Conifolds from D-branes

Abstract

In this note we study the resolution of conifold singularity by D-branes by considering compactification of D-branes on C 3 /(Z 2 × Z 2 ). The resulting vacuum moduli space of D-branes is a toric variety which turns out to be a resolved conifold, that is a nodal variety in C 4 . This has the implication that all the corresponding phases of Type-II string theory are geometrical and are accessible to the D-branes, since they are related by flops.In the aftermath of the second superstring revolution the role of D-branes in the study of space-time has been widely appreciated. It has become quite apparent that one can probe sub-stringy scales in spacetime using D-branes -the BPS states that carry Rammond-Rammond charges in string theory. D-branes have recently been used to understand the short-distance geometry of Calabi-Yau manifolds [1][2][3]. These studies discern rather directly, as compared to the studies with fundamental strings, the phase structure of Calabi-Yau manifolds [1,3]. It has been shown that the topological and geometric properties of Calabi-Yau manifolds studied using both D-branes and fundamental strings corroborate. More specifically, it has been shown [3] that the D0-branes do not allow for non-geometric phases to appear in the theory which is in keeping with the conclusion drawn earlier from the analysis of M-theory and F-theory [4].The analysis of [3] was for C 3 /Γ, where Γ is a discrete Abelian group in SU (3). Specific examples of Γ = Z 3 and Γ = Z 5 were worked out, which could be generalized for any Z n . It was concluded that the Dbranes probe resolved C 3 /Z n singularities. In this scheme one starts with a set of D-branes on C 3 , arranged according to the regular representation of Γ, followed by a truncation to the Γ-invariant sector of the action, thus ending up with a gauged sigma-model whose configuration space, in the low-energy limit, is interpreted as the sub-stringy space-time. This picture has a mathematical counterpart in terms of blowing up a singular Kähler quotient [5,6] in an orbifold, viewed as a toric variety.In this note we shall study another example of this construction, namely a blown-up C 3 /(Z 2 ×Z 2 ), following the lines of [3]. We shall find that, in this case, the D-brane probe senses a double-point singularity, which gets resolved for a generic value of the resolution parameter; and under a smooth variation of the resolution parameter one moves on from one resolution to another, which corresponds to a flop [7,8].

D-brane gauge theories from toric singularities of the form and

“…Hence, the fields that are dynamical are given by the set [p 1 , p 2 , p 3 , p 6 ] while the rest acquire non-zero vev and hence are resolved, and in this case the reduced charge matrix is obtained from (20) to be…”

D-branes on fourfolds with discrete torsion

“…D-branes on the three-dimensional orbifold C 3 /(Z 2 × Z 2 ) have been studied in the absence of discrete torsion [9][10][11], and in its presence as well [4,5,12]. In the absence of discrete torsion, the moduli space of a D-brane on C 3 /(Z 2 × Z 2 ) is a blown down conifold.…”

“…In the absence of discrete torsion, the moduli space of a D-brane on C 3 /(Z 2 × Z 2 ) is a blown down conifold. Adding Fayet-Iliopoulos terms in 2 the gauge theory corresponds to partial [11] or complete [9,10] resolution of the singularity, depending on the non-vanishing combinations of the Fayet-Iliopoulos parameters. However, the scenario is rather different in the presence of discrete torsion [4,5].…”

D-branes on three-dimensional nonabelian orbifolds

“…Abelian orbifolds (Γ = Z n , Z n × Z n ) were investigated in [3,4,5,6]. The phase structure of the Kahler muduli space is investigated and it is shown that only geometric phases appear by using toric methods.…”