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

Abstract

Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, $R_{\theta }^2\times R_{\theta }^2$, we obtain a star-product for the noncommutative (fuzzy) $R_{\lambda }^3$ 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 $R_{\lambda }^3$ 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.

25 pages, Latex file, no figures, v2: minor changes, typos corrected and some references added, v3: Final version to appear in PRD