Hyper-Hermitian Weyl double copy

A bstract Recently the Kerr-Schild double copy of gravitational and gauge fields has been studied at the level of equations of motion: the so called classical double copy. In previous work the self-dual double copy has been generalized from the hyper-Kähler case to the hyper-Hermitian one by replacing the kinematic algebra consisting in the area-preserving diffeomorphism algebra by the diffeomorphisms on a surface algebra. This leads to the hyper-Hermitian double copy in the Kerr-Schild approach. In the present article, we further study the hyper-Hermitian case using the Weyl double copy formalism. In particular, we have found new solutions within this formalism for a family of hyper-Hermitian metrics termed elementary states in the literature. One of the main features of the approach is that the single copy will contain two Maxwell spinors, one of them is source-free, while the other has a source current. In the process it is argued that this result is compatible with the fact that, in general, the hyper-Hermitian spaces are not Ricci-flat.