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Hyperkaehler structures and infinite-dimensional Grassmannians

A. B. Tumpach
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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.

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A. B. Tumpach. 2007-03-13. Hyperkaehler structures and infinite-dimensional Grassmannians. https://doi.org/10.1016/j.jfa.2006.05.019

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