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

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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.