Toric varieties with NC toric actions: NC type IIA geometry

摘要

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 nu(d+1)((nc)). We construct NC complex d dimension Calabi-Yau manifolds embedded in 1,+l 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-Yan backgrounds M-d(nc) embedded in nc, and work out their solutions. The latters depend on the Calabi-Yau condition Sigma(i)q(i)(a) = 0, q(i)(a) being the charges of C*(r); but also on the toric data {q(i)(a), nu(i)(A); p(1)(alpha), nu(iA)*} of the polygons associated to nu(d+1). Moreover, we study fractional D-branes at singularities and show that, due to the complete reducibility property of C` 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. (C) 2003 Elsevier B.V. All rights reserved. [References: 44]