Self-dual Einstein spaces, heavenly metrics, and twistors

Abstract

Four-dimensional quaternion-Kähler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski’s heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kähler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, associated with different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kähler deformations of M and, in the special case where M has a Killing vector field, show that the corresponding variations in h are related to eigenmodes of the conformal Laplacian on M. We exemplify our findings on the four-sphere S4, the hyperbolic plane H4, and on the “universal hypermultiplet,” i.e., the hypermultiplet moduli space in type IIA string compactified on a rigid Calabi–Yau threefold.