The twistor space of ${\mathbb{R}}^{4n}$ and Berezin-Toeplitz operators

TL;DR

This paper constructs a specific quantization for the hyperk"ahler manifold ^{4n} by removing a point from the twistor sphere and provides semiclassical asymptotics for this Berezin-Toeplitz type quantization.

Contribution

It introduces a novel quantization method for ^{4n} based on the twistor space, extending Berezin-Toeplitz quantization to a punctured sphere setting.

Findings

Constructed a quantization for ^{4n} using twistor space

Derived semiclassical asymptotics for the quantization

Replaced the family of Berezin-Toeplitz quantizations with a single one

Abstract

A hyperk\"ahler manifold $M$ has a family of induced complex structures indexed by a two-dimensional sphere $S^2 \cong \mathbb{CP}^1$. The twistor space of $M$ is a complex manifold $Tw(M)$ together with a natural holomorphic projection $Tw(M) \to \mathbb{CP}^1$, whose fiber over each point of $\mathbb{CP}^1$ is a copy of $M$ with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space),and for the case of $M = \mathbb{R}^{4n}$, $n\in\mathbb{N}$, equipped with the standard hyperk\"ahler structure, we construct one quantization that replaces the family of Berezin-Toeplitz quantizations parametrized by $S^2-\{ pt\}$. We provide semiclassical asymptotics for this quantization.