Stability conditions on a non-compact Calabi-Yau threefold

We study the space of stability conditions $\Stab(X)$ on the non-compact Calabi-Yau threefold $X$ which is the total space of the canonical bundle of $\PP^2$. We give a combinatorial description of an open subset of $\Stab(X)$ and state a conjecture relating $\Stab(X)$ to the Frobenius manifold obtained from the quantum cohomology of $\PP^2$. We give some evidence from mirror symmetry for this conjecture.