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

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，在2020年，我们计算了在时期图下Ade奇异性的Milnor晶格的图像。 Otani-takahasi将结果概括为链类型的可逆多项式的情况，但采用了不同的方法。使用Otani-Takahashi发现的链型可逆多项式的米尔诺晶格的基础，我们计算了从镜像的另一侧链型可逆多项式的米尔诺晶格的形象通过将Seifert形式连接的Hertling连接，并在此处进行分析结果。我提到的是通过时期图计算Milnor晶格的图像。我们答案的主要特征是它涉及各种伽玛组成和团结的根源。我们论文的第二个目标是表明，尽管公式看起来很麻烦，但实际上它们背后有一个有趣的结构。我们希望我们的答案可以通过相对K理论优雅地说明，就像我们为Ade Singularity所做的那样。但是，对于一般链类型可逆多项式，等效的相对拓扑K理论的解释要困难得多。