Coherent State Induced Star-Product on $R^3_λ$ and the Fuzzy Sphere

Abstract

Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, $R^2_θ\times R^2_θ$, we obtain a star-product for the noncommutative (fuzzy) $R^3_λ$ defined by $[x^i,x^j]=iλε_{ijk}x^k$. Furthermore, we show that there is a projection function which allows us to reduce the functions on $R^3_λ$ 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 $R_{theta}\times R_θ$ space.