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

TL;DR

This paper develops invariant differential operators on complex domains like the disk and sphere, unifies their study, and applies this framework to derive explicit formulas for Wick star products in deformation quantization.

Contribution

It introduces a unified approach to invariant differential operators on complex domains and derives explicit factorial series formulas for Wick star products.

Findings

Explicit formulas for Wick star products in terms of invariant differential operators.

Recursion identities and coordinate change behaviors of the operators.

Asymptotic expansions of star products in powers of the deformation parameter ar.

Abstract

We introduce and study invariant differential operators acting on the space $\mathcal{H}(\Omega)$ of holomorphic functions on the complement ${\Omega=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ 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 $\Omega$, 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 $\hbar$.