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Abstract

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 K\"ahler moduli space ${\cal M}(X)$. 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 K\"ahler moduli space ${\cal M}(Y)$ contains two copies of ${\cal M}(X)$, one related to the BPS charges and another to the stability conditions. The two sets of data are unified in ${\cal M}(Y)$. 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 $\widetilde{Y}$ containing $Y$.