Linear perturbations of Hyperkahler metrics

TL;DR

This paper investigates linear perturbations of hyperkahler manifolds using twistor methods, revealing how deformations relate to holomorphic functions and affect the manifold's symmetries, with applications to the Atiyah-Hitchin manifold.

Contribution

It introduces a twistor-based framework for encoding and analyzing linear perturbations of hyperkahler metrics via holomorphic functions of specific variables.

Findings

Deformations are encoded by holomorphic functions of 2d+1 variables.

Deformations generally break tri-holomorphic isometries.

Leading exponential deviation of Atiyah-Hitchin manifold from Taub-NUT is determined.

Abstract

We study general linear perturbations of a class of 4d real-dimensional hyperkahler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d+1 variables, as opposed to the functions of d+1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kahler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah-Hitchin manifold away from its negative mass Taub-NUT limit. In a companion paper arXiv:0810.1675, we extend these techniques to quaternionic-Kahler spaces with isometries.