to a surface Σ and to a compact Lie group G, is a probability measure on connections of principal Gbundles over Σ
We will consider the Yang–Mills measure in the case where the surface Σ is fixed and the group G is a classical matrix group of high dimension. The interest of such a set-up from the viewpoint of random matrix theory was first raised in the mathematics literature by Singer [50], who made several conjectures, based on earlier work in physics [26,27,34,35]
The high-dimensional limit of the Yang–Mills measure when Σ is the whole plane has since been studied by Xu [54], Sengupta [49], Lévy [40], Anshelevich and Sengupta [1], Dahlqvist [13] and others [8,24]

Summary

Introduction

The Yang–Mills measure, associated to a (two-dimensional) surface Σ and to a compact Lie group G, is a probability measure on (generalized) connections of principal Gbundles over Σ. The main result of the present work, Theorem 2.2, confirms a conjecture of Singer [50], showing that, under the Yang–Mills measure on the sphere for the unitary group U (N ), the traces of loop holonomies converge as N → ∞ to a deterministic limit. We characterize this limit analytically and derive some further properties. To be explained in a future work, the argument explained here applies to other series of compact groups and with the projective plane in place of the sphere

Setting and Statement of the Main Results

Makeenko–Migdal Equations

Extension to Loops of Finite Length

Further Properties of the Master Field