| 研究課題/領域番号 |
22K03264
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分11010:代数学関連
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| 研究機関 | 筑波大学 |
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研究代表者 |
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| 研究期間 (年度) |
2022-04-01 – 2026-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,160千円 (直接経費: 3,200千円、間接経費: 960千円)
2025年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2024年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2023年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2022年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
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| キーワード | moonshine / vertex algebra / weak Hopf algebra / vertex operator algebra |
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| 研究開始時の研究の概要 |
This research concerns vertex algebras. These are mathematical structures that let us precisely theorize about the physical notion of conformal field theory. Conformal field theories appear in real-world phenomena like spontaneous magnetization, but also in more abstract situations connected to string theory. This project concerns in part a particular vertex algebra with very unusual symmetry, but is also about a more general study of how new kinds of symmetry can be used to study the relations between different vertex algebras.
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| 研究実績の概要 |
1) For the project on Monstrous Moonshine for integral group rings, we have the following results:
a) I am preparing another joint paper with my former student S. Urano that includes the new results of his dissertation. These include a classification of "integral species", or homomorphisms from Green rings of integral group rings to complex numbers, for groups of order pq. In addition, we have a proof of the integral moonshine conjecture for cyclic groups generated by elements in conjugacy classes pqA, namely the "totally Fricke" classes. This resolves a conjecture in an earlier paper of Urano.
b) We have some additional results, such as a classification of species for other groups of square-free order and order 4p, and integral moonshine for "totally Fricke" cyclic groups.
2) For the project on Galois theory of vertex algebra embeddings, I have found that the internal tensor product does not behave particularly well under ordinary composition, but it possesses an additional composition structure. I have found a form of Tannakian reconstruction that seems to work, but it produces an algebraic structure that is somewhat hard to understand. The results I have are not yet publishable.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
There are two main parts of this project. One is slightly ahead, and one is slightly delayed. Overall, I would say the project is slightly delayed.
1) The integral moonshine part of the project is slightly ahead of schedule, because Borcherds's work on the integral no-ghost theorem was useful in producing recursion relations. The results I planned to produce in 2023 are now published.
2) The question of Galois theory for vertex algebra embeddings is still unresolved, but I am making slow progress on the Tannakian reconstruction. The reason it is delayed is that I did not expect the problem to as difficult as it is. Unfortunately, I do not yet see a way to show that C_2 cofiniteness or rationality is preserved under finite index embeddings.
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| 今後の研究の推進方策 |
For the integral moonshine project, I plan to work on decomposing "integral species" for more general groups. Existing results show that we can work over semilocal subrings of the rationals, and that base change to completed local rings reflects isomorphisms. Thus, there exists a gluing procedure for passing from a classification of species over p-adic rings to a classification of species for the semilocal rings, but it is not written. I intend to write such a procedure as explicitly as possible. Additional plans include computational experiments with quasi-replicability, and attempts at taming some groups with infinitely many indecomposable representations.
For the project on Galois theory for vertex algebra embeddings, I plan to write up the Tannakian construction, and work out some easy explicit cases.
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