Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

TL;DR

This paper explores the geometric structures of slices to sums of semisimple adjoint orbits in $GL(n,C)$, revealing connections to well-known hyperkähler manifolds and Hilbert schemes, with implications for monopole moduli spaces.

Contribution

It establishes isomorphisms between slices to sums of orbits and classical hyperkähler manifolds, extending to higher dimensions with links to Hilbert schemes and monopole spaces.

Findings

Slices for $n=2,3,4$ correspond to known hyperkähler manifolds.

Higher $n$ slices relate to open subsets of Hilbert schemes of points.

These subsets carry complete hyperkähler metrics, including the $L^2$-metric on monopole moduli spaces.

Abstract

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on of one the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperk\"ahler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this turns out to be the natural $L^2$-metric on a hyperk\"ahler submanifold of the monopole moduli space.