SYZ mirror symmetry for toric Calabi-Yau manifolds

Kwokwai Chan The Chinese University of Hong Kong Siu-Cheong Lau Boston University Nai-Chung Conan Leung The Chinese University of Hong Kong

Differential Geometry mathscidoc:1702.10002

J. Differential Geom., 90, (2), 177-250, 2012
We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.
SYZ, mirror symmetry, toric Calabi-Yau
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@inproceedings{kwokwai2012syz,
  title={SYZ mirror symmetry for toric Calabi-Yau manifolds},
  author={Kwokwai Chan, Siu-Cheong Lau, and Nai-Chung Conan Leung},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170213234357902326435},
  booktitle={J. Differential Geom.},
  volume={90},
  number={2},
  pages={177-250},
  year={2012},
}
Kwokwai Chan, Siu-Cheong Lau, and Nai-Chung Conan Leung. SYZ mirror symmetry for toric Calabi-Yau manifolds. 2012. Vol. 90. In J. Differential Geom.. pp.177-250. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170213234357902326435.
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