# A K\"ahler Compatible Moyal Deformation of the First Heavenly Equation

## TL;DR

This paper develops a noncommutative Kähler manifold via Moyal deformation of self-dual gravity, preserving key properties and applying it to the Atiyah-Hitchin metric relevant for magnetic monopole interactions.

## Contribution

It introduces a novel noncommutative Kähler deformation of self-dual gravity and derives the associated Kähler potential using Moyal deformation techniques.

## Key findings

- Constructed a noncommutative Kähler manifold preserving commutative properties.
- Derived the noncommutative Kähler potential from Moyal deformed gravity.
- Applied the framework to the Atiyah-Hitchin metric for monopole interactions.

## Abstract

We construct a noncommutative K\"ahler manifold based on a non-linear perturbations of Moyal integrable deformations of $D=4$ self-dual gravity. The deformed K\"ahler manifold preserves all the properties of the commutative one, and we obtain the associated noncommutative K\"ahler potential using the Moyal deformed gravity approach. We apply this construction to the Atiyah-Hitchin metric and its K\"ahler potential, which is useful in the description of interactions among magnetic monopoles at low energies.