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 and in the process introduced for any p >=1 another infinite dimensional Grassmann manifold and a determinant line bundle defined over it. The construction of this determinant line bundle required the notion of a regularized determinant for bounded operators. In this note we specialize to the case p =2 and construct explicitly a connection on the corresponding determinant line bundle and give a simple and explicit formula for its curvature. As an application of our results we give a geometric derivation of the Mickelsson-Rajeev cocycle.