Structured vector bundles define differential K-theory

TL;DR

This paper introduces structured vector bundles with an equivalence relation that preserves Chern-Weil forms, forming a model of differential K-theory that combines vector bundle data with differential form information.

Contribution

It defines structured bundles and constructs a differential K-theory model using the Grothendieck construction, linking ordinary K-theory with differential form data.

Findings

Defines an equivalence relation preserving Chern-Weil forms.

Constructs a ring of structured bundles via Grothendieck completion.

Provides a simple, codified model of differential K-theory.

Abstract

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.