A Hilbert bundle description of differential K-theory

We give an infinite dimensional description of the differential K-theory of a manifold M. The generators are triples [H,A,ω] where H is a Z2-graded Hilbert bundle on M, A is a superconnection on H and ω is a differential form on M. The relations involve eta forms. We show that the ensuing group is the differential K-group Kˇ0(M). In addition, we construct the pushforward of a finite dimensional cocycle under a proper submersion with a Riemannian structure. We give the analogous description of the odd differential K-group Kˇ1(M). Finally, we give a model for twisted differential K-theory.