Coherent State Induced Star-Product on $R^3_{\lambda}$ and the Fuzzy Sphere

TL;DR

This paper constructs a new star-product on noncommutative $R^3_{\lambda}$ and the fuzzy sphere using Hopf fibration, enabling analysis of field theories on these spaces derived from $R^2_{ heta} imes R^2_{ heta}$.

Contribution

It introduces a novel star-product on fuzzy $R^3_{\lambda}$ and the fuzzy sphere, derived from a four-dimensional noncommutative Moyal plane via Hopf fibration.

Findings

New star-product on fuzzy $R^3_{\lambda}$ and the fuzzy sphere.

Projection function linking functions on $R^3_{\lambda}$ to the fuzzy sphere.

Method to extract field theory information from noncommutative spaces.

Abstract

Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, $R^2_{\theta}\times R^2_{\theta}$, we obtain a star-product for the noncommutative (fuzzy) $R^3_{\lambda}$ defined by $[x^i,x^j]=i\lambda\epsilon_{ijk}x^k$. Furthermore, we show that there is a projection function which allows us to reduce the functions on $R^3_{\lambda}$ to that of the fuzzy sphere, and hence we introduce a new star-product on the fuzzy sphere. We will then briefly discuss how using our method one can extract information about the field theory on fuzzy sphere and $\rrlam$ from the corresponding field theories on $R_{theta}\times R_{\theta}$ space.