Quantization on the Cone and Cyon-Oscillator Duality

It is shown that the three–dimensional isotropic oscillator with coordinates belonging to the two–dimensional half-up cone is dual to the cyon , i.e. the planar particle–vortex bound system provided by fractional statistics. Our interest in the paper will be to consider the three– dimensional isotropic oscillator with coordinates belonging to a two– dimensional cone without the tip. We show that this system is dual to the cyon [1], i.e. a planar particle–vortex bound system with the Coulomb attractive interaction. This cyon is a simple prototype for the objects with fractional statistics [2], known also as anyons [3]. Anyons play an important role in the field theory [4], in the fractional quantum Hall effect [5], and in high– T c superconductivity [6].