The Caloron correspondence and odd differential k-theory.

Abstract

The caloron correspondence (introduced in [32] and generalised in [25, 33, 41]) is a tool that gives an equivalence between principal G-bundles based over the manifold M x S¹ and principal LG-bundles on M, where LG is the Frechet Lie group of smooth loops in the Lie group G. This thesis uses the caloron correspondence to construct certain differential forms called string potentials that play the same role as Chern-Simons forms for loop group bundles. Following their construction, the string potentials are used to define degree 1 differential characteristic classes for ΩU(n)-bundles. The notion of an Ω vector bundle is introduced and a caloron correspondence is developed for these objects. Finally, string potentials and Ω vector bundles are used to define an Ω bundle version of the structured vector bundles of [38]. The Ω model of odd differential K-theory is constructed using these objects and an elementary differential extension of odd K-theory appearing in [40].