Two-dimensional Yang-Mills theory via integrable probability

TL;DR

This paper reviews the construction and large N analysis of 2D Yang-Mills theory with U(N) gauge group using probability, combinatorics, and representation theory, highlighting recent advances in asymptotics and expansions.

Contribution

It introduces a probabilistic framework for 2D Yang-Mills theory and derives new large-N asymptotics and 1/N expansions with geometric interpretations.

Findings

Large-N asymptotics of the partition function on all compact surfaces.

1/N expansion of the partition function on a torus.

Connection between expansions and random surface models.

Abstract

In this paper, we review the construction and large $N$ study of the continuous two-dimensional Yang--Mills theory with gauge group $\mathrm{U}(N)$ through probability, combinatorics and representation theory. In the first part, we define the continuous Yang--Mills measure using Markovian holonomy fields, following a construction by L\'evy, then we show in the second part how to derive the character expansion of the partition function for any compact structure group from this setting. We continue with two developments obtained in the last few years by Dahlqvist, Lemoine, L\'evy and Ma\"ida with similar approaches with respect to the partition function: its large-$N$ asymptotics on all compact surfaces for the structure group $\mathrm{U}(N)$, and its $\frac{1}{N}$ expansion on a torus with an interpretation in terms of random surfaces.