1 June 2017 Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds
Kwokwai Chan, Siu-Cheong Lau, Naichung Conan Leung, Hsian-Hua Tseng
Duke Math. J. 166(8): 1405-1462 (1 June 2017). DOI: 10.1215/00127094-0000003X


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.


Download Citation

Kwokwai Chan. Siu-Cheong Lau. Naichung Conan Leung. Hsian-Hua Tseng. "Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds." Duke Math. J. 166 (8) 1405 - 1462, 1 June 2017. https://doi.org/10.1215/00127094-0000003X


Received: 13 July 2015; Revised: 29 June 2016; Published: 1 June 2017
First available in Project Euclid: 25 February 2017

zbMATH: 1371.53090
MathSciNet: MR3659939
Digital Object Identifier: 10.1215/00127094-0000003X

Primary: 53D37
Secondary: 14J33 , 14M25 , 53D12 , 53D20 , 53D40 , 53D45

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

Rights: Copyright © 2017 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.166 • No. 8 • 1 June 2017
Back to Top