Quiver Theories from D6-branes via Mirror Symmetry

TL;DR

This work shows how local mirror symmetry converts D3-branes at del Pezzo singularities into D6-branes on a degenerate $T^{3}$ in the mirror, allowing the quiver gauge theory data to be read from the intersection of 3-cycles $S_i$. By linking affine ${\cal E}_N$ backgrounds and 5-brane webs to the mirror geometry, the authors derive gauge groups, quivers, and RR charges of fractional branes for both toric and non-toric local del Pezzo geometries, including the conifold as a symmetric example. The adjacency matrices arise from 3-cycle intersections (antisymmetric in toric cases; symmetric in the conifold case), and the fractional-brane charges are computed via a precise mirror map, with the sum of brane charges closing to a 0-cycle. The results provide a unified geometric framework, via Picard-Lefschetz monodromy and exceptional collections, to obtain quivers and brane charges across broad local Calabi–Yau geometries, extending from toric to non-toric del Pezzos.

Abstract

We study N=1 four dimensional quiver theories arising on the worldvolume of D3-branes at del Pezzo singularities of Calabi-Yau threefolds. We argue that under local mirror symmetry D3-branes become D6-branes wrapped on a three torus in the mirror manifold. The type IIB (p,q) 5-brane web description of the local del Pezzo, being closely related to the geometry of its mirror manifold, encodes the geometry of 3-cycles and is used to obtain gauge groups, quiver diagrams and the charges of the fractional branes.

Figures (23)

• Figure 1: The cycle $\Delta$ formed by a closed loop in the base and a 1-cycle in the fiber.
• Figure 2: The 3-cycle with topology of $S^{3}$.
• Figure 3: 3-cycles in the mirror of the conifold.
• Figure 4: Quiver diagram of the gauge theory on the D3-brane transverse to a conifold singularity.
• Figure 5: Toric diagrams for the del Pezzo surfaces. a) $\hbox{I}\!\hbox{P}^{2}$, b) $\hbox{I}\!\hbox{P}^{2}$ blown up at one point, c) $\hbox{I}\!\hbox{P}^{2}$ blown up at two points, d) $\hbox{I}\!\hbox{P}^{2}$ blown up at three points, e) $\hbox{I}\!\hbox{P}^{1}\times \hbox{I}\!\hbox{P}^{1}$.