Brane Configurations for Branes at Conifolds

TL;DR

The paper develops a T-dual description of D3 branes at non-orbifold conifold-like singularities in terms of type IIA configurations with rotated NS5 branes and D4 branes, enabling a concrete, brane-based realization of ${\mathcal N}=1$ quiver gauge theories. It shows how quotients ${\bf Z}_k$, ${\bf Z}_k\times{\bf Z}_l$, and isolated singularities yield chiral and non-chiral theories and how Kähler moduli transitions correspond to brane crossings and Seiberg dualities. The analysis provides explicit IIA brane pictures for partial resolutions, Higgs branches, and orientifold projections, tying these to brane-box models and toric diagrams. This framework connects geometric singularities, brane constructions, and four-dimensional gauge dynamics, with potential extensions to M-theory lifts and AdS/CFT contexts.

Abstract

We study the T duality between a set of type IIB D3 branes atnon-orbifold threefold singularities, and type IIA configurations of D4 branes stretched between relatively rotated NS fivebranes. The four-dimensional N=1 field theories on the D3 brane world-volume can be easily described using the IIA brane configuration. These models include families of chiral theories continuously connected to the theories appearing in brane box models (or D3 branes at orbifold singularities). We propose that phase transitions in the Kähler moduli space of the singularity are related to the crossing of rotated NS fivebranes in the T dual picture, and thus to Seiberg's duality in one of the gauge factors. We also comment on the inclusion of orientifold planes in the IIA brane picture.

Figures (7)

• Figure 1: The type IIA brane configuration T dual to the system of D3 branes at the conifold singularity. In this and the following pictures, we represent the NS brane as a continuous vertical line, and the NS$'$ brane as a discontinuous vertical line. Since they are oriented along different directions (45 vs. 89), the D4 branes cannot separate and there is no Coulomb branch. Nevertheless, for the sake of clarity, we have depicted the D4 branes as slightly separated. We also show the chiral fields arising from open strings (depicted as a discontinuous curve) stretching between the D4 branes.
• Figure 2: Brane configuration T dual to a system of D3 branes at the $xy=z^k w^k$ singularity. The model consists in a set of $k$ NS and $k$ NS$'$ branes ordered along $x^6$ in an alternating fashion. The figure depicts the brane configuration in the vicinity of the $i^{th}$ pair of NS, NS$'$ branes. We show the chiral fields in the bi-fundamental representations, and the superpotential quartic interactions mediated by the massive adjoints.
• Figure 3: Comparison between the possible orderings of two NS branes and two NS$'$ branes along $x^6$ and the triangulations of the diagram for the singularity $xy=z^2 w^2$. The NS branes are represented by crosses, and the NS$'$ branes as circles, sitting at points in a circle representing the $x^6$ direction. We also show one example of a transition, and the corresponding exchange of NS and NS$'$ branes in the T dual brane picture.
• Figure 4: Intuitive argument showing that two triangulations corresponding to identical brane configurations differ by a 'rotation' among their ${\rm I P}_1$'s. In the first step, we 'cut' the lower left triangle piece of the toric diagram and 'glue' it on the other side. This amounts to making compact the previously non-compact ${\rm I P}_1$, and making non-compact a previously compact one. The second step is a mere re-drawing of the same toric diagram.
• Figure 5: Brane configuration T dual to a system of D3 branes at the $xy=zw^2$ singularity.