The unwarped, resolved, deformed conifold: fivebranes and the baryonic branch of the Klebanov-Strassler theory

TL;DR

The paper presents a gravity solution for NS-5 branes wrapped on the $S^2$ of the resolved conifold, which continuously interpolates to the deformed conifold with flux, thereby realizing a geometric transition purely within supergravity. Through a chain of dualities, D3-brane charge is generated, connecting this setup to the Klebanov-Strassler baryonic branch and its associated physics. The solution is a torsional, non-Kähler SU(3)-structure background, smooth and flux-supported, offering a concrete holographic realization of the baryonic branch and a vivid bridge between field-theory vacua and gravity. Far along the baryonic branch, a fuzzy two-sphere emerges on the field theory side, corresponding to a fivebrane wrapping the resolved conifold's $S^2$ in the gravity dual, with quantitative matching of spectra, charges, and noncommutativity scales. These results illuminate how geometric transitions, non-Kähler geometry, and holography intertwine in confining quiver theories and suggest broader applicability to similar dualities and non-perturbative phenomena.

Abstract

We study a gravity solution corresponding to fivebranes wrapped on the $S^2$ of the resolved conifold. By changing a parameter the solution continuously interpolates between the deformed conifold with flux and the resolved conifold with branes. Therefore, it displays a geometric transition, purely in the supergravity context. The solution is a simple example of torsional geometry and may be thought of as a non-Kähler analog of the conifold. By U-duality transformations we can add D3 brane charge and recover the solution in the form originally derived by Butti et al. This describes the baryonic branch of the Klebanov-Strassler theory. Far along the baryonic branch the field theory gives rise to a fuzzy two-sphere. This corresponds to the D5 branes wrapping the two-sphere of the resolved conifold in the gravity solution.

Figures (7)

• Figure 1: (a) A picture of the deformed conifold. The $S^2$ shrinks but the $S^3$ does not. (b) The resolved conifold, the $S^2$ does not shrink but the $S^3$ shrinks. In (c) we add five branes wrapping the $S^2$ of the resolved conifold of picture (b). (d) Backreacted geometry. The branes are replaced by geometry and fluxes. The end result is a geometry topologically similar to that of the deformed conifold with flux on the $S^3$. The near brane region is the CV-MN solution volkovonevolkovtwoMN.
• Figure 2: Plots of the solutions for some values of $\gamma^2$. On the left hand side: plots of $c'$ for $\gamma^2=1.01,1.02,1.06,1.25,2,4$. The bottom constant one is the CV-MN value, $\gamma^2=1$. On the right hand side: plot of $e^{2(\phi-\phi_\infty)}$ for $\gamma^2=1.01,1.02,1.06,1.25,2$. The CV-MN profile is not plotted since the dilaton does not asymptote to a constant.
• Figure 3: (a) The moduli space of the conifold with no flux or branes has two branches, denoted here by the vertical and horizontal axes. One is a deformation and the other is the resolution, which has two sides differing by flop transition. When we add flux we have a one parameter family that interpolates continuously between a deformed conifold with flux in region $D$ and a resolved conifold with branes in region $R$. A $\mathbb{Z}_2$ symmetry relates this to another branch that joins the deformed conifold with the flopped resolved conifold. (b) The solution looks like the deformed conifold with flux in region $D$ of (a). (c) In region $R$ of (a) the solution looks like the resolved conifold with some branes, where the branes have been replaced by their near brane geometry. In all cases the topology (but not the geometry) is that of the deformed conifold.
• Figure 4: (a) A D4 brane stretched between two orthogonal NS fivebranes. In (b) we compactify a direction orthogonal to all the branes in (a). In the limit that the size of the 8th circle goes to zero we expect to recover the conifold. In (c) and (d) we have schematically represented the effects of brane bending. The transverse position of branes varies logarithmically. This has the same origin as the dependence of the parameter $\alpha_{eff}^2$ on the radial position.
• Figure 5: (a) IIA configuration with a D4 brane stretching between two orthogonal NS fivebranes. (b) We separate the fivebranes in the compact 8th dimension. In (c) we add the effects of brane bending, the NS fivebranes bend in the non-compact 9th direction. (d) When we include the effects of brane bending the branes now stretch along a slanted direction parametrized by an angle $\delta$.