Wall crossing, quivers and crystals

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. We argue that Seiberg dualities of the quiver correspond to crossing the walls of the second kind of Kontsevich and Soibelman. There is a large class of examples 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. 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. We also show that, in the limit of infinite B-field, the combinatorics of crystals becomes that of the topological vertex, thus re-deriving the Gromov-Witten/Donaldson-Thomas correspondence.