A Hilbert bundle description of differential K-theory

TL;DR

This paper presents an infinite-dimensional Hilbert bundle framework for differential K-theory, including constructions of pushforward maps and models for twisted theories, advancing the mathematical understanding of these topological invariants.

Contribution

It introduces a novel Hilbert bundle-based description of differential K-theory, extending to pushforward maps and twisted theories, providing new tools for geometric and topological analysis.

Findings

Defined generators as triples involving Hilbert bundles, superconnections, and forms.

Established the group structure as the differential K-group rac{K}^0(M).

Constructed pushforward maps and models for twisted differential K-theory.

Abstract

We give an infinite dimensional description of the differential K-theory of a manifold $M$. The generators are triples $[H, A, \omega]$ where $H$ is a ${\bf Z}_2$-graded Hilbert bundle on $M$, $A$ is a superconnection on $H$ and $\omega$ is a differential form on $M$. The relations involve eta forms. We show that the ensuing group is the differential K-group $\check{K}^0(M)$. 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 $\check{K}^1(M)$. Finally, we give a model for twisted differential K-theory.