Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Families of Toeplitz operators, family index and deformation quantization

Clément Cren, Erfan Rezaei

TL;DR

This work develops a comprehensive framework for smooth families of Szegö projections on the fibers of a contact fibration and the associated Toeplitz operator families, enabling deformation quantization of prequantizable symplectic fibrations in the family setting. By extending the Heisenberg calculus to fibered contexts and leveraging Connes–Thom and clutching techniques in bivariant K-theory, the authors derive a family index theorem for Rockland operators and connect it to a fiberwise star-product via Toeplitz calculus. The approach unifies analytic deformation quantization with a robust index theory for families on contact fibrations, providing explicit formulas for the index bundle and its Chern character in terms of principal symbols. The results yield a KK-theoretic formulation of the Toeplitz index problem in families and offer a principled pathway to quantize symplectic fibrations while tracking the associated topological invariants.

Abstract

Given a contact fibration, we construct smooth families of 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. 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.

Families of Toeplitz operators, family index and deformation quantization

TL;DR

This work develops a comprehensive framework for smooth families of Szegö projections on the fibers of a contact fibration and the associated Toeplitz operator families, enabling deformation quantization of prequantizable symplectic fibrations in the family setting. By extending the Heisenberg calculus to fibered contexts and leveraging Connes–Thom and clutching techniques in bivariant K-theory, the authors derive a family index theorem for Rockland operators and connect it to a fiberwise star-product via Toeplitz calculus. The approach unifies analytic deformation quantization with a robust index theory for families on contact fibrations, providing explicit formulas for the index bundle and its Chern character in terms of principal symbols. The results yield a KK-theoretic formulation of the Toeplitz index problem in families and offer a principled pathway to quantize symplectic fibrations while tracking the associated topological invariants.

Abstract

Given a contact fibration, we construct smooth families of 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. 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.
Paper Structure (16 sections, 26 theorems, 118 equations)

This paper contains 16 sections, 26 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Contact and Symplectic Fibrations
  3. Heisenberg Calculus for Contact Fibrations
  4. Heisenberg calculus on a contact manifold
  5. Families of Heisenberg operators on contact fibrations
  6. Szegö Projections and Toeplitz Families
  7. Families of Szegö projections
  8. Families of Toeplitz operators
  9. Applications to Star-products and Deformation Quantization
  10. The Family Index Theorem
  11. Bivariant geometric K-homology
  12. Fundamental classes of a contact fibration
  13. The class of a family of Rockland symbols
  14. Choosing an operator
  15. Clutching constructions
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $E\to M$ be a complex vector bundle over a closed contact manifold and $f\in C^\infty(M,\mathrm{hom}(E))$, which is everywhere invertible. Let $S$ denote a Szegö projection, acting on sections of $E$. The Toeplitz operator is Fredholm with index

Theorems & Definitions (78)

  • Theorem 1.1: Boutet de Monvel BoutetdeMonvelIndex
  • Theorem 1.2: Guillemin Guillemin1995
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 68 more