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Symmetry in Deformation quantization and Geometric quantization

Naichung Conan Leung, Qin Li, Ziming Nikolas Ma

Abstract

In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form $f = f_0 - \frac{\hbar}{4π}(Δf_0 + c)$ for a certain smooth (non-formal) function $f_0$. If $f_0$ is real-valued then $f_0$ corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian $G$-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.

Symmetry in Deformation quantization and Geometric quantization

Abstract

In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form for a certain smooth (non-formal) function . If is real-valued then corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian -symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.

Paper Structure

This paper contains 15 sections, 22 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Fedosov quantization
  3. Fedosov deformation quantization on Kähler manifolds
  4. Quantizable functions
  5. Hilbert spaces and differential operators via Fedosov quantization
  6. Degree $1$ quantizable functions in Berezin-Toeplitz quantization
  7. Degree $1$ formal quantizable functions
  8. Surjectivity of the evaluation map
  9. Quantum Hamiltonian functions as degree $1$ quantizable functions
  10. Quantum Hamiltonian
  11. Which degree $1$ quantizable functions arise from symmetry
  12. Quantum symmetry in deformation and geometric quantization
  13. Quantum symmetry on pre-quantizable Kähler manifolds
  14. Quantum moment map acts as symmetries for $\mathcal{H}_k$
  15. Symmetry in Berezin-Toeplitz geometric quantization

Key Result

Theorem 1.1

For the Berezin-Toeplitz deformation quantization whose Karabegov form is given by $K_{\text{BT}} = -\frac{1}{\hbar} \cdot \omega + \text{Ric}_X$, a degree 1 formal quantizable function $f$ is exactly of the form where $c$ is a constant and $f_0$ is a smooth function such that the $(1,0)$-component $V_{f_0}^{1,0}$ of its Hamiltonian vector field $V_{f_0}$ is holomorphic. Evaluating at $\hbar = \

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1: Theorems 2.17 and 2.25 in CLL
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 3.1
  • ...and 35 more