The Twisted Photon Associated to Hyper-Hermitian Four-Manifolds

Abstract

The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a potential that generalises Plebanski's second heavenly equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a combination of the Nonlinear Graviton Construction with the Ward transform for anti-self-dual Maxwell fields.