A Geometric Model for Odd Differential K-theory

TL;DR

This paper introduces a geometric bundle model for odd differential K-theory, demonstrating its uniqueness and characterizing key forms, thus advancing the understanding of differential refinements in K-theory.

Contribution

It provides a new bundle theoretic model for odd differential K-theory using the caloron correspondence, establishing its uniqueness and characterizing the odd Chern character.

Findings

The model is unique up to a natural isomorphism.

Characterization of the odd Chern character and transgression form.

Proof of a conjecture related to odd K-theory extensions.

Abstract

Odd $K$-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential $K$-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We characterise the odd Chern character and its transgression form in terms of a connection and Higgs field and discuss some applications. Our model can be seen as the odd counterpart to the Simons-Sullivan construction of even differential $K$-theory. We use this model to prove a conjecture of Tradler-Wilson-Zeinalian regarding a related differential extension of odd $K$-theory