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Journal of Mathematical Physics 51 (2010) 073510
Self-dual Einstein Spaces, Heavenly Metrics and Twistors
Sergey Alexandrov1, Boris Pioline2, Stefan Vandoren2, 3

Four-dimensional quaternion-Kahler 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-Kahler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, associated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kahler deformations of M and, in the special case where M has a Killing vector field, show that the corresponding variations of h are related to eigenmodes of the conformal Laplacian on M. We exemplify our findings on the four-sphere S^4, the hyperbolic plane H^4 and on the "universal hypermultiplet", i.e. the hypermultiplet moduli space in type IIA string compactified on a rigid Calabi-Yau threefold.
1:  LPTA - Laboratoire de Physique Théorique et Astroparticules
2:  LPTHE - Laboratoire de Physique Théorique et Hautes Energies
3:  Institute for Theoretical Physics and Spinoza Institute
Physics/High Energy Physics - Theory
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