  We study the spectrum of BPS D-branes on a Calabi-Yau manifold using the 0+1
dimensional quiver gauge theory that describes the dynamics of the branes at
low energies. The results of Kontsevich and Soibelman predict how the
degeneracies change. We argue that Seiberg dualities of the quiver gauge
theories, which change the basis of BPS states, correspond to crossing the
"walls of the second kind." There is a large class of examples, including local
del Pezzo surfaces, where the BPS degeneracies of quivers corresponding to one
D6 brane bound to arbitrary numbers of D4, D2 and D0 branes are counted by
melting crystal configurations. We show that the melting crystals that arise
are a discretization of the Calabi-Yau geometry. The shape of the crystal is
determined by the Calabi-Yau geometry and the background B-field, and its
microscopic structure by the quiver Q. We prove that the BPS degeneracies
computed from Q and Q' are related by the Kontsevich Soibelman formula, using a
geometric realization of the Seiberg duality in the crystal. We also show that,
in the limit of infinite B-field, the combinatorics of crystals arising from
the quivers becomes that of the topological vertex. We thus re-derive the
Gromov-Witten/Donaldson-Thomas correspondence.
