Chern-Simons Solitons, Toda Theories and the Chiral Model

TL;DR

This paper classifies finite charge SU(N) solutions of 2D self-dual Chern-Simons equations by linking them to harmonic maps and chiral models, revealing new relationships with Toda theories.

Contribution

It introduces a classification method for SU(N) solutions by transforming Chern-Simons equations into chiral model equations and applying harmonic map classification, establishing new links with Toda theories.

Findings

Classified all finite charge SU(N) solutions using harmonic map techniques.

Established a new relationship between SU(N) Toda and chiral model solutions.

Provided a framework connecting Chern-Simons equations to integrable models.

Abstract

The two-dimensional self-dual Chern--Simons equations are equivalent to the conditions for static, zero-energy solutions of the $(2+1)$-dimensional gauged nonlinear Schr\"odinger equation with Chern--Simons matter-gauge dynamics. In this paper we classify all finite charge $SU(N)$ solutions by first transforming the self-dual Chern--Simons equations into the two-dimensional chiral model (or harmonic map) equations, and then using the Uhlenbeck--Wood classification of harmonic maps into the unitary groups. This construction also leads to a new relationship between the $SU(N)$ Toda and $SU(N)$ chiral model solutions.