Large $N$ Phase Transition in Continuum QCD$_2$

TL;DR

This paper calculates the exact partition function of continuum QCD in two dimensions at large N, revealing a third-order phase transition with implications for higher-dimensional theories.

Contribution

It provides an exact large N solution for 2D continuum QCD, demonstrating a phase transition and connecting to branched coverings and elliptic integrals.

Findings

Identifies a third-order phase transition at large N.

Expresses the strong coupling partition function using elliptic integrals.

Shows the Wilson loop does not follow a simple area law in the weak coupling phase.

Abstract

We compute the exact partition function for pure continuous Yang-Mills theory on the two-sphere in the large $N$ limit, and find that it exhibits a large $N$ third order phase transition with respect to the area $A$ of the sphere. The weak coupling (small A) partition function is trivial, while in the strong coupling phase (large A) it is expressed in terms of elliptic integrals. We expand the strong coupling result in a double power series in $e^{-g^2 A}$ and $g^2 A$ and show that the terms are the weighted sums of branched coverings proposed by Gross and Taylor. The Wilson loop in the weak coupling phase does not show the simple area law. We discuss some consequences for higher dimensions.