Quantization of K\"ahler manifolds admitting $H$-projective mappings

TL;DR

This paper explores the geometric quantization of certain Kähler manifolds, including those with $H$-projective mappings, which serve as deformations of harmonic oscillators, expanding the understanding of quantization in curved spaces.

Contribution

It introduces a method to quantize Kähler manifolds with $H$-projective mappings, extending geometric quantization to more general curved spaces beyond constant holomorphic curvature.

Findings

Quantization of manifolds with constant holomorphic curvature achieved.

Extension of quantization methods to manifolds with $H$-projective mappings.

Framework for deforming harmonic oscillator systems on curved manifolds.

Abstract

We discuss the quantization of mechanical systems for which the Hamiltonian vector fields of observables form the deformation of $n$-dimensional oscilator algebra. Because of this fact these systems can be considered as "deformations" of the harmonic oscillator. The set of abovementioned mechanical systems are realized at the classical level in the form of K\"ahler manifolds of constant holomorphic curvature. Such mechanical systems are quantized later with the help of the geometric quantization approach. We also discuss the quantization of more general K\"ahler manifolds (not necessarily of constant holomorphic curvature) admitting $H$-projective mappings.