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NC Geometry and Discrete Torsion Fractional Branes:I | | E.H Saidi
; | | Date: |
17 Feb 2002 | | Subject: | hep-th | | Abstract: | Considering the complex n-dimension Calabi-Yau homogeneous hyper-surfaces ${cal H}_{n}$ and using algebraic geometry methods, we develop the crossed product algebra method, introduced by Berenstein et Leigh in hep-th/0105229, and build the non commutative (NC) geometries for orbifolds ${cal O}={cal H}_{n}/{f Z}_{n+2}^{n}$ with a discrete torsion matrix $t_{ab}=exp[{frac{i2pi}{n+2}}{(eta_{ab}-eta_{ba})}]$, $eta_{ab} in SL(n,{f Z})$. We show that the NC manifolds ${cal O}^{(nc)}$ are given by the algebra of functions on the real $(2n+4)$ Fuzzy torus ${cal T}^{2(n+2)}_{eta_{ij}}$ with deformation parameters $eta_{ij}=exp{frac{i2pi}{n+2}}{[(eta^{-1}_{ab}-eta^{-1}_{ba})} q_{i}^{a} q_{j}^{b}]$, $q_{i}^{a}$’s being Calabi-Yau charges of ${f Z}_{n+2}^{n}$. We develop graph rules to represent ${cal O}^{(nc)}$ by quiver diagrams which become completely reducible at singularities. Generic points in these NC geometries are be 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 ${cal Q}^{(nc)}$ are derived with details and general results for complex $n$ dimension orbifolds with discrete torsion are presented. | | Source: | arXiv, hep-th/0202104 | | Services: | Forum | Review | PDF | Favorites | |
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