Crystal melting on toric surfaces

TL;DR

This paper explores the connection between crystal melting models and instanton counting in supersymmetric gauge theories on toric surfaces, developing a vertex formalism to compute related partition functions.

Contribution

It introduces a vertex formalism for the crystal partition function and details how to adapt it for gauge theory partition functions on toric surfaces.

Findings

Established a relationship between crystal melting and gauge theory instanton counting.

Developed a vertex formalism for calculating partition functions.

Outlined modifications needed to relate crystal models to gauge theories.

Abstract

We study the relationship between the statistical mechanics of crystal melting and instanton counting in N=4 supersymmetric U(1) gauge theory on toric surfaces. We argue that, in contrast to their six-dimensional cousins, the two problems are related but not identical. We develop a vertex formalism for the crystal partition function, which calculates a generating function for the dimension 0 and 1 subschemes of the toric surface, and describe the modifications required to obtain the corresponding gauge theory partition function.