Duke Mathematical Journal

Geometry of pseudodifferential algebra bundles and Fourier integral operators

Varghese Mathai and Richard B. Melrose

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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 $\operatorname{PG}(\mathcal{F}^{\mathbb{C}}(Z;V))$ 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 $\mathbf{\Psi}^{\mathbb{Z}}$ 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 $X$.

Article information

Source
Duke Math. J. Volume 166, Number 10 (2017), 1859-1922.

Dates
Received: 23 January 2016
Revised: 22 October 2016
First available in Project Euclid: 21 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1490061610

Digital Object Identifier
doi:10.1215/00127094-0000013X

Subjects
Primary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]
Secondary: 53C08: Gerbes, differential characters: differential geometric aspects 53D22: Canonical transformations

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

Citation

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.


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