Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces

Abstract

In this paper, we compute the open Gromov-Witten invariants for every compact toric surface X which is semi-Fano (i.e. the anticanonical line bundle is nef). Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem. As a consequence, an explicit formula for the Lagrangian Floer superpotential W is obtained, which in turn gives an explicit presentation of the small quantum cohomology ring of X. We also provide a computational verification of the conjectural ring isomorphism between the small quantum cohomology of X and the Jacobian ring of W.

v3: 24 pages, 7 figures, final version, appeared in IMRN; v2: minor changes

References in corpus (3)

• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• Mirror maps equal SYZ maps for toric Calabi-Yau surfaces
• A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry

Cited by in corpus (6)

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• Seidel elements and potential functions of holomorphic disc counting
• Tropical counting from asymptotic analysis on Maurer-Cartan equations
• $A_n$-type surface singularity and nondisplaceable Lagrangian tori
• A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds