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.

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