An index theorem in differential K-theory

We construct a geometric model for differential K-theory, and prove it is isomorphic to the model proposed in [25]. We construct differential K-orientations for families and elucidate the pushforward map given in [25] in detail. We prove a geometric index theorem for odd dimensional manifolds. Finally, using this index theorem and the holonomy theorem of Bismut and Freed from [10], we prove what may be considered a special case of a geometric refinement of the Aityah-Singer index theorem.