| Project/Area Number |
20J20053
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| Research Category |
Grant-in-Aid for JSPS Fellows
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| Allocation Type | Single-year Grants |
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| Section | 国内 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
ZHA Chenghan 東京大学, 数理科学研究科, 特別研究員(DC1)
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| Project Period (FY) |
2020-04-24 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2022: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2021: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2020: ¥900,000 (Direct Cost: ¥900,000)
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| Keywords | 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 |
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| Outline of Research at the Start |
The notion of a Frobenius manifold is very special. It is a source of special functions that have many interesting applications. The work of Costello and Li provides a construction of a Frobenius manifold using harmonic analysis on a CY manifold. My goal is to extend their construction to singularity theory, because it could improve our understanding of the Frobenius structures in singularity theory as well as their applications to the representation theory of vertex algebras and integrable systems. And I will also study generalized periods and mirror symmetry using deformation theory.
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| Outline of Annual Research Achievements |
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.
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| Research Progress Status |
令和4年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
令和4年度が最終年度であるため、記入しない。
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