Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities

TL;DR

This paper introduces a 2D crystal melting model that calculates BPS indices for D2-D0 states bound to D4-branes on toric Calabi-Yau singularities, linking crystal structures to gauge theory moduli spaces.

Contribution

It develops a novel crystal melting framework for computing BPS indices in D-brane systems on toric Calabi-Yau singularities, generalizing instanton constraints.

Findings

Model reproduces BPS indices for various examples.

Crystalline structures correspond to fixed points in gauge theory moduli space.

Consistent with wall-crossing phenomena in BPS state counting.

Abstract

We construct a two-dimensional crystal melting model which reproduces the BPS index of D2-D0 states bound to a non-compact D4-brane on an arbitrary toric Calabi-Yau singularity. The crystalline structure depends on the toric divisor wrapped by the D4-brane. The molten crystals are in one-to-one correspondence with the torus fixed points of the moduli space of the quiver gauge theory on D-branes. The F- and D-term constraints of the gauge theory are regarded as a generalization of the ADHM constraints on instantons. We also show in several examples that our model is consistent with the wall-crossing formula for the BPS index.