The consistency and perfect-matching conjecture for brane tilings

Let a brane tiling have a corresponding toric diagram, either reflexive or non-reflexive. For each boundary point of the toric diagram, consider the coefficient of the associated Newton polynomial and its maximally tempered value, and let perfect matchings be the combinatorial objects associated to the tiling.

Consistency and perfect-matching conjecture. A brane tiling is consistent if and only if the corresponding toric diagram has maximally tempered coefficients for its boundary points. Moreover, the maximally tempered coefficients are equal to the numbers of perfect matchings associated to the boundary points.

This conjecture seeks a common characterization of consistency, Newton-polynomial coefficients, and perfect-matchings data for reflexive and non-reflexive toric diagrams. The paper presents the claim as a natural conjecture based on the examples discussed.