NC Geometry and Discrete Torsion Fractional Branes:I

Considering the complex n-dimension Calabi-Yau homogeneous hyper-surfaces ${\cal H}_{n}$ and using algebraic geometry methods, we develop the crossed product algebra method, introduced by Berenstein et Leigh in hep-th/0105229, and build the non commutative (NC) geometries for orbifolds ${\cal O}={\cal H}_{n}/{\bf Z}_{n+2}^{n}$ with a discrete torsion matrix $t_{ab}=exp[{\frac{i2\pi}{n+2}}{(\eta_{ab}-\eta_{ba})}]$, $\eta_{ab} \in SL(n,{\bf Z})$. We show that the NC manifolds ${\cal O}^{(nc)}$ are given by the algebra of functions on the real $(2n+4)$ Fuzzy torus ${\cal T}^{2(n+2)}_{\beta_{ij}}$ with deformation parameters $\beta_{ij}=exp{\frac{i2\pi}{n+2}}{[(\eta^{-1}_{ab}-\eta^{-1}_{ba})} q_{i}^{a} q_{j}^{b}]$, $q_{i}^{a}$'s being Calabi-Yau charges of ${\bf Z}_{n+2}^{n}$. We develop graph rules to represent ${\cal O}^{(nc)}$ by quiver diagrams which become completely reducible at singularities. Generic points in these NC geometries are be represented by polygons with $(n+2)$ vertices linked by $(n+2)$ edges while singular ones are given by $(n+2)$ non connected loops. We study the various singular spaces of quintic orbifolds and analyze the varieties of fractional $D$ branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic ${\cal Q}^{(nc)}$ are derived with details and general results for complex $n$ dimension orbifolds with discrete torsion are presented.