Self-dual Einstein Spaces, Heavenly Metrics and Twistors

Abstract

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.

44 pages, 1 figure; misprints corrected

References in corpus (8)

• D-instantons and twistors
• Membrane Instantons and de Sitter Vacua
• Quantum mirror symmetry and twistors
• Linear perturbations of Hyperkahler metrics
• Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and G_2(2)
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Cited by in corpus (19)

• Twistor Approach to String Compactifications: a Review
• Twistor theory at fifty: from contour integrals to twistor strings
• Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces
• Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence
• Hypermultiplet metric and D-instantons
• Arithmetic and Hyperbolic Structures in String Theory
• Quaternion-Kaehler four-manifolds and Przanowski's function
• Hypermultiplet metric and NS5-instantons
• Killing Symmetries in $\mathcal{H}$-Spaces with $\Lambda$
• Metastable de Sitter vacua in N=2 to N=1 truncated supergravity
• c-map as c=1 string
• Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons
• Membrane Instantons from Toda Field Theory
• A Kähler Compatible Moyal Deformation of the First Heavenly Equation
• Compact Osserman manifolds with neutral metric
• On Walker and para-Hermite Einstein spaces
• Integrability of anti-self-dual vacuum Einstein equations with nonzero cosmological constant: an infinite hierarchy of nonlocal conservation laws
• Warped Product Einstein Manifolds in Four Dimensions
• Calabi-Yau compactification of type II string theories