3D Oscillator and Coulomb Systems reduced from Kahler spaces

TL;DR

This paper explores the reduction of 4D Kahler oscillator and Coulomb systems to 3D systems with monopoles, identifying their geometric origins and constructing superintegrable models on curved spaces.

Contribution

It introduces a method to derive 3D oscillator and Coulomb systems with monopoles from 4D Kahler spaces, including superintegrable models on curved geometries.

Findings

Identified Kahler spaces with conic singularities as origins of 3D systems.

Constructed superintegrable oscillator models on curved spaces with monopoles.

Extended results to Kahler spaces with conic singularities.

Abstract

We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kahler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kahler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid are originated. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to monopole, and find their four-dimensional origins. In the latter case the metric of configuration space is non-Kahler one. Finally, we extend these results to the family of Kahler spaces with conic singularities.