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Self-dual Einstein Spaces, Heavenly Metrics and Twistors
Alexandrov S., Pioline B., Vandoren S.
Journal of Mathematical Physics 51 (2010) 073510 - http://hal.archives-ouvertes.fr/hal-00442066
Physique/Physique des Hautes Energies - Théorie
Self-dual Einstein Spaces, Heavenly Metrics and Twistors
Sergey Alexandrov ()1, Boris Pioline2, Stefan Vandoren2, 3
1 :  LPTA - Laboratoire de Physique Théorique et Astroparticules
CNRS : UMR5207 – IN2P3 – Université Montpellier II - Sciences et techniques
Bât 13- 1er Et. - CC 070 Place Eugène Bataillon 34095 MONTPELLIER CEDEX 5
2 :  LPTHE - Laboratoire de Physique Théorique et Hautes Energies
CNRS : UMR7589 – Université Pierre et Marie Curie (UPMC) - Paris VI – Université Paris VII - Paris Diderot
3 :  Institute for Theoretical Physics and Spinoza Institute
Spinoza Institute
Postbus 80.195, 3508 TD Utrecht, The Netherlands
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.

Journal of Mathematical Physics (J. Math. Phys.)
Publisher American Institute of Physics (AIP)
ISSN 0022-2488 
Articles dans des revues avec comité de lecture

44 pages, 1 figure

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