Crystal Melting, BPS Quivers and Plethystics

TL;DR

This paper investigates the structure of BPS partition functions for D-brane bound states on toric Calabi-Yau threefolds, revealing their product forms, wall crossing behaviors, and connections to algebraic functions and quivers.

Contribution

It provides explicit product formulas for crystal and BPS partition functions across various geometries, explores wall crossing phenomena, and links these functions to algebraic and quiver representations.

Findings

Partition functions expressed as products of MacMahon functions.

Wall crossing effects analyzed for different chambers.

Connections established between partition functions and Kac polynomials.

Abstract

We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions in terms of plethystic exponentials. For $\mathbb{C}^3$ and tripled affine quivers, we find their connections to nilpotent Kac polynomials. Similarly, the partition functions of D4-D2-D0 brane bound states can be obtained by replacing the (generalized) MacMahon functions with the inverse of (generalized) Euler functions.