Self-Dual Yang-Mills and the Hamiltonian Structures of Integrable Systems

TL;DR

This paper explores how integrable systems derived from self-dual Yang-Mills equations naturally possess bihamiltonian structures and introduces a gauge-invariant formulation that unifies various notions of gauge equivalence among these systems.

Contribution

It presents a gauge-invariant formulation of the self-dual Yang-Mills hierarchy and discusses a unifying concept of gauge equivalence for integrable systems.

Findings

Integrable systems inherit bihamiltonian structures from self-dual Yang-Mills.

A simple gauge-invariant formulation of the hierarchy is proposed.

The notion of gauge equivalence is expanded to unify diverse systems.

Abstract

In recent years it has been shown that many, and possibly all, integrable systems can be obtained by dimensional reduction of self-dual Yang-Mills. I show how the integrable systems obtained this way naturally inherit bihamiltonian structure. I also present a simple, gauge-invariant formulation of the self-dual Yang-Mills hierarchy proposed by several authors, and I discuss the notion of gauge equivalence of integrable systems that arises from the gauge invariance of the self-duality equations (and their hierarchy); this notion of gauge equivalence may well be large enough to unify the many diverse existing notions.