  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.
