Classical Solutions for Two Dimensional QCD on the Sphere

TL;DR

This paper analyzes two-dimensional QCD on a sphere, revealing classical saddle points, phase transitions, and new stringy solutions in the large N limit, using fermionic and lattice approaches.

Contribution

It provides an alternative derivation of Witten's partition function sum over saddle points and introduces new stringy solutions with charge constraints for large N.

Findings

Reformulation of the partition function via free fermions

Identification of phase transition mechanisms

Discovery of new solutions characterized by U(1) charge

Abstract

We consider $U(N)$ and $SU(N)$ gauge theory on the sphere. We express the problem in terms of a matrix element of $N$ free fermions on a circle. This allows us to find an alternative way to show Witten's result that the partition function is a sum over classical saddle points. We then show how the phase transition of Douglas and Kazakov occurs from this point of view. By generalizing the work of Douglas and Kazakov, we find other `stringy' solutions for the $U(N)$ case in the large $N$ limit. Each solution is described by a net $U(1)$ charge. We derive a relation for the maximum charge for a given area and we also describe the critical behavior for these new solutions. Finally, we describe solutions for lattice $SU(N)$ which are in a sense dual to the continuum $U(N)$ solutions. (Parts of this paper were presented at the Strings '93 Workshop, Berkeley, May 1993.)