Mathematics – Symplectic Geometry
Scientific paper
2010-06-19
J. Differential Geom. 90 (2012), no. 2, 177-250
Mathematics
Symplectic Geometry
v3: final version, published in JDG 90 (2012), no. 2, 177-250. 71 pages, 14 figures; substantially revised and expanded
Scientific paper
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 $K_{\proj^1\times\proj^1}$.
Chan Kwokwai
Lau Siu-Cheong
Leung Naichung Conan
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