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

TL;DR

This paper constructs noncommutative Calabi-Yau manifolds within toric varieties using algebraic geometry methods, revealing an infinite spectrum of fractional branes at singularities due to noncommutative toric group representations.

Contribution

It introduces a method to realize noncommutative Calabi-Yau manifolds embedded in noncommutative toric varieties, extending previous algebraic geometry approaches.

Findings

Constructed NC extensions of Calabi-Yau manifolds in singular toric varieties.

Derived constraint equations and solutions for NC Calabi-Yau manifolds.

Found an infinite number of fractional branes at singularities due to group representation properties.

Abstract

Using the algebraic geometry method of Berenstein and Leigh (BL), hep-th/0009209 and hep-th/0105229), and considering singular toric varieties ${\cal V}_{d+1}$ with NC irrational torus fibration, we construct NC extensions ${\cal M}_{d}^{(nc)}$ of complex d dimension Calabi-Yau (CY) manifolds embedded in ${\cal V}_{d+1}^{(nc)}$. We give realizations of the NC $\mathbf{C}^{\ast r}$ toric group, derive the constraint eqs for NC Calabi-Yau (NCCY) manifolds ${\cal M}^{nc}_d$ embedded in ${\cal V}_{d+1}^{nc}$ 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.