Enumerative meaning of mirror maps for toric Calabi-Yau manifolds

TL;DR

This paper establishes a precise mathematical relationship between mirror maps and genus 0 open Gromov-Witten invariants for toric Calabi-Yau manifolds, confirming a conjecture linking mirror symmetry and disk counting invariants.

Contribution

It proves that the inverse mirror map for certain toric Calabi-Yau manifolds can be explicitly expressed using genus 0 open Gromov-Witten invariants, confirming a conjecture by Gross, Siebert, and others.

Findings

Inverse mirror map expressed via generating functions of open Gromov-Witten invariants

Confirmation of conjectured relation between mirror maps and disk counting invariants

Mathematical proof of the conjecture in the toric Calabi-Yau setting

Abstract

We prove that the inverse of a mirror map for a toric Calabi-Yau manifold of the form $K_Y$, where $Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono \cite{FOOO10}. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert \cite[Conjecture 0.2 and Remark 5.1]{GS11} as part of their program, and was later formulated in terms of Fukaya-Oh-Ohta-Ono's invariants in the toric Calabi-Yau case in \cite[Conjecture 1.1]{CLL12}.