The Dynamics of Classical Chiral $QCD_{2}$ Currents

TL;DR

This paper analyzes the classical dynamics of chiral QCD2 currents, revealing integrability, conserved quantities, and geometric interpretations, along with explicit soliton solutions via inverse scattering methods.

Contribution

It introduces an integrable framework for classical chiral QCD2, including Lax pairs, conserved charges, geometric insights, and explicit soliton solutions.

Findings

Existence of an infinite set of conserved quantities.

Demonstration of Poisson-commuting conserved charges.

Explicit reflectionless soliton solutions obtained.

Abstract

In this paper the dynamics of the classical chiral $QCD_{2}$ currents is studied. We describe how the dynamics of the theory can be summarized in an equation of the Lax form, thereby demonstrating the existence of an infinite set of conserved quantities. Next, the $r$ matrix of a fundamental Poisson relation is obtained and used to demonstrate that the conserved charges Poisson commute. An underlying diffeomorphism symmetry of the equations of motion which is not a symmetry of the action is used to provide a geometric interpretation for the case of gauge group SU(2). This enables us to show that the solutions to the classical equations of motion can be identified with a large class of curves, to demonstrate an auto-B\"acklund transformation and to demonstrate a non linear superposition principle. A link between the spectral problem for $QCD_{2}$ and the solution to the closed curve problem is also demonstrated. We then go on to provide a systematic inverse scattering treatment. This formalism is used to obtain the reflectionless single boundstate eigenvalue soliton solution.