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

TL;DR

The paper studies four-dimensional hyper-Kähler manifolds with $SO(3)$ symmetry by reformulating the problem in terms of the $3$-dimensional continual Toda equation, $(\partial_x^2+\partial_y^2)\Psi+\partial_z^2(e^{\Psi})=0$, to obtain the Toda potential $\Psi$ for the complete, non-singular examples Eguchi–Hanson, Taub–NUT, and Atiyah–Hitchin. It shows that the Eguchi–Hanson and Taub–NUT metrics admit explicit free-field realizations of the Toda potential, while the Atiyah–Hitchin metric does not, signaling a genuinely new class of continual Toda solutions governed by topological features of area-preserving diffeomorphisms. The work clarifies how rotational versus translational isometries manifest in the Toda framework and discusses implications for potential series of purely rotational HK4 metrics, as well as connections to monopole moduli spaces and Nahm data. The Note Added highlights recent progress proposing new families of purely rotational HK4 manifolds, underscoring ongoing exploration of this geometric landscape.

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.