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-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.

arXiv commentv4: 44 pages, 3 figures, minor modifications, final version to appear in DMJ

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• Gauged Linear Sigma Model in Geometric Phases. I
• Localized mirror functor constructed from a Lagrangian torus
• SYZ mirror symmetry for toric varieties
• Tropical counting from asymptotic analysis on Maurer-Cartan equations
• An open quantum Kirwan map
• $T$-equivariant disc potentials for toric Calabi-Yau manifolds
• Non-contractible Hamiltonian loops in the kernel of Seidel's representation
• Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period
• A Note on Disk Counting in Toric Orbifolds
• The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map
• The extended tropical vertex group
• A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds