Chirality and Dirac Operator on Noncommutative Sphere

TL;DR

This paper constructs the Dirac operator on the noncommutative 2-sphere, revealing two spectral types and their algebraic implications, including a new restriction on the Planck constant in Berezin's quantization.

Contribution

It provides a novel derivation of the Dirac operator on the noncommutative sphere and explores its spectral properties within Connes' framework, introducing new quantization restrictions.

Findings

Two distinct spectra of the Dirac operator identified

Two classes of quantized algebras correspond to spectral types

New restriction on the Planck constant in Berezin's quantization

Abstract

We give a derivation of the Dirac operator on the noncommutative $2$-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed.