可逆多项式Burnside环之间的Saito对偶

增强的等变Saito对偶

Enhanced equivariant Saito duality

在以前的论文中，作者定义了单值zeta函数之间的所谓Saito对偶的等变版本，作为阿贝尔群的Burnside环与其特征群之间的一种傅里叶变换。这里定义了有限群$G$的增强Burnside环$\widehat{B}(G)$。它的一个元素表示为一个有限的$G$ -集，该集具有$G$ -等变变换和与所有点相关联的稳定子群的特征标。一个给一个增强版本的等变齐藤对偶。对于一个复解析$G$ -流形，它的$G$ -等变变换有一个增强的等变欧拉示性数，其值在$\widehat{B}(G)$的完备中。证明了Berglund-Hübsch对偶可逆多项式的Milnor纤维的（约化）增强等变欧拉示性数在符号上一致，并指出这蕴涵了以前关于Berglund-Hübsch-Henningson对偶的轨道zeta函数的结果.

对偶可逆多项式的单值性

Monodromy of dual invertible polynomials

第一作者和A.Takahashi将Arnold的奇异对偶推广到三元可逆多项式，包括以下关系。对于某些可逆多项式$f$，$f$的约化单值zeta函数的Saito对偶与其Berglund-Hübsch转置$f^T$的约化单值zeta函数的形式“根”重合。在这里，我们给出了一个几何解释的“根”的monodromyzeta函数和推广上述关系的所有非退化可逆多项式在三个变量和一些多项式在任意数量的变量的形式，包括“根”的monodromyzeta函数的$f$和$f^T$。

A version of the Berglund-Hübsch-Henningson duality with non-abelian groups

A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality property for invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition"). An inspect…

Categorical Saito theory, II: Landau-Ginzburg orbifolds

Let $W\in \mathbb{C}[x_1,\cdots,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {\sf SL}_N\cap (\mathbb{C}^*)^N$ be a finite diagonal and special linear symmetry group of $W$. In this paper, we use the category ${\sf MF}_G(W)$ of $G$-equivariant matrix factorizations and its associated VSHS to construct a $G$-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of ${\sf MF}_G(W)$ characterized by $G_W^{\sf max}$-equivariance. Conjecturally, this $G$-equivariant Saito theory is equivale…

Landau-Ginzburg Mirror Symmetry Conjecture

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.

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。