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 Plebański’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-Kähler 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.
Highlights
A crucial idea, which fostered the development of the double copy, is the colourkinematics duality of Bern, Carrasco and Johansson (BCJ) [2]
This ‘duality’ remains a conjecture at loop level, but is well established for tree-level gauge theory. It states that the scattering amplitudes can be written in such a way that the kinematic dependence has the same algebraic properties as the colour dependence, hinting that there is a kinematic analogue of the colour Lie algebra in gauge theory
Upon taking the double copy, the colour information is replaced by another copy of its kinematic analogue, so that two kinematic algebras appear in gravity scattering amplitudes
Summary
We review salient details of the double copy in the self-dual sector described in ref. [83]. We set the different coupling constants (y, g, κ) to one This ‘heavenly’ example of the double copy is a different viewpoint to the well-known story in the integrability literature where self-dual gravity is recovered as a symmetry reduction of self-dual Yang-Mills theory, considering the gauge group as the area-preserving diffeomorphism group on the two-surface Σ, SDiff(Σ) [108,109,110]. This fact has been used to build hyper-Kähler metrics from solutions of two-dimensional reduced SDYM equations [111,112,113]. It would be interesting to explore these and other alternative formulations, in the search for further insights into the double copy
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