Consistently melting crystals

TL;DR

This paper proves that the consistency conditions for melting crystal models of BPS states are universally satisfied by physical brane tilings, linking them to conformal field theory and perfect matchings.

Contribution

It demonstrates that the conditions for melting crystal models are inherently met by all physical brane tilings due to conformality, providing a direct proof of their perfect matchings.

Findings

Consistency conditions follow from conformality.

All physical brane tilings satisfy these conditions.

Any physical brane tiling has a perfect matching.

Abstract

Recently Ooguri and Yamazaki proposed a statistical model of melting crystals to count BPS bound states of certain D-brane configurations on toric Calabi--Yau manifolds [arXiv:0811.2801]. This construction relied on a set of consistency conditions on the corresponding brane tiling, and in this note I show that these conditions are satisfied for any physical brane tiling; they follow from the conformality of the low energy field theory on the D-branes. As a byproduct I also provide a simple direct proof that any physical brane tiling has a perfect matching.