The Period Map of a Two-Dimensional Semi-Simple Frobenius Manifold

二维半简弗罗贝尼乌斯流形的周期图

基本信息

批准号：
20J20053
负责人：
ZHA Chenghan
金额：
$ 1.6万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2020
资助国家：
日本
起止时间：
2020-04-24 至 2023-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20J20053/
关键词：
mirror symmetry period map chain type polynomials Seifert form equivariant K-theory singularity theory invertible polynomial relative K-theory matrix factorizations integrable hierarchies topological K-theory simple singularities

项目摘要

Recall that in 2020, we computed the image of the Milnor lattice of an ADE singularity under a period map. Otani-Takahashi generalized the result to the case of invertible polynomials of chain type but in a different method. Using the basis of Milnor lattice of chain type invertible polynomials that was found by Otani-Takahashi, we calculated the image of the Milnor lattice of chain type invertible polynomials from the other side of the mirror following our original method.As an application, an important topological invariant of the basis called Seifert form, which is related to a more well-known topological invariant called intersection form, was calculated following a significant formula by Hertling connecting Seifert form and somewhat analytical result here.As I mentioned our goal is to compute the image of the Milnor lattice via the period map. The main feature of our answer is that it involves various gamma-constants and roots of unity. The second goal of our paper was to show that although the formulas look cumbersome, in fact there is an interesting structure behind them. We expected that our answer can be stated quite elegantly via relative K-theory as what we did for ADE singularity. However, as for the general chain type invertible polynomials, equivariant relative topological K-theory interpretation is far more difficult.

回想一下，2020 年，我们计算了周期图下 ADE 奇点的 Milnor 晶格图像。大谷高桥将结果推广到链式可逆多项式的情况，但方法不同。利用大谷高桥发现的链式可逆多项式 Milnor 格子的基础，我们按照我们原来的方法从镜子的另一侧计算了链式可逆多项式 Milnor 格子的图像。称为 Seifert 形式的基础的拓扑不变量，与一种更著名的称为交集形式的拓扑不变量相关，是根据 Herling 连接 Seifert 形式和某种分析结果的重要公式计算的正如我提到的，我们的目标是通过周期图计算米尔诺晶格的图像。我们答案的主要特点是它涉及各种伽马常数和单位根。我们论文的第二个目标是表明，虽然这些公式看起来很麻烦，但实际上它们背后有一个有趣的结构。我们期望我们的答案可以通过相对 K 理论非常优雅地表述，就像我们为 ADE 奇点所做的那样。然而，对于一般链式可逆多项式，等变相对拓扑K理论解释要困难得多。