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

Kwokwai Chan The Chinese University of Hong Kong Siu-Cheong Lau Boston University Naichung Conan Leung The Chinese University of Hong Kong Hsian-Hua Tseng Ohio State Unviersity

mathscidoc:1608.01001

Best Paper Award in 2019

Duke Mathematical Journal
Let X be a compact toric Kaehler 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.
mirror symmetry, toric, Gromov-Witten, SYZ, Seidel representation
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@inproceedings{kwokwaiopen,
  title={Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds},
  author={Kwokwai Chan, Siu-Cheong Lau, Naichung Conan Leung, and Hsian-Hua Tseng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160810163054226054047},
  booktitle={Duke Mathematical Journal},
}
Kwokwai Chan, Siu-Cheong Lau, Naichung Conan Leung, and Hsian-Hua Tseng. Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds. In Duke Mathematical Journal. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160810163054226054047.
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