Geometry of Pseudodifferential Algebra Bundles and Fourier Integral Operators

We study the geometry and topology of (filtered) algebra-bundles ΨZ over a smooth manifold X with typical fibre ΨZ(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 PG(FC(Z;V )), is precisely the automorphism group, Aut(ΨZ(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. We define a natural class of connections and B-fields on the principal bundle to which ΨZ 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 ΨZ/Ψ−∞. Some examples of pseudodifferential bundles with non-torsion Dixmier-Douady class are given; in general such a bundle is not associated to a finite dimensional fibre bundle over X.