Hyperkaehler structures and infinite-dimensional Grassmannians

TL;DR

This paper constructs a hyperkaehler structure on an infinite-dimensional Grassmannian via a Banach manifold quotient, revealing deep geometric relations and computing Kähler potentials.

Contribution

It demonstrates a hyperkaehler quotient construction yielding a Hilbert manifold from a non-diffeomorphic Banach manifold, linking it to the cotangent bundle and complexification of the Grassmannian.

Findings

The quotient space is a Hilbert manifold.

The quotient is isomorphic to the cotangent space of the Grassmannian.

Kähler potentials are explicitly computed.

Abstract

In this paper, we describe an example of a hyperkaehler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can furthermore be identified either with the cotangent space of a connected component of the restricted Grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkaehler manifolds. In addition, Kaehler potentials are computed using Kostant-Souriau's theory of prequantization.