The orbifold quantum cohomology of $\mathbb{C}^{2}/\mathbb{Z}_3$ and Hurwitz-Hodge integrals

Let $\mathbb{Z}_3$ act on $\mathbb{C}^2$ by non-trivial opposite characters. Let $\mathcal{X}=[\mathbb{C}^{2}/\mathbb{Z}_3]$ be the orbifold quotient, and let $Y$ be the unique crepant resolution. We show that the equivariant genus 0 Gromov-Witten potentials $F^{\mathcal{X}}$ and $F^{Y}$ are equal after a change of variables--verifying the Crepant Resolution Conjecture for the pair $(\mathcal{X},Y)$. Our computations involve Hodge integrals on trigonal Hurwitz spaces, which are of independent interest. In a self-contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.