Gauge Theories from Toric Geometry and Brane Tilings

TL;DR

This paper develops a systematic method to derive quiver gauge theories from toric Calabi-Yau singularities using geometric, topological, and combinatorial tools, and verifies the results through R-charge calculations.

Contribution

It introduces a unified approach combining geometry, topology, and brane tilings to construct and analyze quiver gauge theories from toric singularities.

Findings

Constructed quiver gauge theories from toric data using tilings.

Performed a-maximisation and Z-minimisation to find exact R-charges.

Analyzed examples including the L^{a,b,c} family and Suspended Pinch Point.

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.