Eigenvalue density of Wilson loops in 2D SU(N) YM

TL;DR

This paper investigates the eigenvalue density of Wilson loop matrices in 2D SU(N) Yang-Mills theory, analyzing finite N smooth extensions of the known infinite N phase transition and comparing different smoothed expressions.

Contribution

It provides explicit finite N smoothed expressions for eigenvalue densities and compares various approaches to the infinite N phase transition in 2D SU(N) YM.

Findings

Finite N eigenvalue densities are smooth and differ from the infinite N limit.

Multiple smoothed expressions converge to the Durhuus-Olesen singularity at infinite N.

Comparison of different finite N extensions enhances understanding of phase transition behavior.

Abstract

In 1981 Durhuus and Olesen (DO) showed that at infinite N the eigenvalue density of a Wilson loop matrix W associated with a simple loop in two-dimensional Euclidean SU(N) Yang-Mills theory undergoes a phase transition at a critical size. The averages of det(z-W), 1/det(z-W), and det(1+uW)/(1-vW) at finite N lead to three different smoothed out expressions, all tending to the DO singular result at infinite N. These smooth extensions are obtained and compared to each other.