Toric varieties with NC toric actions: NC type IIA geometry

Abstract

Extending the usual C ∗r actions of toric manifolds by allowing asymmetries between the various C ∗ factors, we build a class of non-commutative (NC) toric varieties V d+1 ( nc) . We construct NC complex d dimension Calabi–Yau manifolds embedded in V d+1 ( nc) by using the algebraic geometry method. Realizations of NC C ∗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 equations for NC Calabi–Yau backgrounds M d nc embedded in V d+1 nc and work out their solutions. The latters depend on the Calabi–Yau condition ∑ i q i a =0, q i a being the charges of C ∗r ; but also on the toric data {q i a,ν i A;p I α,ν iA ∗} of the polygons associated to V d+1 . Moreover, we study fractional D-branes at singularities and show that, due to the complete reducibility property of C ∗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 C ∗r representation spectrums. An illustrating example is presented.