NC Calabi-Yau Manifolds in Toric Varieties with NC Torus fibration

Abstract

Using the algebraic geometry method of Berenstein and Leigh (BL), hep-th/0009209 and hep-th/0105229), and considering singular toric varieties ${\mathcal{V}}_{d+1}$ with NC irrational torus fibration, we construct NC extensions ${\mathcal{M}}_d^{(nc)}$ of complex d dimension Calabi-Yau (CY) manifolds embedded in ${\mathcal{V}}_{d+1}^{(nc)}$. We give realizations of the NC ${\mathbf{C}}^{\ast r}$ toric group, derive the constraint eqs for NC Calabi-Yau (NCCY) manifolds ${\mathcal{M}}_d^{nc}$ embedded in ${\mathcal{V}}_{d+1}^{nc}$ and work out solutions for their generators. We study fractional $D$ branes at singularities and show that, due to the complete reducibility property of ${\mathbf{C}}^{\ast r}$ group representations, there is an infinite number of non compact fractional branes at fixed points of the NC toric group.

12 pages, LaTex, no figure