The Non-commutative Topological Vertex and Wall Crossing Phenomena

Abstract

We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed by the ordinary topological vertex. We prove an identify expressing the non-commutative topological vertex of a toric Calabi-Yau manifold X as a specialization of the closed BPS partition function of an orbifold of X, thus giving a closed expression for our vertex. We also clarify the action of the Weyl group of an affine A_L Lie algebra on chambers, and comment on the generalization of our results to the case of refined BPS invariants.

33 pages, 10 figures

References in corpus (5)

• Refined, Motivic, and Quantum
• Crystal Melting and Toric Calabi-Yau Manifolds
• Wall crossing in local Calabi Yau manifolds
• Wall Crossing and M-theory
• Emergent Calabi-Yau Geometry

Cited by in corpus (7)

• Open BPS Wall Crossing and M-theory
• Wall-crossing of D4-D2-D0 and flop of the conifold
• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions
• Non-commutative Donaldson-Thomas theory and vertex operators
• Geometry and Combinatorics of Crystal Melting
• Notes On U(1) Instanton Counting On $A_{l-1}$ ALE Spaces