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

Abstract

Let $X$ be a compact toric Kähler 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, 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 ${\mathbb{P}}^1$ 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.