Geometry of pseudodifferential algebra bundles and Fourier integral operators

Abstract

We study the geometry and topology of (filtered) algebra bundles ${\mathbf{\Psi }}^{\mathbb{Z}}$ over a smooth manifold $X$ with typical fiber ${\Psi }^{\mathbb{Z}}(Z;V)$, the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle $V$ over the compact manifold $Z$ and of integral order. First, a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators $PG\left({\mathcal{F}}^{\mathbb{C}}\right(Z;V\left)\right)$ is precisely the automorphism group of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their work by microlocal ones, thereby removing the topological assumption. We define a natural class of connections and $B$-fields on the principal bundle to which ${\mathbf{\Psi }}^{\mathbb{Z}}$ 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 ${\mathbf{\Psi }}^{\mathbb{Z}}/{\mathbf{\Psi }}^{-\infty }$. Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over $X$.