From Principal Chiral Model to Self-dual Gravity

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.