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October, 2001 Local Formula for the Index of a Fourier Integral Operator
Eric Leichtnam, Ryszard Nest, Boris Tsygan
J. Differential Geom. 59(2): 269-300 (October, 2001). DOI: 10.4310/jdg/1090349429

Abstract

Let X and Y be two closed connected Riemannian manifolds of the same dimension and φ : S* XS* Y a contact diffeomorphism. We show that the index of an elliptic Fourier operator Φ associated with φ is given by ∫B*(X) eθ0 Â(T*X) − ∫B*(Y) eθ0 Â(T*Y) where θ0 is a certain characteristic class depending on the principal symbol of Φ and, B*(X) and B*(Y) are the unit ball bundles of the manifolds X and Y. The proof uses the algebraic index theorem of Nest-Tsygan for symplectic Lie Algebroids and an idea of Paul Bressler to express the index of Φ as a trace of 1 in an appropriate deformed algebra.

In the special case when X = Y we obtain a different proof of a theorem of Epstein-Melrose conjectured by Atiyah and Weinstein.

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Eric Leichtnam. Ryszard Nest. Boris Tsygan. "Local Formula for the Index of a Fourier Integral Operator." J. Differential Geom. 59 (2) 269 - 300, October, 2001. https://doi.org/10.4310/jdg/1090349429

Information

Published: October, 2001
First available in Project Euclid: 20 July 2004

zbMATH: 1030.58015
MathSciNet: MR1908824
Digital Object Identifier: 10.4310/jdg/1090349429

Rights: Copyright © 2001 Lehigh University

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Vol.59 • No. 2 • October, 2001
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