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

TL;DR

This paper computes open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces, revealing explicit formulas and verifying conjectural ring isomorphisms, advancing understanding in symplectic geometry and mirror symmetry.

Contribution

It provides explicit formulas for open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces, including verification of mirror symmetry conjectures.

Findings

Explicit superpotential formulas for semi-Fano toric surfaces

Verification of the quantum cohomology and Jacobian ring isomorphism

Handling non-trivial obstructions in the moduli problem

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.