K-Theories for Certain Infinite Rank Bundles

TL;DR

This paper develops K-theories for various classes of infinite rank vector bundles arising in topology and physics, enabling the detection of nontrivial elements via Chern characters.

Contribution

It introduces new K-theories tailored for infinite rank bundles such as those in the families index theorem and pseudodifferential operator bundles.

Findings

Constructed K-theories for infinite rank bundles

Developed formalism and Chern character applications

Detected nontrivial elements in these K-theories

Abstract

Several authors have recently constructed characteristic classes for classes of infinite rank vector bundles appearing in topology and physics. These include the tangent bundle to the space of maps between closed manifolds, the infinite rank bundles in the families index theorem, and bundles with pseudodifferential operators as structure group. In this paper, we construct the corresponding K-theories for these types of bundles. We develop the formalism of these theories and use their Chern character to detect a large class of nontrivial elements.