Geometry of Pseudodifferential algebra bundles and Fourier Integral Operators

TL;DR

This paper explores the geometry and topology of algebra bundles of pseudodifferential operators, generalizing key theorems, defining connections, and computing characteristic classes with applications to non-finite-dimensional bundles.

Contribution

It extends Duistermaat and Singer's theorem to vector bundle sections, introduces natural connections and B-fields, and provides a de Rham formula for the Dixmier-Douady class.

Findings

Generalized automorphism group characterization

Defined natural connections and B-fields

Derived de Rham representative of the Dixmier-Douady class

Abstract

We study the geometry and topology of (filtered) algebra-bundles ${\bf\Psi}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $\Psi^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators ${\rm PGL}({\mathcal F}^\bullet(Z; V))$, is precisely the automorphism group, ${\rm Aut}(\Psi^{\mathbb Z}(Z; V)),$ of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their paper by microlocal ones, thereby removing the topological assumption well as extending their result to sections of a vector bundle. We define a natural class of connections and B-fields the principal bundle to which ${\bf\Psi}^{\mathbb Z}$ is associated and obtain a de Rham representative of the Dixmier-Douady class, in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki; the resulting formula only depends on the formal symbol algebra ${\bf\Psi}^{\mathbb Z}/{\bf\Psi}^{-\infty}.$ Examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fibre bundle over $X.$