Duke Mathematical Journal
- Duke Math. J.
- Volume 166, Number 8 (2017), 1405-1462.
Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds
Let be a compact toric Kähler manifold with nef. Let be a regular fiber of the moment map of the Hamiltonian torus action on . Fukaya, Oh, Ohta, and Ono defined open Gromov–Witten (GW) invariants of as virtual counts of holomorphic disks with Lagrangian boundary condition . We prove a formula that equates such open GW invariants with closed GW invariants of certain -bundles over used by Seidel and McDuff earlier to construct Seidel representations for . We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disk potential of , an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya, Oh, Ohta, and Ono.
Duke Math. J., Volume 166, Number 8 (2017), 1405-1462.
Received: 13 July 2015
Revised: 29 June 2016
First available in Project Euclid: 25 February 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 14J33: Mirror symmetry [See also 11G42, 53D37] 53D40: Floer homology and cohomology, symplectic aspects 14M25: Toric varieties, Newton polyhedra [See also 52B20] 53D12: Lagrangian submanifolds; Maslov index 53D20: Momentum maps; symplectic reduction
Chan, Kwokwai; Lau, Siu-Cheong; Leung, Naichung Conan; Tseng, Hsian-Hua. Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds. Duke Math. J. 166 (2017), no. 8, 1405--1462. doi:10.1215/00127094-0000003X. https://projecteuclid.org/euclid.dmj/1487991810