Toda Fields of SO(3) Hyper-Kahler Metrics

TL;DR

This paper explores the Toda frame formulation of SO(3)-invariant hyper-Kahler 4-metrics, explicitly constructing complex structures for Eguchi-Hanson, Taub-NUT, and Atiyah-Hitchin metrics using a rotational SO(2) symmetry.

Contribution

It provides an explicit construction of complex structures for key hyper-Kahler metrics leveraging their SO(2) isometry, including the purely rotational Atiyah-Hitchin metric.

Findings

Explicit Toda frame formulations for Eguchi-Hanson, Taub-NUT, and Atiyah-Hitchin metrics.

Construction of complex structures as a singlet and doublet under SO(2).

Identification of the Atiyah-Hitchin metric as purely rotational.

Abstract

We examine the Toda frame formulation of the SO(3)-invariant hyper-Kahler 4-metrics, namely Eguchi-Hanson, Taub-NUT and Atiyah-Hitchin. Our method exploits the presence of a rotational SO(2) isometry, leading to the explicit construction of all three complex structures as a singlet plus a doublet. The Atiyah-Hitchin metric on the moduli space of BPS SU(2) monopoles with magnetic charge 2 is purely rotational.