OPEN GROMOV-WITTEN INVARIANTS, MIRROR MAPS, AND SEIDEL REPRESENTATIONS FOR TORIC MANIFOLDS

摘要

Let X be a compact toric Kahler manifold with -K-X nef. Let L subset of 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 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.