BPS states, crystals and matrices

TL;DR

This paper reviews various mathematical models like free fermions, crystals, and matrices to understand BPS state counting and wall-crossing phenomena in toric Calabi-Yau manifolds, connecting them to topological string amplitudes.

Contribution

It provides a comprehensive review of the matrix model, crystal, and fermion representations for BPS states, including refined and unrefined cases, and their relation to topological string theory.

Findings

Matrix model representations of BPS counting are established.

Connections between crystal models and wall-crossing phenomena are elucidated.

Refined and unrefined topological string amplitudes are derived from these models.

Abstract

We review free fermion, melting crystal and matrix model representations of wall-crossing phenomena on local, toric Calabi-Yau manifolds. We consider both unrefined and refined BPS counting of closed BPS states involving D2 and D0-branes bound to a D6-brane, as well as open BPS states involving open D2-branes ending on an additional D4-brane. Appropriate limit of these constructions provides, among the others, matrix model representation of refined and unrefined topological string amplitudes.