Refined Wall-Crossing, Free Fermions and Crystal Melting

TL;DR

This paper explores the refined wall-crossing phenomena and their relation to free fermions and crystal melting, proposing a conjecture connecting refined topological strings with BPS states for toric Calabi-Yau threefolds.

Contribution

It introduces a conjecture linking refined topological string partition functions with BPS states and demonstrates how to reproduce these functions using 2d free fermion vertex operators.

Findings

Conjectured a refined OSV formula for toric Calabi-Yau threefolds.

Reproduced refined BPS partition functions for specific geometries.

Derived wall-crossing formulas using free fermion techniques.

Abstract

In this paper, we use an M-theory model to conjecture the refined reminiscence of the OSV formula connecting the refined topological string partition function with the refined BPS states partition function for the toric Calabi-Yau threefolds without any compact four cycles. Further, we show how to use the vertex operators in 2d free fermions to reproduce the refined BPS states partition function for the $\mathbb{C}^3$ case and the wall-crossing formulas of the refined BPS states partition function for the resolved conifold and $\mathcal{O}(-2)\oplus\mathcal{O}\rightarrow \mathbb{P}^1$ cases.