Some questions about the index of quantized contact transformations

TL;DR

This paper proposes a generalized index formula for quantized contact transformations, extending classical results and conjectures to broader geometric contexts involving CR structures and symplectic fillings.

Contribution

It introduces a new index formula that generalizes the Atiyah-Singer index theorem for contact transformations with CR structures and symplectic fillings.

Findings

Proposes a unified index formula for contact transformations.

Extends the Atiyah-Singer index theorem to new geometric settings.

Provides a conjectured index formula for Fourier integral operators.

Abstract

An index formula is proposed for contact transformations between contact manifolds equipped with CR structures or with fillings by symplectic manifolds. The formula generalizes the Atiyah-Singer formula and gives a conjectured formula for the index of Fourier integral operators, as well as Epstein's relative index for CR structures.