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

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.