Gauge Theories from Toric Geometry and Brane Tilings

TL;DR

The paper develops a geometry-driven framework to extract complete quiver gauge theories for toric Calabi–Yau singularities, using toric data, Sasaki–Einstein volumes, a-maximisation, and the dimer/brane tiling approach. It shows how to identify a distinguished set of fields, their multiplicities, and global charges directly from the toric diagram, and how to assemble these into a consistent superpotential and gauge-group structure. By applying this to the L^{a,b,c} family and its subfamily L^{a,b,a}, the authors verify exact R-charges and central charges through both field-theory (a-maximisation) and geometric (Z-minimisation) methods, and provide explicit brane tilings illustrating Seiberg duality between toric phases. The work delivers a practical, metric-free method to construct gauge theories dual to toric Sasaki–Einstein spaces, with broad implications for AdS/CFT and toric geometry. It also clarifies the role of brane tilings as an organizing principle for complex quivers and sets the stage for extensions to more general toric geometries.

Abstract

We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki-Einstein manifolds L^{a,b,c} is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily L^{a,b,a}, whose smallest member is the Suspended Pinch Point.

Figures (21)

• Figure 1: A four--faceted polyhedral cone in ${\rm I R}^3$.
• Figure 2: Toric diagram for the $L^{a,b,c}$ geometries.
• Figure 3: (p,q)-web for the $L^{a,b,c}$ theories.
• Figure 4: a) Toric diagram and b) $(p,q)$ web for the $L^{a,b,a}$ sub--family.
• Figure 5: The four building blocks for the construction of brane tilings for $L^{a,b,c}$.