Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds

TL;DR

This paper establishes a formula linking open Gromov-Witten invariants of toric manifolds with closed invariants of associated bundles, enabling explicit calculations and insights into mirror symmetry and Seidel representations.

Contribution

It provides a new formula connecting open and closed GW invariants for toric manifolds, facilitating explicit computations and deeper understanding of mirror maps and Seidel representations.

Findings

Derived explicit formulas for open GW invariants

Provided an enumerative interpretation of mirror maps

Described the inverse of a key ring isomorphism in Floer theory

Abstract

Let $X$ be a compact toric K\"ahler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.