# The large-N limit for two-dimensional Yang-Mills theory

## TL;DR

This paper rigorously analyzes the large-N limit of Wilson loop functionals in 2D Yang-Mills theory on a sphere, establishing existence and variance vanishing for loops with crossings, and extends results to arbitrary surfaces with conjectures.

## Contribution

It provides a rigorous proof for the second stage of large-N analysis on the 2-sphere and introduces conjectures about Wilson loops on general surfaces.

## Key findings

- Existence of large-N limit for loops with crossings on the 2-sphere.
- Variance of Wilson loop functionals tends to zero in the large-N limit.
- Extension of results to arbitrary surfaces under certain conjectures.

## Abstract

The analysis of the large-$N$ limit of $U(N)$ Yang-Mills theory on a surface proceeds in two stages: the analysis of the Wilson loop functional for a simple closed curve and the reduction of more general loops to a simple closed curve. In the case of the 2-sphere, the first stage has been treated rigorously in recent work of Dahlqvist and Norris, which shows that the large-$N$ limit of the Wilson loop functional for a simple closed curve in $S^{2}$ exists and that the associated variance goes to zero. We give a rigorous treatment of the second stage of analysis in the case of the 2-sphere. Dahlqvist and Norris independently performed such an analysis, using a similar but not identical method. Specifically, we establish the existence of the limit and the vanishing of the variance for arbitrary loops with (a finite number of) simple crossings. The proof is based on the Makeenko-Migdal equation for the Yang-Mills measure on surfaces, as established rigorously by Driver, Gabriel, Hall, and Kemp, together with an explicit procedure for reducing a general loop in $S^{2}$ to a simple closed curve. The methods used here also give a new proof of these results in the plane case, as a variant of the methods used by L\'{e}vy. We also consider loops on an arbitrary surface $\Sigma$. We put forth two natural conjectures about the behavior of Wilson loop functionals for topologically trivial simple closed curves in $\Sigma.$ Under the weaker of the conjectures, we establish the existence of the limit and the vanishing of the variance for topologically trivial loops with simple crossings that satisfy a "smallness" assumption. Under the stronger of the conjectures, we establish the same result without the smallness assumption.