Geometry of infinite dimensional Grassmannians and the Mickelsson–Rajeev cocycle

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 Gr p and a determinant line bundle Det p over it, generalizing the constructions of Pressley and Segal. The definition of the line bundle Det p 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 Det 2 . We construct explicitly a connection on Det 2 and give a simple formula for its curvature. From our results we obtain a geometric derivation of the Mickelsson–Rajeev cocycle.