Open Access
February 2012 SYZ mirror symmetry for toric Calabi-Yau manifolds
Kwokwai Chan, Siu-Cheong Lau, Naichung Conan Leung
J. Differential Geom. 90(2): 177-250 (February 2012). DOI: 10.4310/jdg/1335230845


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 Kähler parameters of X have integral coefficients. Applying the results in "A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry," to appear in Pacific J. Math., and "A relation for Gromov-Witten invariants of local Calabi-Yau threefolds," to appear in Math. Res. Lett., 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 $K_{\mathbb{P}^2}$ and $K_{\mathbb{P}^1}$.


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Kwokwai Chan. Siu-Cheong Lau. Naichung Conan Leung. "SYZ mirror symmetry for toric Calabi-Yau manifolds." J. Differential Geom. 90 (2) 177 - 250, February 2012.


Published: February 2012
First available in Project Euclid: 24 April 2012

zbMATH: 1297.53061
MathSciNet: MR2899874
Digital Object Identifier: 10.4310/jdg/1335230845

Rights: Copyright © 2012 Lehigh University

Vol.90 • No. 2 • February 2012
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