Loop Differential K-theory

TL;DR

This paper introduces loop differential K-theory, an equivariant extension of differential K-theory that incorporates holonomy information via an equivariant Chern-Simons form on the free loop space, providing a refined invariant.

Contribution

It defines loop differential K-theory using an equivariant Chern-Simons form, extending differential K-theory to include holonomy and loop space data.

Findings

Loop differential K-theory refines differential K-theory.

It incorporates holonomy information into K-theory classes.

The paper computes the loop differential K-theory of the circle.

Abstract

In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [SS]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.