Hyperkähler structures and infinite-dimensional Grassmannians

Abstract

In this paper, we describe an example of a hyperkähler 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 be furthermore identified either with the cotangent space of a connected component Gr j res (j ∈ Z), of the restricted Grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkähler manifolds. Moreover, Kähler potentials associated with the natural complex structure of the cotangent space of Gr j res and with the natural complex structure of the complexification of Gr j res are computed using Kostant-Souriau's theory of prequantization.

Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits

“…A first step toward the generalization of the results of O. Biquard and P. Gauduchon mentioned above to the infinite-dimensional setting has been carry out by the author in [30]. An infinite-dimensional hyperkähler quotient of a Banach manifold by a Banach Lie group has been used to construct hyperkähler structures on a natural complexification of the restricted Grassmannian and on the cotangent space of the restricted Grassmannian.…”

“…Explicit formulas of the metric in terms of the complex orbit and of the cotangent space are given. As a particular case, we obtain the one-parameter family of hyperkähler structures on a natural complexification of the restricted Grassmannian and on the cotangent space of the restricted Grassmannian constructed by hyperkähler reduction in [30] .…”

“…Des formules explicites de la métriques en termes de l'orbite complexifiée et de l'espace cotangent sont données. Comme cas particulier, nous retrouvons la famille à un paramètre de structures hyperkählériennes sur une complexification naturelle de la grassmannienne restreinte et sur l'espace cotangent de la grassmannienne restreinte obtenue par réduction hyperkählérienne en [30].…”

“…The Kähler structures on Gr 0 res (H + , H − ) obtained by these identifications are proportional to the standard one as defined in [26] or [37]. The complexified , k = 0, was obtained by hyperkähler reduction in [30].…”

On Complex Infinite-Dimensional Grassmann Manifolds

“…The occurrence of quaternionic structures on this level is a fairly known phenomenon in finite dimensions; see for instance the complexifications of Hermitian symmetric spaces of compact type studied in [9] and [10]. An infinite-dimensional version of this phenomenon was discussed in [29] in the special case of the restricted Grassmann manifold. The latter manifold is modeled on a Hilbert space and is endowed with a Riemannian structure which allows one to construct almost complex structures on the tangent bundle by identifying it with the cotangent bundle.…”