Quiver theories, soliton spectra and Picard-Lefschetz transformations

TL;DR

The paper develops a mirror-symmetric framework in which 4d ${ m N}=1$ quiver gauge theories from D3-branes at Calabi–Yau singularities are mapped to D6-branes on 3-cycles in the Type IIA mirror. Orbifold quivers correspond to soliton spectra of ${ m N}=(2,2)$ theories with weighted projective targets, while local del Pezzo cases yield quivers connected by Picard–Lefschetz monodromy, reproducing Seiberg duality and revealing fractional dualities. The authors derive Diophantine equations from mirror geometry and Ramond-charge constraints that classify quivers related by PL moves, and show how eight independent pieces of data (cycles, monodromy, and exceptional collections) encode both matter content and, partially, superpotential structure. They further illustrate how global symmetries constrain superpotentials and relate Seiberg-dual phases to Markov-type Hurwitz equations, highlighting deep connections between geometry, number theory, and gauge dualities with potential for broad extensions to non-toric and more complex singularities.

Abstract

Quiver theories arising on D3-branes at orbifold and del Pezzo singularities are studied using mirror symmetry. We show that the quivers for the orbifold theories are given by the soliton spectrum of massive 2d N=2 theory with weighted projective spaces as target. For the theories obtained from the del Pezzo singularities we show that the geometry of the mirror manifold gives quiver theories related to each other by Picard-Lefschetz transformations, a subset of which are simple Seiberg duals. We also address how one indeed derives Seiberg duality on the matter content from such geometrical transitions and how one could go beyond and obtain certain ``fractional Seiberg duals.'' Moreover, from the mirror geometry for the del Pezzos arise certain Diophantine equations which classify all quivers related by Picard-Lefschetz. Some of these Diophantine equations can also be obtained from the classification results of Cecotti-Vafa for the 2d N=2 theories.

Figures (11)

• Figure 1: Toric diagram of $\mathbb{P}^{2}_{[1,a,b]}$.
• Figure 2: The web diagram of the resolution of $\mathbb{C}^{3}/\mathbb{Z}_{5}$.
• Figure 3: Three cycles mirror to the fractional branes in the resolution of $\mathbb{C}^{3}/\mathbb{Z}_{N}$.
• Figure 4: The $i$-th node of the quiver diagram for $\mathbb{C}^3/\mathbb{Z}_N$. We have marked all the nodes linked to $i$.
• Figure 5: The quiver diagram of $\mathbb{C}^{2}/\mathbb{Z}_{5}$. Here we use $[1]$ to denote the rank of that node is 1.