Toric varieties with NC toric actions: NC type IIA geometry Original Research Article

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 Vd+1(nc). We construct NC complex d dimension Calabi–Yau manifolds embedded in Vd+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 Mdnc embedded in Vd+1nc and work out their solutions. The latters depend on the Calabi–Yau condition ∑iqia=0, qia being the charges of C∗r; but also on the toric data {qia,νiA;pIα,νiA∗} of the polygons associated to Vd+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.