Anti-self-dual four–manifolds with a parallel real spinor

Anti-self-dual metrics in the (++ --) signature that admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth-order integrable partial differential equation (PDE), and some examples are given. The corresponding twistor space is characterized by existence of a preferred non-zero real section of κ-1/4, where κ is the canonical line bundle of the twistor space. It is demonstrated that if the parallel spinor is preserved by a Killing vector, then the fourth-order PDE reduces to the dispersionless Kadomtsev-Petviashvili equation and its linearization. Einstein-Weyl structures on the space of trajectories of the symmetry are characterized by the existence of a parallel weighted null vector.