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15 July 2017 Geometry of pseudodifferential algebra bundles and Fourier integral operators
Varghese Mathai, Richard B. Melrose
Duke Math. J. 166(10): 1859-1922 (15 July 2017). DOI: 10.1215/00127094-0000013X


We study the geometry and topology of (filtered) algebra bundles ΨZ over a smooth manifold X with typical fiber ΨZ(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(FC(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 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. Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over X.


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Varghese Mathai. Richard B. Melrose. "Geometry of pseudodifferential algebra bundles and Fourier integral operators." Duke Math. J. 166 (10) 1859 - 1922, 15 July 2017.


Received: 23 January 2016; Revised: 22 October 2016; Published: 15 July 2017
First available in Project Euclid: 21 March 2017

zbMATH: 06773294
MathSciNet: MR3679883
Digital Object Identifier: 10.1215/00127094-0000013X

Primary: 58J40
Secondary: 53C08 , 53D22

Keywords: automorphisms of pseudodifferential operators , central extension , derivations of pseudodifferential operators , Dixmier–Douady invariant , Fourier integral operators , gerbes , pseudodifferential algebra bundles , regularized trace , residue trace , twisted (fiber) cosphere bundles

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 10 • 15 July 2017
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