Families of Toeplitz operators, family index and deformation quantization

TL;DR

This paper develops a framework for families of Toeplitz operators on contact fibrations, enabling deformation quantization and a generalized family index, extending previous index formulas to new settings.

Contribution

It introduces a method to construct smooth families of Toeplitz operators on contact fibrations and generalizes an index formula to families, advancing the analytic approach to deformation quantization.

Findings

Constructed smooth families of Szeg"o projections and Toeplitz operators.

Achieved a deformation quantization of symplectic fibrations.

Generalized Baum and van Erp's index formula to families.

Abstract

Given a contact fibration, we construct smooth families of Szeg\"o 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. 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.