2020 Fiscal Year Annual Research Report
The Period Map of a Two-Dimensional Semi-Simple Frobenius Manifold
| Project/Area Number |
20J20053
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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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| Keywords | mirror symmetry / topological K-theory / period map / simple singularities |
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| Outline of Annual Research Achievements |
We computed the image of the Milnor lattice of an ADE singularity under a period map. We also proved that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity.
Givental-Milanov and Frenkel-Givental-Milanov proved that the total descendant potential is a tau-function of the principal Kac-Wakimoto hierarchy of the same type A, D, or E as the singularity f. The outcome of the above work is that the generating function of FJRW invariants is a tau-function of an appropriate Kac-Wakimoto hierarchy. However, there is still a small gap in this statement. Namely, while the state space of FJRW theory is identified explicitly with the Milnor ring of the singularity, the identification of the Milnor ring and the Cartan subalgebra of the corresponding simple Lie algebra is given by a period map and it was not explicit. In order to obtain an explicit identification, we need to determine the image of the root lattice in the Milnor ring of the singularity. This is exactly the problem that we solved in this paper.
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 is to show that although the formulas look cumbersome, in fact there is an interesting structure behind them. It turns out that our answer can be stated quite elegantly via relative K-theory.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
According to FJRW theory and some previous works by Chiodo-Iritani-Ruan and Frenkel-Givental-Milanov, Our proposal makes sense not only for Fermat type polynomials or ADE singularities, but more generally for an arbitrary invertible polynomial. And it enough to prove it for chain type cases and loop type cases respectively.
We focused on chain type cases. However, the image of the Milnor lattice of a chain type singularity under a period map was solved by Otani-Takahashi. So we have been working on the equivariant topological relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. The whole project is about to finish.
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| Strategy for Future Research Activity |
We have not formulated a conceptual definition of the map in our main theorem, i.e., Theorem 1.3 on page 5. Our definition is on a case by case basis. We expect that relative orbifold cohomology has a natural identification with the state space of FJRW-theory under which the map in our main theorem is identified with the mirror map of Fan-Jarvis-Ruan.
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