Self-dual Einstein Spaces, Heavenly Metrics and Twistors

TL;DR

This paper explores the relationship between quaternion-Kahler metrics, twistor theory, and solutions to Przanowski's Heavenly equation, revealing multiple complex structures and deformations in self-dual Einstein spaces with applications to string theory.

Contribution

It clarifies how quaternion-Kahler spaces can be described by various solutions to Heavenly's equation linked to different complex structures, and analyzes their deformations and eigenmodes.

Findings

Multiple solutions h correspond to different complex structures on M.

Deformations of M relate to eigenmodes of the conformal Laplacian.

Applications to S^4, H^4, and hypermultiplet moduli space in string theory.

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.