Toric Varieties with NC Toric Actions: NC Type IIA Geometry

Abstract

Extending the usual ${\mathbf{C}}^{\ast r}$ actions of toric manifolds by allowing asymmetries between the various ${\mathbf{C}}^{\ast }$ factors, we build a class of non commutative (NC) toric varieties $\mathcal{V}$. We construct NC complex $d$ dimension Calabi-Yau manifolds embedded in ${\mathcal{V}}_{d+1}^{(nc)}$ by using the algebraic geometry method. Realizations of NC ${\mathbf{C}}^{\ast r}$ toric group are given in presence and absence of quantum symmetries and for both cases of discrete or continuous spectrums. We also derive the constraint eqs for NC Calabi-Yau backgrounds ${\mathcal{M}}_d^{nc}$ embedded in ${\mathcal{V}}_{d+1}^{nc}$ and work out their solutions. The latters depend on the Calabi-Yau condition , $q_i^a$ being the charges of ${\mathbf{C}}^{\ast r}$% ; but also on the toric data $q_i^a,{\nu }_i^A;p_I^{\alpha },{\nu }_{iA}^{\ast }$ of the polygons associated to $\mathcal{V}$. Moreover, 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 fractional $D$ branes. We also give the generalized Berenstein and Leigh quiver diagrams for discrete and continuous ${\mathbf{C}}^{\ast r}$ representation spectrums. An illustrating example is presented.

25 pages, no figures

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Cited by in corpus (3)

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