Fuzzy Sphere and Hyperbolic Space from Deformation Quantization

TL;DR

This paper constructs explicit noncommutative products on symmetric 2D spaces using Fedosov's deformation quantization, resulting in algebraic structures suitable for noncommutative field theories on curved spaces.

Contribution

It provides explicit * products for S^2 and H^2 spaces, connecting deformation quantization with noncommutative geometry on curved manifolds.

Findings

Derived su(2) algebra for S^2

Derived su(1,1) algebra for H^2

Applicable to noncommutative field theories on curved spaces

Abstract

We explicitly construct noncommutative * products on circularly symmetric two dimensional space by using the technique of Fedosov's deformation quantization. Especially, on constant curvature spaces i.e., S^2 and H^2, we get su(2) and su(1,1) algebra respectively. These are candidates of * products applicable to noncommutative field theories or noncommutative gauge theories on spaces with nontrivial symplectic structure.