# Self-dual gravity via Hitchin's equations

## TL;DR

This paper derives half-flat metrics from Hitchin's equations, explores the SU(∞) case, and constructs solutions using integrable equations and Moyal deformations, contributing to the understanding of self-dual gravity.

## Contribution

It introduces a method to obtain half-flat metrics from Hitchin's equations, including the SU(∞) case, and constructs explicit solutions via integrable equations and deformations.

## Key findings

- Derived half-flat metrics from Hitchin's equations.
- Connected SU(∞) Hitchin's equations to the Husain-Park equation.
- Constructed solutions using Liouville, sinh-Gordon, and Painlevé III equations.

## Abstract

In this work half-flat metrics are obtained from Hitchin's equations. The SU$(\infty)$ Hitchin's equations are obtained and as a consequence of them, the Husain-Park equation is found. Considering that the gauge group is SU$(2)$, some solutions associated to Liouville, sinh-Gordon and Painlev\'e III equations are taken and, through Moyal deformations, solutions of the SU$(\infty)$ Hitchin's equations are obtained. From these solutions, hamiltonian vector fields are determined, which in turn are used to construct the half-flat metrics. Following an approach of Dunajski, Mason and Woodhouse, it is also possible to construct half-flat metrics on ${\cal M} \times\mathbb{CP}^{1}$.