Wall Crossing As Seen by Matrix Models

The number of BPS bound states of D-branes on a Calabi-Yau manifold depends on two sets of data, the BPS charges and the stability conditions. For D0 and D2-branes bound to a single D6-brane wrapping a Calabi-Yau 3-fold X , both are naturally related to the Kähler moduli space M(X) . We construct unitary one-matrix models which count such BPS states for a class of toric Calabi-Yau manifolds at infinite 't Hooft coupling. The matrix model for the BPS counting on X  turns out to give the topological string partition function for another Calabi-Yau manifold Y , whose Kähler moduli space M(Y)  contains two copies of M(X), one related to the BPS charges and another to the stability conditions. The two sets of data are unified in M(Y) . The matrix models have a number of other interesting features. They compute spectral curves and mirror maps relevant to the remodeling conjecture. For finite 't Hooft coupling they give rise to yet more general geometry Y^~ containing Y.