Duke Mathematical Journal
- Duke Math. J.
- Volume 166, Number 10 (2017), 1859-1922.
Geometry of pseudodifferential algebra bundles and Fourier integral operators
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 .
Duke Math. J. Volume 166, Number 10 (2017), 1859-1922.
Received: 23 January 2016
Revised: 22 October 2016
First available in Project Euclid: 21 March 2017
Permanent link to this document
Digital Object Identifier
Primary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]
Secondary: 53C08: Gerbes, differential characters: differential geometric aspects 53D22: Canonical transformations
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
Mathai, Varghese; Melrose, Richard B. Geometry of pseudodifferential algebra bundles and Fourier integral operators. Duke Math. J. 166 (2017), no. 10, 1859--1922. doi:10.1215/00127094-0000013X. https://projecteuclid.org/euclid.dmj/1490061610