An Elliptic Yangian-Invariant, `Leading Singularity'

$\widetilde {\bf E_6}$型和Jacobi型简单椭圆奇点的平坦结构

Flat structure for the simple elliptic singularity of type $\widetilde {\bf E_6}$ and Jacobi form

为了构造简单椭圆奇点的半泛变形的本原型的周期映射的逆映射，K.Saito在[5]中引入了广义仿射根系的“平坦结构”。在第三节中，我们利用Wirthmüller [7]引入的Jacobi形式，明确地构造了$E_6$型情形下的平坦θ不变量。结合Kato [3]，Noumi [4]的结果（平面坐标的显式描述），给出了关于$\tilde E_6$型简单椭圆奇点的周期映射的Jacobi反演问题（直到线性同构）的一个答案（也见[6]）.细节将在其他地方公布。

可逆简单椭圆奇点的整体镜像对称性

Global mirror symmetry for invertible simple elliptic singularities

$E_N^{(1,1)}$（$N=6,7,8$）型的简单椭圆奇点可以用可逆多项式$W$的边缘变形来描述。在论文\cite{KS}和\cite{MR}中，作者证明了$W$的某些特定选择的镜像对称性陈述，并利用它证明了某些椭圆轨道$\mathbb{P}^1$ s的Gromov-Witten不变量的拟模性。然而，多项式$W$及其边缘变形$ϕ_μ$的选择不是唯一的。本文研究了单参数$W+σϕ_μ$族的整体镜像对称现象。在每种情况下，镜像对称都由某个超几何方程组决定。我们猜想，$W+σϕ_μ$的Saito-Givental理论在任何一个特殊极限$σ$上都与椭圆轨道$\mathbb{P}^1$的Gromov-Witten理论或具有对角对称性的可逆简单椭圆奇点的Fan-Jarvis-Ruan-Witten理论是镜像的，极限由奇点的Milnor数和$j$ -不变量来分类.证明了当$W$是Fermat多项式时的猜想。我们还证明了在Gepner点$σ=0$在所有其他情况下，猜想是正确的。

Extremal 1/2 Calabi-Yau 3-folds and six-dimensional F-theory applications

We discuss a method for classifying the singularity types of 1/2 Calabi-Yau 3-folds, a family of rational elliptic 3-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi-Yau 3-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi-Yau 3-folds yield elliptic Calabi-Yau 3-folds and applications to six-dimensiona…

非微扰共形块在其奇异因子处的匹配分支

Matching branches of non-perturbative conformal block at its singularity divisor

共形块是多变量的函数，通常表示为形式级数，其系数是手征（例如Virasoro）代数中的某些矩阵元素。非微扰共形块是一个多值函数，在维度空间上全局定义，有许多分支，也许还有额外的自由参数，在微扰水平上看不到。我们讨论额外的复杂性的非微扰描述，造成的事实，即所有最好的研究的例子共形块位于奇点轨迹的模空间（在因子的系数，或者简单地说，在零的Kac行列式）。一个典型的例子是Ashkin-Teller点，其中椭圆Dotsenko-Fateev积分和著名的Zamolodchikov公式至少提供了两个朴素的非微扰表达式，它们是不同的。伊辛和其他极小模型点的情况有些类似。

Traintrack Calabi-Yaus from Twistor Geometry

We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in $\mathbb{P}^{3}$. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in $\mathbb{P}^{1} \times \mathbb{P}^{1}$. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction.