Some Properties of Large $N$ Two Dimensional Yang--Mills Theory

TL;DR

This paper explores large N two-dimensional Yang--Mills theory on different manifolds, establishing a relation between saddle-point and collective field descriptions, and analyzing phase transitions, instantons, and boundary effects.

Contribution

It establishes a connection between saddle-point and collective field theories in 2D QCD and derives an exact formula for phase transitions with arbitrary boundary holonomy.

Findings

The Douglas--Kazakov phase transition is linked to eigenvalue gaps in Wilson loops.

An exact formula for phase transition on a disc with boundary holonomy is derived.

Partition function on a vertex manifold vanishes unless boundary conditions meet a specific selection rule.

Abstract

Large $N$ two-dimensional QCD on a cylinder and on a vertex manifold (a sphere with three holes) is investigated. The relation between the saddle-point description and the collective field theory of QCD$_2$ is established. Using this relation, it is shown that the Douglas--Kazakov phase transition on a cylinder is associated with the presence of a gap in the eigenvalue distributions for Wilson loops. An exact formula for the phase transition on disc with an arbitrary boundary holonomy is found. The role of instantons in inducing such transitions is discussed. The zero-area limit of the partition function on a vertex manifold is studied. It is found that this partition function vanishes unless the boundary conditions satisfy a certain selection rule which is an analogue of momentum conservation in field theory.