New heavenly double copies

TL;DR

This paper extends the double copy framework to a broader class of heavenly equations by incorporating Moyal deformations and algebraic generalizations, linking gauge theories to hyper-Hermitian geometries and exploring classical integrability.

Contribution

It introduces a generalized double copy construction for heavenly equations, including Moyal deformations and algebraic extensions, connecting gauge theories with hyper-Hermitian manifolds.

Findings

Extended double copy to hyper-Hermitian manifolds.

Introduced a double-Moyal deformation of the heavenly equation.

Constructed Lax pairs consistent with Ward's conjecture.

Abstract

The double copy relates scattering amplitudes and classical solutions in Yang-Mills theory, gravity, and related field theories. Previous work has shown that this has an explicit realisation in self-dual YM theory, where the equation of motion can be written in a form that maps directly to Pleba\'nski's heavenly equation for self-dual gravity. The self-dual YM equation involves an area-preserving diffeomorphism algebra, two copies of which appear in the heavenly equation. In this paper, we show that this construction is a special case of a wider family of heavenly-type examples, by (i) performing Moyal deformations, and (ii) replacing the area-preserving diffeomorphisms with a less restricted algebra. As a result, we obtain a double-copy interpretation for hyper-Hermitian manifolds, extending the previously known hyper-K\"ahler case. We also introduce a double-Moyal deformation of the heavenly equation. The examples where the construction of Lax pairs is possible are manifestly consistent with Ward's conjecture, and suggest that the classical integrability of the gravity-type theory may be guaranteed in general by the integrability of at least one of two gauge-theory-type single copies.