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.