Stevenson, D.2014-09-022014-09-022010Journal of Geometry and Physics, 2010; 60(4):664-6770393-0440http://hdl.handle.net/2440/84873In 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 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 Gr<inf>p</inf> and a determinant line bundle Det<inf>p</inf> over it, generalizing the constructions of Pressley and Segal. The definition of the line bundle Det<inf>p</inf> 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 dim M = 3) and consider the geometry of the determinant line bundle Det<inf>2</inf>. We construct explicitly a connection on Det<inf>2</inf> and give a simple formula for its curvature. From our results we obtain a geometric derivation of the Mickelsson-Rajeev cocycle. © 2010 Elsevier B.V. All rights reserved.enCopyright © 2010 Elsevier B.V. All rights reserved.Infinite dimensional Grassmannians; Regularized determinant line bundles; Schatten ideals; Lie algebra cocycleJournal article002013373210.1016/j.geomphys.2009.12.0102-s2.0-7734908530716889Stevenson, D. [0000-0003-4399-7632]