HyperK\"ahler Potentials in Cohomogeneity Two

TL;DR

This paper explicitly computes hyperK"ahler potentials for cohomogeneity two nilpotent orbits in complex simple Lie algebras, revealing a family of metrics generalizing Eguchi-Hanson in four dimensions.

Contribution

It provides explicit formulas for hyperK"ahler potentials on certain nilpotent orbits, extending known metrics and exploring their parameter spaces.

Findings

Explicit hyperK"ahler potentials for cohomogeneity two orbits

Identification of a one-parameter family of metrics

Generalization of Eguchi-Hanson metrics in higher dimensions

Abstract

A hyperK\"ahler potential is a function rho that is a K\"ahler potential for each complex structure compatible with the hyperK\"ahler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperK\"ahler metrics admitting such potentials. In this paper, we explicitly calculate the hyperK\"ahler potential when the orbit is of cohomogeneity two. In some cases, we find that this structure lies in a one-parameter family of hyperK\"ahler metrics with K\"ahler potentials, generalising the Eguchi-Hanson metrics in dimension four.