Duke Math. J. 166 (10), 1859-1922, (15 July 2017) DOI: 10.1215/00127094-0000013X
Varghese Mathai, Richard B. Melrose
KEYWORDS: pseudodifferential algebra bundles, Fourier integral operators, automorphisms of pseudodifferential operators, derivations of pseudodifferential operators, twisted (fiber) cosphere bundles, regularized trace, residue trace, central extension, gerbes, Dixmier–Douady invariant, 58J40, 53C08, 53D22
We study the geometry and topology of (filtered) algebra bundles over a smooth manifold with typical fiber , the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle over the compact manifold 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 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 -fields on the principal bundle to which 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 . Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over .