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

Abstract

Let X be a compact toric Kähler manifold with −KX nef. Let L⊂X be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya, Oh, Ohta, and Ono defined open Gromov–Witten (GW) invariants of X as virtual counts of holomorphic disks with Lagrangian boundary condition L. We prove a formula that equates such open GW invariants with closed GW invariants of certain X-bundles over ℙ1 used by Seidel and McDuff earlier to construct 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 disk potential of X, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya, Oh, Ohta, and Ono.