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Toeplitz Operators on Contact Manifolds and Equivariant K-homology

Alexander Gorokhovsky, Erik van Erp

Abstract

We present an equivariant generalization of Boutet de Monvel's index theorem for Toeplitz operators on contact manifolds. We prove that the Dirac operator and the Szegö projection determine the same class in equivariant $K$-homology, generalizing a theorem of Baum-Douglas-Taylor. We do not assume that the contact manifold is the boundary of a strictly pseudoconvex domain. The proof proceeds by a deformation linking the principal symbols of the classical and Heisenberg pseudodifferential calculi. At the level of symbols, the projection defining the Dirac class deforms to the principal Heisenberg symbol of the Szegö projection. This deformation implies equality of the corresponding classes in K-homology. This, in turn, gives an equivariant generalization of Boutet de Monvel's index formula for Toeplitz operators.

Toeplitz Operators on Contact Manifolds and Equivariant K-homology

Abstract

We present an equivariant generalization of Boutet de Monvel's index theorem for Toeplitz operators on contact manifolds. We prove that the Dirac operator and the Szegö projection determine the same class in equivariant -homology, generalizing a theorem of Baum-Douglas-Taylor. We do not assume that the contact manifold is the boundary of a strictly pseudoconvex domain. The proof proceeds by a deformation linking the principal symbols of the classical and Heisenberg pseudodifferential calculi. At the level of symbols, the projection defining the Dirac class deforms to the principal Heisenberg symbol of the Szegö projection. This deformation implies equality of the corresponding classes in K-homology. This, in turn, gives an equivariant generalization of Boutet de Monvel's index formula for Toeplitz operators.
Paper Structure (38 sections, 26 theorems, 203 equations)

This paper contains 38 sections, 26 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. The Heisenberg Group and the Weyl algebra
  3. Symplectic vector spaces
  4. Convolution on the Heisenberg group
  5. The Weyl algebra
  6. Weyl pseudodifferential calculus
  7. Equivariant Bott Periodicity as Deformation Quantization
  8. Deformations in $K$-theory
  9. Quantization
  10. Clifford multiplication
  11. The Bott generator
  12. The Bargmann-Fock representation
  13. The vacuum projection
  14. Quantization of the Bott element
  15. The equivariant Thom isomorphism
  16. ...and 23 more sections

Key Result

Theorem 1

Let $M$ be a compact contact manifold, and let $G$ be a compact Lie group that acts smoothly on $M$, preserving the contact form. Let $D$ be a $G$-equivariant Spin$^c$ Dirac operator on $M$, and let $S$ be a $G$-equivariant Szegö projection on $M$. Then in $G$-equivariant $K$-homology,

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Remark 7
  • Example 8
  • Example 9
  • Lemma 10
  • ...and 49 more