From Principal Chiral Model to Self-dual Gravity

TL;DR

This paper shows how the SU(N) principal chiral model converges to Husain's heavenly equation in the large N limit and introduces a new solution method using Lie algebra representations.

Contribution

It establishes a connection between the principal chiral model and self-dual gravity equations, and proposes a novel approach to find solutions via Lie algebra representations.

Findings

The principal chiral model leads to Husain's heavenly equation as N approaches infinity.

The chiral equation is equivalent to a Moyal deformation of Husain's equation.

A new method for solving the deformed equation using Lie algebra representations is introduced.

Abstract

It is demonstrated that the action of SU$(N)$ principal chiral model leads in the limit $N \to {\infty}$ to the action for Husain's heavenly equation. The principal chiral model in the Hilbert space $L^2(\Re^1)$ is considered and it is shown, that in this case the chiral equation is equivalent to the Moyal deformation of Husain's heavenly equation. New method of searching for solutions to this latter equation, via Lie algebra representations in $L^2(\Re^1)$ is given.