Geometry of pseudodifferential algebra bundles and Fourier integral operators

Abstract

We study the geometry and topology of (filtered) algebra bundles Ψ ℤ over a smooth manifold X with typical fiber Ψ 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(ℱ ℂ .(Z;V)) 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 Ψ 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 Ψ ℤ /Ψ -∞ . Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over X.