Families of Toeplitz operators, family index and deformation quantization

Abstract

Given a contact fibration, we construct smooth families of formal Szegö projections on the fibers. This allows us to define smooth families of Toeplitz operators. We apply these operators to construct a deformation quantization of prequantizable symplectic fibrations, recovering a result of Kravchenko in an analytic way. This also leads to a comparison between various forms of prequantizations and their geometric implications. We also derive a family index for these families of Toeplitz operators. To this end, we generalize an index formula of Baum and van Erp to families.

Correction of false statement that continuous families of Szegö projections always existed. Szegö projections replaced by projections modulo smoothing. Added a whole new section comparing our notion of quantization with others from Poisson geometry. 56 pages, comments are welcome !