Wall-crossings in toric {G}romov–{W}itten theory {I}: crepant examples

Abstract

Let $\mathcal{X}$ be a Gorenstein orbifold with projective coarse moduli space $X$ and let $Y$ be a crepant resolution of $X$. We state a conjecture relating the genus-zero Gromov–Witten invariants of $\mathcal{X}$ to those of $Y$, which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan–Graber, and prove our conjecture when $\mathcal{X}=\mathbb{P}\left(1,1,2\right)$ and $\mathcal{X}=\mathbb{P}\left(1,1,1,3\right)$. As a consequence, we see that the original form of the Bryan–Graber Conjecture holds for $\mathbb{P}\left(1,1,2\right)$ but is probably false for $\mathbb{P}\left(1,1,1,3\right)$. Our methods are based on mirror symmetry for toric orbifolds.