Branes at Generalized Conifolds and Toric Geometry

TL;DR

This paper leverages toric geometry to connect D3-branes at generalized conifold singularities with Type IIA configurations of D4-branes between rotated NS5 and NS5' branes. It shows the Higgs branch reproduces the generalized conifold and that FI-terms realize geometric resolutions, analyzing explicit examples such as $xy=z^{2} w^{2}$ and $xy=z^{3} w^{3}$ and exploring Seiberg duality via brane interchange. A key finding is a precise mapping of FI-parameters under duality with matching singularity sizes, though no toric signature of flop transitions is observed, questioning certain flop-based conjectures. Overall, the work demonstrates the utility of toric methods for elucidating brane-singularity correspondences and lays groundwork for studying more complex generalized conifolds.

Abstract

We use toric geometry to investigate the recently proposed relation between a set of D3 branes at a generalized conifold singularity and type IIA configurations of D4 branes stretched between a number of relatively rotated NS5 branes. In particular we investigate how various resolutions of the singularity corresponds to moving the NS branes and how Seiberg's duality is realized when two relatively rotated NS-branes are interchanged.

Figures (4)

• Figure 1: The brane configuration
• Figure 2: The singularity
• Figure 3: A more complicated example
• Figure 4: The dual configuration