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

Daniel R. Gulotta

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.

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

Abstract

The 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.

Paper Structure

This paper contains 22 sections, 20 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Consistency of dimer field theories
  4. Criteria for consistency and inconsistency
  5. Some perfect matchings of properly ordered dimers
  6. Zigzag paths and $(p,q)$-legs
  7. Unique corner perfect matchings
  8. $R$-charges and cubic anomalies
  9. Unitarity bound
  10. Bounds on $a$
  11. Bounds on $a$ for toric theories
  12. Comparison to non-toric field theories
  13. Merging zigzag paths
  14. Deleting an edge of the dimer
  15. Making multiple deletions
  16. ...and 7 more sections

Key Result

Theorem 3.1

A connected dimer is properly ordered iff it has no valence one nodes and it has a number of faces equal to twice the area of the convex polygon whose $(p,q)$-legs are the winding numbers of the zigzag paths of the dimer.

Theorems & Definitions (38)

  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 28 more