SYZ mirror symmetry for toric Calabi-Yau manifolds
SYZ mirror symmetry for toric Calabi-Yau manifolds
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DOI：
10.4310/jdg/1335230845
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发表时间：
2010-06
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影响因子：
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通讯作者：
Kwokwai Chan;Siu-Cheong Lau;N. Leung
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文献类型：
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作者：
Kwokwai Chan;Siu-Cheong Lau;N. Leung
我们从 SYZ 猜想的角度研究了环面 Calabi-Yau 流形的镜像对称性。从复曲面 Calabi-Yau 流形 $X$ 上的非复曲面特殊拉格朗日环面纤维开始，我们使用经过量子校正修改的 T-对偶性构造一个复流形 $\check{X}$。这些校正通过某些开放 Gromov-Witten 不变量的生成函数的傅立叶变换进行编码。我们推测，这个复杂流形 $\check{X}$ 属于 Hori-Iqbal-Vafa 镜像家族，本质上是用规范平面坐标编写的。特别是，我们获得了（逆）镜像映射的枚举意义，这给出了为什么它们的泰勒级数展开以 $X$ 的卡勒参数表示具有积分系数的几何原因。应用 \cite{Chan10} 和 \cite{LLW10} 中的结果，我们根据局部 BPS 不变量计算开 Gromov-Witten 不变量，并为包括 $K_{\proj^2}$ 和 $K_{\proj^1\times\proj^1}$ 在内的几个 3 维示例提供我们的猜想证据。
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 Kahler 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 $K_{\proj^1\times\proj^1}$.