Large-N limit of the generalized 2D Yang-Mills theory on cylinder

TL;DR

This paper investigates the large-N limit of generalized 2D Yang-Mills theory on a cylinder, revealing a connection between the classical collective field equations and a generalized Hopf equation, and deriving the large-N free energy via the Itzykson-Zuber integral.

Contribution

It introduces a novel approach linking the classical collective field equations to a generalized Hopf equation in the large-N limit of 2D Yang-Mills theory on a cylinder.

Findings

Classical collective field obeys a generalized Hopf equation.

Large-N free energy is determined by the inverse of the Hopf equation solution.

Established a connection between the saddle-point density and the Hopf equation.

Abstract

Using the collective field theory approach of large-N generalized two-dimensional Yang-Mills theory on cylinder, it is shown that the classical equation of motion of collective field is a generalized Hopf equation. Then, using the Itzykson-Zuber integral at the large-N limit, it is found that the classical Young tableau density, which satisfies the saddle-point equation and determines the large-N limit of free energy, is the inverse of the solution of this generalized Hopf equation, at a certain point.