Solutions in Self-Dual Gravity Constructed Via Chiral Equations

TL;DR

This paper explores solutions in self-dual gravity using chiral equations, identifying a Lie algebra of symplectic diffeomorphisms and expressing self-dual metrics through harmonic maps and algebraic bases.

Contribution

It introduces a finite-dimensional Lie algebra of symplectic diffeomorphisms and constructs explicit self-dual gravity solutions via chiral equations.

Findings

Identified a Lie algebra related to symplectic diffeomorphisms.

Expressed self-dual metrics explicitly using harmonic maps.

Connected chiral models to self-dual gravity solutions.

Abstract

The chiral model for self-dual gravity given by Husain in the context of the chiral equations approach is discussed. A Lie algebra corresponding to a finite dimensional subgroup of the group of symplectic diffeomorphisms is found, and then use for expanding the Lie algebra valued connections associated with the chiral model. The self-dual metric can be explicitly given in terms of harmonic maps and in terms of a basis of this subalgebra.