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 Pressley and Segal to obtain representations of the groups Map(M,G) where MM is an odd dimensional spin manifold. In the course of their work, Mickelsson and Rajeev introduced for any p≥1p≥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=2p=2 (which is relevant for the case when dimM=3dimM=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.