SYZ mirror symmetry for toric Calabi-Yau manifolds

TL;DR

This paper explores mirror symmetry for toric Calabi-Yau manifolds via the SYZ conjecture, constructing mirror manifolds using quantum-corrected T-duality and relating open Gromov-Witten invariants to mirror maps.

Contribution

It introduces a method to construct mirror manifolds from special Lagrangian fibrations using quantum corrections linked to open Gromov-Witten invariants, providing a geometric interpretation of mirror maps.

Findings

Constructed mirror manifolds using quantum-corrected T-duality.

Linked open Gromov-Witten invariants to mirror maps.

Provided evidence for the conjecture in specific examples.

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