Symbolic calculus for Toeplitz operators with half-forms

TL;DR

This paper develops a symbolic calculus framework for Berezin-Toeplitz operators on Kähler manifolds using half-form bundles, connecting Bohr-Sommerfeld conditions, trace formulas, and deformation quantization.

Contribution

It introduces a symbolic calculus for Toeplitz operators with half-forms and relates it to Bohr-Sommerfeld conditions and characteristic classes in deformation quantization.

Findings

Established Bohr-Sommerfeld conditions for Toeplitz operators

Derived trace formulas relating to Toeplitz operators

Developed symbolic calculus for Lagrangian sections with subprincipal estimates

Abstract

This paper is devoted to the use of half-form bundles in the symbolic calculus of Berezin-Toeplitz operators on Kahler manifolds. We state the Bohr-Sommerfeld conditions and relate them to the functional calculus of Toeplitz operators, a trace formula and the characteristic classes in deformation quantization. We also develop the symbolic calculus of Lagrangian sections, with the crucial estimate of the subprincipal terms.