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 the action for the Park-Husain (P-H) heavenly equation. The principal chiral model in the Hilbert space L²(&R;¹) is considered and it is shown, that in this case the chiral equation is equivalent to the Moyal deformation of the P-H heavenly equation. A new method of searching for solutions to this latter equation, via Lie algebra representations in L²(&R;¹) is given.