Local formula for the index of a Fourier Integral Operator

Abstract

We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism $ϕ$ of cosphere bundles of two Riemannian manifolds X and Y is given by $\int_{B^*X}\hat{A}(T^*X)\expθ - \int_{B^*Y}\hat{A}(T^*Y)\expθ$. Here $B^*$ stands for the unit coball bundle and $θ$ is a certain characteristic class depending on the principal symbol of the Fourier integral operator. In the special case when X=Y we obtain a different proof of the theorem of Epstein and Melrose.