Journal of Differential Geometry

SYZ mirror symmetry for toric Calabi-Yau manifolds

Abstract

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 $K_{\mathbb{P}^1}$.

Article information

Source
J. Differential Geom., Volume 90, Number 2 (2012), 177-250.

Dates
First available in Project Euclid: 24 April 2012

https://projecteuclid.org/euclid.jdg/1335230845

Digital Object Identifier
doi:10.4310/jdg/1335230845

Mathematical Reviews number (MathSciNet)
MR2899874

Zentralblatt MATH identifier
1297.53061

Citation

Chan, Kwokwai; Lau, Siu-Cheong; Leung, Naichung Conan. SYZ mirror symmetry for toric Calabi-Yau manifolds. J. Differential Geom. 90 (2012), no. 2, 177--250. doi:10.4310/jdg/1335230845. https://projecteuclid.org/euclid.jdg/1335230845