NC Calabi-Yau Orbifolds in Toric Varieties with Discrete Torsion

Abstract

Using the algebraic geometric approach of Berenstein et {\it al} (hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with discrete torsion. We first develop a new way of getting complex $d$ mirror Calabi-Yau hypersurfaces $H_Δ^{\ast d}$ in toric manifolds $M_{Δ}^{\ast (d+1)}$ with a $C^{\ast r}$ action and analyze the general group of the discrete isometries of $H_Δ^{\ast d}$. Then we build a general class of $d$ complex dimension NC mirror Calabi-Yau orbifolds where the non commutativity parameters $θ_{μν}$ are solved in terms of discrete torsion and toric geometry data of $M_Δ^{(d+1)}$ in which the original Calabi-Yau hypersurfaces is embedded. Next we work out a generalization of the NC algebra for generic $d$ dimensions NC Calabi-Yau manifolds and give various representations depending on different choices of the Calabi-Yau toric geometry data. We also study fractional D-branes at orbifold points. We refine and extend the result for NC $% (T^{2}\times T^{2}\times T^{2})/(\mathbf{Z_{2}\times {Z_{2})}}$ to higher dimensional torii orbifolds in terms of Clifford algebra.