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Quivers, Tilings, Branes and Rhombi

Amihay Hanany, David Vegh

TL;DR

The paper introduces the Fast Inverse Algorithm to derive brane tilings, quivers, and superpotentials directly from toric diagrams of toric Calabi–Yau 3-fold singularities, enabling AdS/CFT duals without requiring explicit metrics. It develops a geometric framework where R-charges map to angles in isoradial brane tilings via a rhombus lattice and uses rhombus loops/zig-zag paths to connect tiling geometry with toric data, including reading off $(p,q)$-legs. Through concrete examples (C^3, conifold, L^{131}, L^{152}), it demonstrates how the tilings reproduce known quivers and superpotentials and how Kasteleyn methods validate toric data, while discussing toric vs Seiberg dualities via loop-moves. The work outlines broader implications and future directions, such as integrating with integrable structures, canonical toric phases, and higher-genus generalizations, highlighting rich intersections between geometry, combinatorics, and gauge theory.

Abstract

We describe a simple algorithm that computes the recently discovered brane tilings for a given generic toric singular Calabi-Yau threefold. This therefore gives AdS/CFT dual quiver gauge theories for D3-branes probing the given non-compact manifold. The algorithm solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. We study the parameter space of a-maximization; this study is made possible by identifying the R-charges of bifundamental fields as angles in the brane tiling. We also study Seiberg duality from a new perspective.

Quivers, Tilings, Branes and Rhombi

TL;DR

The paper introduces the Fast Inverse Algorithm to derive brane tilings, quivers, and superpotentials directly from toric diagrams of toric Calabi–Yau 3-fold singularities, enabling AdS/CFT duals without requiring explicit metrics. It develops a geometric framework where R-charges map to angles in isoradial brane tilings via a rhombus lattice and uses rhombus loops/zig-zag paths to connect tiling geometry with toric data, including reading off -legs. Through concrete examples (C^3, conifold, L^{131}, L^{152}), it demonstrates how the tilings reproduce known quivers and superpotentials and how Kasteleyn methods validate toric data, while discussing toric vs Seiberg dualities via loop-moves. The work outlines broader implications and future directions, such as integrating with integrable structures, canonical toric phases, and higher-genus generalizations, highlighting rich intersections between geometry, combinatorics, and gauge theory.

Abstract

We describe a simple algorithm that computes the recently discovered brane tilings for a given generic toric singular Calabi-Yau threefold. This therefore gives AdS/CFT dual quiver gauge theories for D3-branes probing the given non-compact manifold. The algorithm solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. We study the parameter space of a-maximization; this study is made possible by identifying the R-charges of bifundamental fields as angles in the brane tiling. We also study Seiberg duality from a new perspective.

Paper Structure

This paper contains 17 sections, 22 equations, 40 figures.

Table of Contents

  1. Introduction
  2. Brane tilings and quivers
  3. The basics
  4. Superconformal fixed point and R--charges
  5. Isoradial embeddings and R-charges
  6. Rhombus loops and zig--zag paths
  7. Inconsistent theories
  8. Conjecture of $(p,q)$--legs and rhombus loops
  9. Parameter space of a--maximization
  10. Fast Inverse Algorithm
  11. $\mathbb{C}^3$ ($\mathcal{N}=4$)
  12. Conifold
  13. $L^{131}$
  14. $L^{152}$
  15. Toric duality and Seiberg duality
  16. ...and 2 more sections

Figures (40)

  • Figure 1: The logical flowchart.
  • Figure 2: The $\bf{dP}_0$ periodic quiver. The nodes denote $U(N)$ gauge groups, the directed edges between them are bifundamental fields. The plaquettes of the quiver graph are terms in the superpotential. This example has three gauge groups, they are labelled by numbers. If we identify the nodes with the same numbers (i. e. we "compactify" the periodic quiver), then we arrive at the usual quiver diagram.
  • Figure 3: $\bf{dP}_0$ brane tiling & quiver. The unit cell of the lattice is shown in red. The theory has three gauge groups (faces in the tiling) and six cubic terms in the superpotential (valence three nodes of the tiling).
  • Figure 4: Brane tiling & quiver for the conifold. The green arrows indicate the directions of the bifundamental arrows in the quiver
  • Figure 5: Isoradially embedded part of an arbitrary brane tiling (in green).
  • ...and 35 more figures