Conifolds and geometric transitions

Abstract

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-Tseytlin͒ and resolved ͑Pando Zayas-Tseytlin͒ 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.

The evolution of conifolds

“…There also exists the deformed conifold solution, which uses another method to rectify the singularity of the singular conifold [5,22,6,25]. It involves expanding the singularity to the form of an S 3 .…”

“…We added momentum of the following forṁ 6) to initiate the dynamics, with the momenta for the A and B functions being determined by the constraint equations in the appendix. By choosing the C and D momenta to related we are making the choice of maintaining the form of the 3-sphere, at least initially.…”

“…One way to picture these transitions between manifolds is to study the cycles within them, for example it may be that certain cycles collapse to zero size on one side of the transition and expand as different cycles on the other. Transitions may change the Hodge numbers, such as a conifold transition [5,6] where an S 2 collapses and reappears as an S 3 or they may maintain the Hodge numbers but alter the linking numbers of the cycles, such as the flop transition [4]. Remarkably, string theory is able to make sense of these singular geometries by the appearance of new light states corresponding to D-branes wrapping the collapsing cycles [7].…”

Topics in gauge/gravity dualities

“…So we can begin with the KS (the deformed) solution at the bottom of the cascade, pass through the singular solution of the KW solution, to attain the PT (the resolved) solution by blowing up the S 2 . This is known as conifold transition [135]. The geometric transition between conifold geometries is an example of a string theory duality between compactifications on different geometrical backgrounds.…”

“…with 135) are known as the supersymmetry conditions. (3.133) and (3.134) allow to obtain the Killing spinors of the theory and then, the number of preserved supersymmetries in the background.…”