NC geometry and fractional branes

Considering complex n-dimension Calabi-Yau homogeneous hypersurfaces Hn with discrete torsion and using the Berenstein and Leigh algebraic geometry method, we study fractional D-branes that result from stringy resolution of singularities. We first develop the method introduced by Berenstein and Leigh (Preprint hep-th/0105229) and then build the non-commutative (NC) geometries for orbifolds O = Hn/Zn+2n with a discrete torsion matrix tab exp[i2π/n+2(ηab - ηba)], ηab element of SL(n, Z). We show that the NC manifolds O(nc) are given by the algebra of functions on the real (2n + 4) fuzzy torus Tβij2(n+2) with deformation parameters βij = exp i2π/n+2[(ηab-1 - ηba-1)qai qbj] with qai being charges of Znn+2. We also give graphic rules to represent O(nc) by quiver diagrams which become completely reducible at orbifold singularities. It is also shown that regular points in these NC geometries are 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 analyse the varieties of fractional D-branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic Q(nc) are derived with details and general results for complex n-dimension orbifolds with discrete torsion are presented