Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds

Abstract

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Y. Ruan's original CRC ["The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten theory of spin curves and orbifolds, 117-126, Contemp. Math., 403, Amer. Math. Soc., Providence, RI, 2006]. We prove the open CRC for the weighted projective spaces $\mathcal{X}=\mathbb{P}(1,\ldots,1,n)$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.