NC Geometry and Fractional Branes

TL;DR

This paper develops a framework for non-commutative geometries of Calabi-Yau orbifolds with discrete torsion, analyzing fractional D-branes and singularities using algebraic and quiver diagram methods.

Contribution

It introduces a method to construct NC geometries for Calabi-Yau orbifolds with discrete torsion and explores fractional branes and singularities in these spaces.

Findings

NC manifolds are described by fuzzy tori with specific deformation parameters.

Quiver diagrams represent the NC geometries and their reducibility at singularities.

Explicit solutions for the NC quintic and general results for complex orbifolds are provided.

Abstract

Considering complex $n$-dimension Calabi-Yau homogeneous hyper-surfaces $% \mathcal{H}_{n}$ with discrete torsion and using Berenstein and Leigh algebraic geometry method, we study Fractional D-branes that result from stringy resolution of singularities. We first develop the method introduced in hep-th/0105229 and then build the non commutative (NC) geometries for orbifolds $\mathcal{O}=\mathcal{H}_{n}/\mathbf{Z}_{n+2}^{n}$ with a discrete torsion matrix $t_{ab}=exp[{\frac{i2\pi}{n+2}}{(\eta_{ab}-\eta_{ba})}]$, $\eta_{ab} \in SL(n,\mathbf{Z})$. We show that the NC manifolds $% \mathcal{O}^{(nc)}$ are given by the algebra of functions on the real $% (2n+4) $ Fuzzy torus $\mathcal{T}_{\beta_{ij}}^{2(n+2)}$ with deformation parameters $\beta_{ij}=exp{\frac{i2\pi}{n+2}}{[(\eta_{ab}^{-1}-\eta_{ba}^{-1})} q_{i}^{a} q_{j}^{b}]$ with $q_{i}^{a}$'s being charges of $% \mathbf{Z}_{n+2}^{n}$. We also give graphic rules to represent $\mathcal{O}% ^{(nc)}$ 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 analyze the varieties of fractional $D$ branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic $\mathcal{Q}^{(nc)}$ are derived with details and general results for complex $n$ dimension orbifolds with discrete torsion are presented.