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Brane Tilings

Kristian D. Kennaway

TL;DR

This work develops a unifying, combinatorial framework for the four-dimensional N=1 SCFTs arising from D3-branes at toric Calabi–Yau singularities by introducing brane tilings and dimer models. It shows these theories are completely determined by toric data via planar, bipartite tilings on a T^2 and connects them to mirror symmetry through D6-branes on the mirror Calabi–Yau, with the moduli space, BPS spectrum, and R-charges computable from dimer combinatorics and isoradial embeddings. Key contributions include a detailed dictionary between quivers, dimers, and toric geometry, a GLSM-based construction of the classical moduli space from dimers, and a mirror-symmetry picture that explains the emergence of the tiling structure and the superpotential from holomorphic disk instantons. The framework enables a systematic analysis of anomalous vs anomaly-free U(1) symmetries, BPS operators, and the R-symmetry via $a$-maximization, providing a powerful, metric-independent route to study AdS/CFT duals of toric geometries and guiding extensions to more general geometries and higher-dimensional dualities.

Abstract

We review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the world-volume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an ``orbifold point'' in Kaehler moduli space. These provide an infinite class of four-dimensional N=1 superconformal field theories which may be studied in the context of the AdS/CFT correspondence. It is now understood that these gauge theories are completely specified by certain two-dimensional torus graphs, called brane tilings, and the combinatorics of the dimer models on these graphs. In particular, knowledge of the dual Sasaki-Einstein metric is not required to determine the gauge theory, only topological and symplectic properties of the toric Calabi-Yau cone. By analyzing the symmetries of the toric quiver theories we derive the dimer models and use them to construct the moduli space of the theory both classically and semiclassically. Using mirror symmetry the brane tilings are shown to arise in string theory on the world-volumes of the fractional D6-branes that are mirror to the stack of D3-branes at the tip of the cone.

Brane Tilings

TL;DR

This work develops a unifying, combinatorial framework for the four-dimensional N=1 SCFTs arising from D3-branes at toric Calabi–Yau singularities by introducing brane tilings and dimer models. It shows these theories are completely determined by toric data via planar, bipartite tilings on a T^2 and connects them to mirror symmetry through D6-branes on the mirror Calabi–Yau, with the moduli space, BPS spectrum, and R-charges computable from dimer combinatorics and isoradial embeddings. Key contributions include a detailed dictionary between quivers, dimers, and toric geometry, a GLSM-based construction of the classical moduli space from dimers, and a mirror-symmetry picture that explains the emergence of the tiling structure and the superpotential from holomorphic disk instantons. The framework enables a systematic analysis of anomalous vs anomaly-free U(1) symmetries, BPS operators, and the R-symmetry via -maximization, providing a powerful, metric-independent route to study AdS/CFT duals of toric geometries and guiding extensions to more general geometries and higher-dimensional dualities.

Abstract

We review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the world-volume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an ``orbifold point'' in Kaehler moduli space. These provide an infinite class of four-dimensional N=1 superconformal field theories which may be studied in the context of the AdS/CFT correspondence. It is now understood that these gauge theories are completely specified by certain two-dimensional torus graphs, called brane tilings, and the combinatorics of the dimer models on these graphs. In particular, knowledge of the dual Sasaki-Einstein metric is not required to determine the gauge theory, only topological and symplectic properties of the toric Calabi-Yau cone. By analyzing the symmetries of the toric quiver theories we derive the dimer models and use them to construct the moduli space of the theory both classically and semiclassically. Using mirror symmetry the brane tilings are shown to arise in string theory on the world-volumes of the fractional D6-branes that are mirror to the stack of D3-branes at the tip of the cone.

Paper Structure

This paper contains 38 sections, 87 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Toric quiver gauge theories
  3. Brane Tilings and Dimer Models
  4. Planar quivers and brane tilings
  5. Example: Brane tiling for del Pezzo 1
  6. Quiver gauge theories and dimer models
  7. The classical $U(1)$ symmetries of the toric quiver theories
  8. Example: Dimer model for del Pezzo 1
  9. Review of toric geometry
  10. Dimer models and toric geometry
  11. Preferred bases of matchings
  12. Baryonic and mesonic symmetries
  13. Representation as difference of matchings
  14. Zig-zag symmetries
  15. Example: Charges for del Pezzo 1
  16. ...and 23 more sections

Figures (22)

  • Figure 1: The quiver gauge theory/planar quiver/brane tiling dictionary.
  • Figure 2: The quiver for D3-branes at a conical del Pezzo 1 singularity.
  • Figure 3: The planar quiver for the del Pezzo 1 quiver.
  • Figure 4: The brane tiling for the del Pezzo 1 quiver.
  • Figure 5: a) The Newton polygon $\Delta$ for the toric del Pezzo 1 surface, with the labels corresponding to perfect matchings. b) A complete triangulation of the toric diagram, corresponding to the resolution of the singular del Pezzo cone, and the dual $(p,q)$ web.
  • ...and 17 more figures