Toda fields of SO(3) hyper-Kahler metrics and free field realizations

TL;DR

This paper explores SO(3)-invariant hyper-Kahler metrics in four dimensions, focusing on solutions to the Toda equation and their free field realizations, highlighting unique features of the Atiyah-Hitchin metric.

Contribution

It provides explicit Toda potentials for known hyper-Kahler metrics and investigates their free field expansions, offering insights into their topological and geometric properties.

Findings

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

Identification of unique features in the Atiyah-Hitchin metric related to diffeomorphism groups.

Analysis of free field series expansions of the solutions.

Abstract

The Eguchi-Hanson, Taub-NUT and Atiyah-Hitchin metrics are the only complete non-singular SO(3)-invariant hyper-Kahler metrics in four dimensions. The presence of a rotational SO(2) isometry allows for their unified treatment based on solutions of the 3-dim continual Toda equation. We determine the Toda potential in each case and examine the free field realization of the corresponding solutions, using infinite power series expansions. The Atiyah-Hitchin metric exhibits some unusual features attributed to topological properties of the group of area preserving diffeomorphisms. The construction of a descending series of SO(2)-invariant 4-dim regular hyper-Kahler metrics remains an interesting question.