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

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 ${\mathbb{F}}_2$.

v3: many minor changes, published in Pacific J. Math.; v2: 16 pages. Completely rewritten and improved

Cited by in corpus (6)

• Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds
• Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
• Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds
• Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces
• Seidel elements and potential functions of holomorphic disc counting
• Categorical mirror symmetry on cohomology for a complex genus 2 curve