The Implicit Metric on a Deformation of the Atiyah-Hitchin Manifold

TL;DR

This paper derives a hyperkähler metric on a deformed Atiyah-Hitchin manifold using twistor methods, linking monopole moduli spaces with deformations of rational singularities in complex geometry.

Contribution

It introduces a new explicit generating function for the metric on a deformed Atiyah-Hitchin space, connecting monopole moduli spaces with complex surface singularity deformations.

Findings

Derived a generating function for the deformed metric

Connected monopole moduli spaces to rational singularity deformations

First explicit metric for this deformation family

Abstract

Using twistor methods we derive a generating function which leads to the hyperk\" ahler metric on a deformation of the Atiyah-Hitchin monopole moduli space. This deformation was first considered by Dancer through the quotient construction and is related to a charge two monopole configuration in a completely broken SU(3) gauge theory. The manifold and metric are the first members of a family of hyperk\" ahler manifolds which are deformations of the $D_k$ rational singularities of $C^2$.