# Yang-Mills measure and the master field on the sphere

## TL;DR

This paper investigates the high-dimensional limit of the Yang-Mills measure on the sphere with unitary groups, establishing the convergence to a deterministic master field characterized by variational and differential equations.

## Contribution

It introduces the concept of the master field on the sphere, characterizes it via variational and Makeenko--Migdal equations, and links it to the high-dimensional limit of the Brownian loop in unitary groups.

## Key findings

- Traces of loop holonomies converge to the master field.
- The master field is characterized by a variational problem and differential equations.
- Identifies the high-dimensional limit of the Brownian loop in unitary matrices.

## 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 loop in the group of unitary matrices.