A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry

TL;DR

This paper establishes a formula linking open and closed Gromov-Witten invariants for certain toric manifolds, enabling explicit mirror superpotential computations and providing a new proof for the Hirzebruch surface case.

Contribution

It proves an equality between open and closed Gromov-Witten invariants for semi-Fano toric manifolds of a specific form, facilitating mirror symmetry calculations.

Findings

Computed mirror superpotentials for semi-Fano toric manifolds.

Provided a simple proof for the superpotential of the Hirzebruch surface _2.

Established a formula connecting open and closed Gromov-Witten invariants.

Abstract

We prove that open Gromov-Witten invariants for semi-Fano toric manifolds of the form $X=\mathbb{P}(K_Y\oplus\mathcal{O}_Y)$, where $Y$ is a toric Fano manifold, are equal to certain 1-pointed closed Gromov-Witten invariants of $X$. As applications, we compute the mirror superpotentials for these manifolds. In particular, this gives a simple proof for the formula of the mirror superpotential for the Hirzebruch surface $\mathbb{F}_2$.