| Project/Area Number |
22K03264
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2022-04-01 – 2026-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2025: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2024: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2023: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | moonshine / vertex algebra / weak Hopf algebra / vertex operator algebra |
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| Outline of Research at the Start |
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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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
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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| Strategy for Future Research Activity |
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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