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 noncompact 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.