Geometry of infinite dimensional Grassmannians and the Mickelsson-Rajeev cocycle

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 Pressely 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 note we specialize to the case when p = 2 (which is relevant for the case when dimM = 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.