The Fuzzy S^4 by Quantum Deformation

TL;DR

This paper explores the quantum deformation of the fuzzy algebra of S^4 by embedding it in a Kaehler coset space, constructing harmonic functions, and applying Fedosov formalism to generate a fuzzy algebra, with potential generalizations to higher dimensions.

Contribution

It introduces a novel approach to quantizing the algebra of S^4 using quantum deformation and Fedosov formalism within a Kaehler coset space framework.

Findings

Construction of harmonic functions on S^4 in complex coordinates

Generation of fuzzy algebra ___(S^4) via Fedosov formalism

Extension to higher even-dimensional fuzzy spheres

Abstract

The fuzzy algebra of S^4 is discussed by quantum deformation. To this end we embed the classical S^4 in the Kaehler coset space SO(5)/U(2). The harmonic functions of S^4 are constructed in terms of the complex coordinates of SO(5)/U(2). Being endowed with the symplectic structure they can be deformed by the Fedosov formalism. We show that they generate the fuzzy algebra \hat A_\infty (S^4) under the * product defined therein, by using the Darboux coordinate system. The fuzzy spheres of higher even dimensions can be discussed similarly. We give basic arguments for the generalization as well.