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Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond

Michael Heins, Annika Moucha, Oliver Roth, Toshiyuki Sugawa

Abstract

We introduce and study invariant differential operators acting on the space $\mathcal{H}(Ω)$ of holomorphic functions on the complement ${Ω=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the "complexified unit circle" $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w =1\}$. We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for investigating conformally invariant differential operators on the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$, which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply the machinery to a problem in deformation quantization by deriving explicit formulas for the canonical Wick-type star products on $Ω$, the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$ in terms of such invariant differential operators. These formulas are given in form of factorial series which depend holomorphically on a complex deformation parameter $\hbar$ and lead to asymptotic expansions of the star products in powers of $\hbar$.

Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond

Abstract

We introduce and study invariant differential operators acting on the space of holomorphic functions on the complement of the "complexified unit circle" . We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for investigating conformally invariant differential operators on the unit disk and the Riemann sphere , which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply the machinery to a problem in deformation quantization by deriving explicit formulas for the canonical Wick-type star products on , the unit disk and the Riemann sphere in terms of such invariant differential operators. These formulas are given in form of factorial series which depend holomorphically on a complex deformation parameter and lead to asymptotic expansions of the star products in powers of .
Paper Structure (7 sections, 28 theorems, 155 equations, 2 figures)

This paper contains 7 sections, 28 theorems, 155 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Möbius--type automorphisms of $\Omega$
  3. Peschl--Minda Derivatives
  4. Pure Peschl--Minda Derivatives
  5. Comparison with invariant derivatives in one variable
  6. The Wick Star Product on $\mathcal{H}(\Omega)$
  7. Continuous module structures induced by the Wick Star Product

Key Result

Lemma 2.1

The group $\mathcal{M}$ is generated by the flip $\mathcal{F}$ and the mappings defined by with $(z,w) \in \Omega \cap \mathbb C^2$ and $\gamma \in \mathbb C^*$. The mappings eq:MoebiusGenerators generate the subgroup $\mathcal{M}^+$. More precisely, for every $T \in \mathcal{M}$ there exist $(z,w) \in \Omega \cap \mathbb C^2$ and $\gamma \in \mathbb C^*$ such that

Figures (2)

  • Figure 1: Schematic picture of the domain $\Omega$
  • Figure 2: Schematic picture of the domains $\Omega$ (left), $\Omega_+$ (center) and $\Omega_-$ (right) with points at infinity.

Theorems & Definitions (53)

  • Lemma 2.1
  • proof
  • Definition 3.1: Peschl--Minda derivatives
  • Lemma 3.2
  • Remark 1: Swap symmetry
  • Proposition 3.3: $D$ vs. $\tilde{D}$
  • proof
  • Proposition 3.4
  • proof : Second proof of Proposition \ref{['prop:DvsDtilde']} and proof of Proposition \ref{['rem:PMexplicit']}
  • Corollary 3.5
  • ...and 43 more