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

回想一下，2020年，我们计算了周期图下ADE奇点的米尔诺晶格图像。大谷高桥推广的结果的情况下可逆多项式的链型，但在不同的方法。利用Otani-Takahashi发现的链式可逆多项式的Milnor格的基，按照我们原来的方法，从镜像的另一面计算了链式可逆多项式的Milnor格的像，作为应用，给出了该基的一个重要拓扑不变量Seifert形式，它与一个更著名的拓扑不变量相交形式有关，是根据Hertling的一个重要公式计算的，它将Seifert形式和这里的一些分析结果联系起来。正如我提到的，我们的目标是通过周期图计算Milnor晶格的图像。我们的答案的主要特点是，它涉及到各种伽玛常数和单位根。我们论文的第二个目标是表明，虽然公式看起来很麻烦，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理论非常优雅地陈述，就像我们对ADE奇点所做的那样。然而，对于一般的链型可逆多项式，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.