Geometry of infinite dimensional Grassmannians and the Mickelsson-Rajeev cocycle

TL;DR

This paper constructs an explicit connection and curvature formula for a determinant line bundle over a specific infinite dimensional Grassmannian, providing a geometric derivation of the Mickelsson-Rajeev cocycle.

Contribution

It explicitly constructs a connection and computes the curvature on the determinant line bundle for p=2, and derives the Mickelsson-Rajeev cocycle geometrically.

Findings

Explicit connection on the determinant line bundle for p=2

Simple formula for the curvature of the bundle

Geometric derivation of 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.