Generalized Coherent State Approach to Star Products and Applications to the Fuzzy Sphere

TL;DR

This paper introduces a generalized coherent state method to construct star products for arbitrary two-dimensional Poisson structures, enabling applications to fuzzy geometries like the sphere and torus.

Contribution

It develops a unified approach to derive star products for various fuzzy spaces using coherent states, including new formulations for the fuzzy sphere.

Findings

Constructed star product for arbitrary 2D Poisson structures

Recovered star products for fuzzy torus and sphere

Defined fuzzy stereographic projection and integration measure

Abstract

We construct a star product associated with an arbitrary two dimensional Poisson structure using generalized coherent states on the complex plane. From our approach one easily recovers the star product for the fuzzy torus, and also one for the fuzzy sphere. For the latter we need to define the `fuzzy' stereographic projection to the plane and the fuzzy sphere integration measure, which in the commutative limit reduce to the usual formulae for the sphere.