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

回想一下，在2020年，我们计算了一个周期映射下ADE奇点的米尔诺晶格的图像。Otani-Takahashi用不同的方法将结果推广到链式可逆多项式的情况。利用Otani-Takahashi发现的链式可逆多项式的Milnor格的基础，我们按照原来的方法计算了链式可逆多项式的Milnor格在镜面另一侧的像。作为应用，基的一个重要的拓扑不变量称为Seifert形式，它与一个更著名的拓扑不变量称为交点形式有关，根据Hertling将Seifert形式与这里的一些分析结果联系起来的一个重要公式计算。正如我提到的我们的目标是通过周期图来计算米尔诺晶格的图像。我们的答案的主要特点是它涉及到各种常数和单位根。我们论文的第二个目标是表明，尽管这些公式看起来很繁琐，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很优雅地通过相对k理论表述就像我们对ADE奇点所做的那样。然而，对于一般的链式可逆多项式，等变相对拓扑k理论的解释要困难得多。

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.