Geometry and Combinatorics of Crystal Melting

TL;DR

This paper surveys the geometric and combinatorial aspects of crystal melting models related to Donaldson-Thomas invariants, highlighting the role of plane partitions, vicious walker models, and matrix models in understanding BPS invariants for toric Calabi-Yau manifolds.

Contribution

It provides a comprehensive overview of the combinatorial structures underlying crystal melting models and their connections to various mathematical physics frameworks.

Findings

Plane partitions are central to the combinatorial description of BPS invariants.

Crystal melting models are equivalent to vicious walker models.

Partition functions can be represented via matrix models.

Abstract

We survey geometrical and especially combinatorial aspects of generalized Donaldson-Thomas invariants (also called BPS invariants) for toric Calabi-Yau manifolds, emphasizing the role of plane partitions and their generalizations in the recently proposed crystal melting model. We also comment on equivalence with a vicious walker model and the matrix model representation of the partition function.