Inverse algorithm and triple point diagrams

TL;DR

This paper explores how triple point diagrams can systematize the inverse algorithm for constructing dimer models on a torus, aiding in the design of models with specific symmetries and substructures.

Contribution

It introduces the use of triple point diagrams to improve the inverse algorithm for dimer model construction, including symmetry implementation and substructure inclusion.

Findings

Developed a systematic method using triple point diagrams for dimer model construction.

Constructed the Octagon dimer model with specified constraints.

Proposed a new criterion for symmetry implementation in dimer models.

Abstract

Dimer models (also known as brane tilings) are special bipartite graphs on a torus $\mathbb{T}^2$. They encode the structure of the 4d $\mathcal{N} = 1$ worldvolume theories of D3 branes probing toric affine Calabi-Yau singularities. Constructing dimer models from a singularity can in principle be done via the so-called inverse algorithm, however it is hard to implement in practice. We discuss how combinatorial objects called triple point diagrams systematize the inverse algorithm, and show how they can be used to construct dimer models satisfying some symmetry or containing particular substructures. We present the construction of the Octagon dimer model which satisfies both types of constraints. Eventually we present a new criterion concerning possible implementations of symmetries in dimer models, in order to illustrate how the use of triple point diagrams could strengthen such statements.