  We describe wall-crossing for local, toric Calabi-Yau manifolds without
compact four-cycles, in terms of free fermions, vertex operators, and crystal
melting. Firstly, to each such manifold we associate two states in the free
fermion Hilbert space. The overlap of these states reproduces the BPS partition
function corresponding to the non-commutative Donaldson-Thomas invariants,
given by the modulus square of the topological string partition function.
Secondly, we introduce the wall-crossing operators which represent crossing the
walls of marginal stability associated to changes of the B-field through each
two-cycle in the manifold. BPS partition functions in non-trivial chambers are
given by the expectation values of these operators. Thirdly, we discuss crystal
interpretation of such correlators for this whole class of manifolds. We
describe evolution of these crystals upon a change of the moduli, and find
crystal interpretation of the flop transition and the DT/PT transition. The
crystals which we find generalize and unify various other Calabi-Yau crystal
models which appeared in literature in recent years.
