On pseudo-hyperk\"ahler prepotentials

TL;DR

This paper introduces a new method to explicitly parameterize and construct pseudo-hyperk"ahler metrics using holomorphic functions called prepotentials, extending previous work with a coordinate-free approach.

Contribution

It constructs a surjective mapping from prepotentials to pseudo-hyperk"ahler metrics, providing a complete parameterization and explicit construction method.

Findings

Establishes a bijection between prepotentials and pseudo-hyperk"ahler metrics.

Provides a coordinate-free formulation of the construction.

Includes a reformulation of real analytic G-structures in holomorphic terms.

Abstract

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-hyperk\"ahler metrics of signature (4p,4q) in dimension 4n is constructed. The holomorphic functions, called prepotentials, are analogues of K\"ahler potentials for K\"ahler 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-hyperk\"ahler 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.