D-string fluid in conifold: II. Matrix model for D-droplets on S³ and S²

Motivated by similarities between Fractional Quantum Hall (FQH) systems and aspects of topological string theory on conifold, we continue in the present paper our previous study (hep-th/0604001, hep-th/0601020) concerning FQH droplets on conifold. Here we focus our attention on the conifold sub-varieties $\mathbb{S}^{3}$\textbf{\}and\textbf{\}$\mathbb{S}^{2}$ and study the non commutative quantum dynamics of D1 branes wrapped on a circle. We give a matrix model proposal for FQH droplets of $N$ point like particles on $\mathbb{S}^{3}$\textbf{\}and\textbf{\}$\mathbb{S}^{2}$ with filling fraction $\nu =\frac{1}{k}$. We show that the ground state of droplets on $% \mathbb{S}^{3}$ carries an isospin $j=k\frac{N(N-1)}{2}$ and gives remarkably rise to $2j+1$ droplets on $\mathbb{S}^{2}$ with Cartan-Weyl charge $| j_{z}| \leq j$.