Stability conditions on a non-compact Calabi-Yau threefold

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