Two-dimensional QCD on the sphere and on the cylinder

TL;DR

This paper explores the representation of 2D QCD on simple surfaces, especially the cylinder, using a matrix model approach, and investigates phase transitions and large N limits with new analytical results.

Contribution

It introduces a novel matrix model formulation of QCD2 on the cylinder and provides new insights into phase transitions and large N behavior.

Findings

Alternative description of the Douglas-Kazakov phase transition on the sphere

Evidence for an analogous phase transition on the cylinder

Explicit large N limit of the partition function and Itzykson-Zuber integral

Abstract

The partition functions of QCD2 on simple surfaces admit representations in terms of exponentials of the inverse coupling, that are modular transforms of the usual character expansions. We review the construction of such a representation in the case of the cylinder, and show how it leads to a formulation of QCD2 as a $c=1$ matrix model of the Kazakov-Migdal type. The eigenvalues describe the positions of $N$ Sutherland fermions on a circle, while their discretized momenta label the representations in the corresponding character expansion. Using this language, we derive some new results: we give an alternative description of the Douglas-Kazakov phase transition on the sphere, and we argue that an analogous phase transition exists on the cylinder. We calculate the large $N$ limit of the partition function on the cylinder with boundary conditions given by semicircular distributions of eigenvalues, and we find an explicit expression for the large $N$ limit of the Itzykson-Zuber integral with the same boundary conditions. (Talk given at the ``Workshop on high energy physics and cosmology'' at Trieste, July 1993.)