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 View the MathML source 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 View the MathML source and a determinant line bundle View the MathML source over it, generalizing the constructions of Pressley and Segal. The definition of the line bundle View the MathML source 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 dimM=3) and consider the geometry of the determinant line bundle View the MathML source. We construct explicitly a connection on View the MathML source and give a simple formula for its curvature. From our results we obtain a geometric derivation of the Mickelsson–Rajeev cocycle