Local Formula for the Index of a Fourier Integral Operator

Abstract

Let X and Y be two closed connected Riemannian manifolds of the same dimension and φ : S^* X → S^* Y a contact diffeomorphism. We show that the index of an elliptic Fourier operator Φ associated with φ is given by ∫_B*(X) e^θ₀ Â(T^*X) − ∫_B*(Y) e^θ₀ 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.