Characteristic Classes and Zeroth Order Pseudodifferential Operators

TL;DR

This paper investigates the vanishing of Wodzicki-Chern classes for bundles with the group of invertible zeroth order pseudodifferential operators, showing they vanish under certain conditions and relate to de Rham cohomology.

Contribution

It provides evidence for the conjecture that Wodzicki-Chern classes vanish for all such bundles and proves vanishing when the structure group reduces to identity leading order symbols.

Findings

Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators.

Vanishing is proven when the structure group reduces to operators with identity leading order symbol.

Leading order Chern classes are nonzero and detect elements of de Rham cohomology groups.

Abstract

We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the identity. The leading order Chern classes are nonzero in general, and they detect elements of the de Rham cohomology groups of the classifying space BZ.