Chern-Simons Field Theory and Completely Integrable Systems

TL;DR

This paper establishes a connection between non-abelian Chern-Simons theory and integrable systems, revealing new insights into their relationship through gauge reductions and transformations.

Contribution

It demonstrates how Chern-Simons actions relate to integrable systems like the Davey-Stewartson hierarchy via gauge reductions and interprets Bäcklund Transformations within this framework.

Findings

Chern-Simons theory relates to integrable systems through gauge reductions.

Bäcklund Transformations are interpreted via Chern-Simons equations.

A mapping between Chern-Simons theory and nonlinear sigma-models is discussed.

Abstract

We show that the classical non-abelian pure Chern-Simons action is related in a natural way to completely integrable systems of the Davey-Stewartson hyerarchy, via reductions of the gauge connection in Hermitian spaces and by performing certain gauge choices. The B\"acklund Transformations are interpreted in terms of Chern-Simons equations of motion or, on the other hand, as a consistency condition on the gauge. A mapping with a nonlinear $\sigma$-model is discussed.