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

Abstract

Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The corresponding twistor space is characterised by existence of a preferred non-zero real section of ${\kappa }^{-1/4}$, where $\kappa$ 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 linearisation. Einstein--Weyl structures on the space of trajectories of the symmetry are characterised by the existence of a parallel weighted null vector.

18 pages, Final version, to appear in Proc. Roy. Soc. Lond. A

References in corpus (1)

• Einstein-Weyl geometry, the dKP equation and twistor theory

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