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Type: Journal article
Title: Geometry of infinite dimensional Grassmannians and the Mickelsson-Rajeev cocycle
Author: Stevenson, D.
Citation: Journal of Geometry and Physics, 2010; 60(4):664-677
Publisher: Elsevier B.V.
Issue Date: 2010
ISSN: 0393-0440
Statement of
Danny Stevenson
Abstract: In their study of the representation theory of loop groups, Pressley and Segal introduced a determinant line bundle over an infinite dimensional Grassmann manifold. Mickelsson and Rajeev subsequently generalized the work of Pressley and Segal to obtain representations of the groups Map (M, G) where M is an odd dimensional spin manifold. In the course of their work, Mickelsson and Rajeev introduced for any p ≥ 1, an infinite dimensional Grassmannian Grp and a determinant line bundle Detp over it, generalizing the constructions of Pressley and Segal. The definition of the line bundle Detp requires the notion of a regularized determinant for bounded operators. In this paper we specialize to the case when p = 2 (which is relevant for the case when dim M = 3) and consider the geometry of the determinant line bundle Det2. We construct explicitly a connection on Det2 and give a simple formula for its curvature. From our results we obtain a geometric derivation of the Mickelsson-Rajeev cocycle. © 2010 Elsevier B.V. All rights reserved.
Keywords: Infinite dimensional Grassmannians; Regularized determinant line bundles; Schatten ideals; Lie algebra cocycle
Rights: Copyright © 2010 Elsevier B.V. All rights reserved.
DOI: 10.1016/j.geomphys.2009.12.010
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Mathematical Sciences publications

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