Beyond fuzzy spheres

TL;DR

This paper explores polynomial deformations of the fuzzy sphere, deriving the Higgs algebra through quantization, and develops tools like the star product to facilitate field theory formulations on these non-commutative geometries.

Contribution

It introduces a method to derive the Higgs algebra from Poisson structures and quantizes singular surfaces using coherent states, advancing fuzzy space models.

Findings

Derived the Higgs algebra from Poisson structures.

Quantized singular surfaces with coherent states.

Constructed the star product for the nonlinear algebra.

Abstract

We study polynomial deformations of the fuzzy sphere, specifically given by the cubic or the Higgs algebra. We derive the Higgs algebra by quantizing the Poisson structure on a surface in $\mathbb{R}^3$. We find that several surfaces, differing by constants, are described by the Higgs algebra at the fuzzy level. Some of these surfaces have a singularity and we overcome this by quantizing this manifold using coherent states for this nonlinear algebra. This is seen in the measure constructed from these coherent states. We also find the star product for this non-commutative algebra as a first step in constructing field theories on such fuzzy spaces.