Properly ordered dimers, $R$-charges, and an efficient inverse algorithm

TL;DR

This paper introduces an efficient algorithm to construct dimer models from Calabi-Yau geometries, ensuring the resulting superconformal field theories are consistent and satisfy physical bounds, with the concept of properly ordered dimers as a key tool.

Contribution

It presents a novel, efficient algorithm for deriving dimer models from toric Calabi-Yau geometries, including consistency checks and bounds on physical quantities.

Findings

The algorithm produces consistent superconformal field theories.

The models satisfy anomaly and unitarity constraints.

Bounds relate the central charge to the toric diagram area.

Abstract

The $\mathcal{N}=1$ superconformal field theories that arise in AdS-CFT from placing a stack of D3-branes at the singularity of a toric Calabi-Yau threefold can be described succinctly by dimer models. We present an efficient algorithm for constructing a dimer model from the geometry of the Calabi-Yau. Since not all dimers produce consistent field theories, we perform several consistency checks on the field theories produced by our algorithm: they have the correct number of gauge groups, their cubic anomalies agree with the Chern-Simons coefficients in the AdS dual, and all gauge invariant chiral operators satisfy the unitarity bound. We also give bounds on the ratio of the central charge of the theory to the area of the toric diagram. To prove these results, we introduce the concept of a properly ordered dimer.