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 \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 $K_{\proj^1\times\proj^1}$.

arXiv commentv3: final version, published in JDG 90 (2012), no. 2, 177-250. 71 pages, 14 figures; substantially revised and expanded

Cited by in corpus (15)

• Smoothing toroidal crossing spaces
• Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds
• Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
• Homological mirror symmetry for log Calabi-Yau surfaces
• Lagrangian torus fibrations and homological mirror symmetry for the conifold
• Non-Kaehler SYZ mirror symmetry
• Open Gromov-Witten invariants and SYZ under local conifold transitions
• Categorical mirror symmetry on cohomology for a complex genus 2 curve
• Homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Disk counting and wall-crossing phenomenon via family Floer theory
• SYZ mirror symmetry for hypertoric varieties
• Pseudotoric structures and special Lagrangian torus fibrations on certain flag varieties
• Family Floer mirror space for local SYZ singularities
• Homological mirror symmetry for local Calabi-Yau manifolds via SYZ
• Geometric transitions and SYZ mirror symmetry