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 $\mathcal{X}=\mathbb{P}(1,\dots ,1,n)$ 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.

48 pages, 1 figure; v2: references added and updated, final version, to appear in CMP

References in corpus (11)

• A Mirror Theorem for Toric Stacks
• The Crepant Transformation Conjecture for Toric Complete Intersections
• Crepant resolutions and open strings
• Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds
• Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
• On the Crepant Resolution Conjecture in the Local Case
• Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds
• Toric degeneration and non-displaceable Lagrangian tori in $S^2\times S^2$
• The cohomological crepant resolution conjecture for P(1,3,4,4)
• Chen-Ruan cohomology of ADE singularities
• Chern-Weil Maslov index and its orbifold analogue

Cited by in corpus (9)

• A Mirror Theorem for Toric Stacks
• Crepant resolutions and open strings
• Noncommutative homological mirror functor
• Open Gromov-Witten invariants and SYZ under local conifold transitions
• SYZ mirror symmetry for toric varieties
• $A_n$-type surface singularity and nondisplaceable Lagrangian tori
• Big Quantum cohomology of orbifold spheres
• A Note on Disk Counting in Toric Orbifolds
• Quantum McKay correspondence for disc invariants of toric Calabi-Yau 3-orbifolds