NC Calabi–Yau manifolds in toric varieties with NC torus fibration

Using the algebraic geometry method of Berenstein and Leigh (BL) (hep-th/0009209 and hep-th/0105229), and considering singular toric varieties Vd+1 with NC irrational torus fibration, we construct NC extensions Mdnc of complex d dimension Calabi–Yau (CY) manifolds embedded in Vd+1nc. We give realizations of the NC C∗r toric group, derive the constraint equations for NC Calabi–Yau (NCCY) manifolds Mncd embedded in Vd+1nc and work out solutions for their generators. We study fractional D branes at singularities and show that, due to the complete reducibility property of C∗r group representations, there is an infinite number of noncompact fractional branes at fixed points of the NC toric group.