Wall-crossing, free fermions and crystal melting

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.

61 pages, 14 figures, published version

References in corpus (7)

• The Refined Topological Vertex
• Refined, Motivic, and Quantum
• Non-commutative Donaldson-Thomas theory and the conifold
• Crystal Melting and Toric Calabi-Yau Manifolds
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Matrix models for 2* theories
• Crystal Model for the Closed Topological Vertex Geometry

Cited by in corpus (24)

• Webs of W-algebras
• Quiver Yangian from Crystal Melting
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Wall Crossing, Quivers and Crystals
• Wall Crossing As Seen By Matrix Models
• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities
• Refined matrix models from BPS counting
• Statistical model and BPS D4-D2-D0 counting
• Multiple D4-D2-D0 on the Conifold and Wall-crossing with the Flop
• A Note on Quiver Yangians and $\mathcal{R}$-Matrices
• The Origin of Calabi-Yau Crystals in BPS States Counting
• Wall-crossing, open BPS counting and matrix models
• Wall-Crossing Effects on Quiver BPS Algebras
• Open string amplitudes of closed topological vertex
• Algorithms for representations of quiver Yangian algebras
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Evidence for Duality of Conifold from Fundamental String
• BPS states, crystals and matrices
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants
• Weyl Mutations in Quiver Yangians