On pseudo-hyperkähler prepotentials

An explicit surjection from a set of (locally defined) unconstrained holomorphic functions on a certain submanifold of (Sp_1(C) \times C^{4n}) onto the set HK_{p,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 HK_{p,q}. In particular, there exists a bijection between HK_{p,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. An appendix provides a vital tool for this construction: a reformulation of real analytic G-structures in terms of holomorphic frame fields on complex manifolds.

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