Types of gauge groups in six-dimensional F-theory on double covers of rational elliptic 3-folds

$\frac{1}{2}$ Calabi-Yau 3-folds, Calabi-Yau 3-folds as double covers, and F-theory with U(1)s

In this study, we introduce a new class of rational elliptic 3-folds, which we refer to as "1/2 Calabi-Yau 3-folds". We construct elliptically fibered Calabi-Yau 3-folds by utilizing these rational elliptic 3-folds. The construction yields a novel approach to build elliptically fibered Calabi-Yau 3-folds of various Mordell-Weil ranks. Our construction of Calabi-Yau 3-folds can be considered as a three-dimensional generalization of the operation of gluing pairs of 1/2 K3 surfaces to yield elliptic K3 surfaces. From one to seven $U(1)$s form in six-dimensional $N=1$ F-theory on the constructed C…

Calabi-Yau三重和四重的椭圆亏格、乘积公式和广义卡茨-穆迪代数

The elliptic genus of Calabi-Yau 3- and 4-folds, product formulae and generalized Kac-Moody algebras

本文导出了一般Calabi-Yau四重图的椭圆亏格。Kawai最近的工作计算N=2杂种优势串单圈阈值校正与威尔逊线打开扩展到一个类似的计算，其中K3被替换为一般的卡-丘3-或4-倍。在所有情况下，似乎都涉及一个广义卡茨-穆迪代数，其分母公式出现在结果中。

具有各种Mordell-Weil秩的K3曲面上的F-理论模型-使用有理椭圆曲面的二次基变换的构造

F-theory models on K3 surfaces with various Mordell-Weil ranks -constructions that use quadratic base change of rational elliptic surfaces

我们构造了几个族的椭圆K3曲面与Mordell-Weil群的秩从1到4。我们研究了这些椭圆K3曲面乘以K3曲面上的F-理论紧化。具有非零Mordell-Weil秩的全同有理椭圆曲面的胶合对产生椭圆K3曲面，其Mordell-Weil群具有非零秩。任何具有整体截面的有理椭圆曲面的奇点型秩和Mordell-Weil群秩之和为8。利用这一性质，通过选择适当的奇异类型，可以得到具有各种非零Mordell-Weil秩的有理椭圆曲面族。胶合对这些合理的椭圆表面产生家庭的椭圆K3表面与各种非零Mordell-Weil秩。我们还确定了所产生的K3曲面乘以K3曲面上的F-理论紧化中出现的规范群的全局结构。$U(1)$规范场在这些紧化中产生。

$\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors

In this study, four-dimensional $N=1$ F-theory models with multiple U(1) gauge group factors are constructed. A class of rational elliptic 4-folds, which we call as "$\frac{1}{2}$Calabi-Yau 4-folds," is introduced, and we construct the elliptically fibered 4-folds by utilizing them. This yields a novel approach for building families of elliptically fibered Calabi-Yau 4-folds with positive Mordell-Weil ranks. The introduced $\frac{1}{2}$Calabi-Yau 4-folds possess the characteristic property wherein the sum of the ranks of the singularity type and the Mordell-Weil group is always equal to six. T…

复曲面Calabi-Yau 3-折叠的等变椭圆亏格

On Equivariant Elliptic Genera of Toric Calabi-Yau 3-folds

证明了复曲面Calabi-Yau 3-folds的等变椭圆亏格是广义弱Jacobi型。我们还引入了复曲面Calabi-Yau 3-folds的平均等变椭圆亏格的概念，并证明了它们是由江口和菅原预测的显式公式给出的普通弱Jacobi形式。

1/2 Calabi-Yau 4-folds的奇点与四维F-theory中规范群的分类方案

Singularities of 1/2 Calabi-Yau 4-folds and classification scheme for gauge groups in four-dimensional F-theory

本文提出了一种针对1/2 Calabi-Yau 4-folds奇点类型的分类方法，用于对四维$N=1$ F-theory中形成的非阿贝尔规范群进行分类，并通过显式构建包含两个U(1)因子的4D模型对该方法进行了演示。