Conifolds and geometric transitions

Conifold geometries have recieved much attention in string theory and string-inspired cosmology recently, in particular the Klebanov-Strassler background that is known as the “warped throat.” This paper provides a pedagogical explanation for the singularity resolution in this geometry and emphasizes its connection to geometric transitions. The first part focuses on the gauge theory dual to the Klebanov-Strassler background, including the T-dual intersecting branes description. Then, a connection to the Gopakumar-Vafa conjecture for open-closed string duality is presented and a series of papers verifying this model on the supergravity level is summarized. An appendix provides extensive background material about conifold geometries. Special attention is given to their complex structures and the supersymmetry conditions on the background flux in constructions with fractional D3-branes on the singular (Klebanov-Tseytlin) and resolved (Pando Zayas–Tseytlin) conifolds are reevaluated. In agreement with earlier results, it is shown that only the singular solution allows a supersymmetric flux. However, the importance of using the correct complex structure to reach this conclusion is emphasized.

Figure 1

The Klebanov-Strassler model.

Figure 2

Vafa’s model.

Figure 3

The conifold transition. All three geometries share the base $S^2\times S^3$, but the $S^3$ of the deformed conifold (left) remains finite. In the conifold transition it can be shrunk to zero size to give the singular conifold (center), from which blowing up the $S^2$ gives the resolved conifold (right).

Figure 4

Comparing the Klebanov-Witten and Klebanov-Strassler setups.

Figure 5

Vafa’s duality chain. Following the arrows through a series of mirror symmetries and a flop transition, we can verify the geometric transition as conjectured for IIA and IIB.

Figure 6

The IIA brane configuration dual to the Klebanov-Witten setup.

Figure 7

The IIA brane configuration dual to the Klebanov-Strassler setup.

Figure 8

A Seiberg duality transformation in the dual brane picture. Note that here $N=1$ and $M=2$ so in the final picture two of the branes are actually antibranes (denoted by dashed lines).

Figure 9

The IIA brane configuration dual to the Vafa setup.

Figure 10

The modified duality. The backgrounds in IIA have to be replaced by non-Kähler versions of the deformed and resolved conifolds and the $\mathcal{M}$-theory lift does not possess $G_2$ holonomy anymore.

Figure 11

The heterotic duality chain. Following the arrows one can construct non-Kähler backgrounds in type I and heterotic theory that are dual to the type IIB backgrounds before and after transition. This implies that the new backgrounds are also transition duals in some sense.