Wall-crossing, free fermions and crystal melting

TL;DR

This paper presents a novel framework connecting wall-crossing phenomena in toric Calabi-Yau manifolds with free fermion models, vertex operators, and crystal melting, providing new insights into BPS invariants and phase transitions.

Contribution

It introduces a free fermion and crystal melting approach to describe wall-crossing and BPS invariants in toric Calabi-Yau manifolds without compact four-cycles, unifying previous models.

Findings

Overlap of fermion states reproduces BPS partition functions.

Wall-crossing operators encode changes across stability walls.

Crystal models describe moduli evolution and phase transitions.

Abstract

We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal interpretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years.