Duke Mathematical Journal

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

Kwokwai Chan, Siu-Cheong Lau, Naichung Conan Leung, and Hsian-Hua Tseng

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Abstract

Let X be a compact toric Kähler manifold with KX nef. Let LX 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.

Article information

Source
Duke Math. J., Volume 166, Number 8 (2017), 1405-1462.

Dates
Received: 13 July 2015
Revised: 29 June 2016
First available in Project Euclid: 25 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1487991810

Digital Object Identifier
doi:10.1215/00127094-0000003X

Mathematical Reviews number (MathSciNet)
MR3659939

Zentralblatt MATH identifier
1371.53090

Subjects
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

Keywords
open Gromov–Witten invariants mirror maps Seidel representations toric manifolds mirror symmetry Landau–Ginzburg models quantum cohomology Lagrangian Floer theory superpotential

Citation

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


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