On pseudo-hyperkähler prepotentials

Abstract

An explicit surjection from a set of (locally defined) unconstrained holomorphic functions on a certain submanifold of Sp1(ℂ) × ℂ4n onto the set HKp,q of local isometry classes of real analytic pseudo-hyperkähler metrics of signature (4p, 4q) in dimension 4n is constructed. The holomorphic functions, called prepotentials, are analogues of Kähler potentials for Kähler metrics and provide a complete parameterisation of HKp,q. In particular, there exists a bijection between HKp,q and the set of equivalence classes of prepotentials. This affords the explicit construction of pseudo-hyperkähler metrics from specified prepotentials. The construction generalises one due to Galperin, Ivanov, Ogievetsky, and Sokatchev. Their work is given a coordinate-free formulation and complete, self-contained proofs are provided. The Appendix provides a vital tool for this construction: a reformulation of real analytic G-structures in terms of holomorphic frame fields on complex manifolds.