Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions
Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions
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通过 conifold 过渡构造新的 Calabi-Yau 3 折及其镜子
DOI：
10.4310/atmp.2010.v14.n3.a3
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发表时间：
2008
影响因子：
1.5
通讯作者：
M. Kreuzer
中科院分区：
文献类型：
--
作者：
V. Batyrev;M. Kreuzer
我们构造了一类具有小Picard数的新Calabi-Yau 3-fold $X$，并提出了使用具有conconfold奇点的环面超曲面的光滑构造它们的镜像$X^*$。这些新例子通过折叠转换与以前已知的例子相关。我们的结果通过环向退化推广了Grassmannians和flag流形中Calabi-Yau完全交点的镜像构造。有198849个自反的4-多面体，其2面仅为最小体积的三角形或平行四边形。每一个这样的多面体都会产生一组Calabi-Yau超曲面，它们在最坏的情况下具有折叠奇点。利用Namikawa判据，我们得到了30241个自反的4-多面体，使得相应的Calabi-Yau超曲面通过平面变形光滑。特别是，我们发现了210个自反的4-多面体，定义了68个拓扑结构不同的Calabi- Yau 3-fold， $h_{11}=1$。我们解释了镜面的构造，并计算了镜面Calabi-Yau 3-fold各自的1参数族的几个新的Picard—Fuchs算子。
We construct a surprisingly large class of new Calabi-Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^*$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi-Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive 4-polytopes whose 2-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi-Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive 4-polytopes such that the corresponding Calabi-Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive 4-polytopes defining 68 topologically different Calabi--Yau 3-folds with $h_{11}=1$. We explain the mirror construction and compute several new Picard--Fuchs operators for the respective 1-parameter families of mirror Calabi-Yau 3-folds.