| 研究課題/領域番号 |
17K14152
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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) |
2017-04-01 – 2020-03-31
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| キーワード | 代数 / 頂点作用素代数 / モジュラー形式 / ムーンシャイン |
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| 研究実績の概要 |
I have 3 main results from the past year, all of which have been placed on the ArXiv.
1) 51 constructions of the Moonshine module (accepted): I establish an orbifold duality correspondence between non-Fricke elements of the Monster simple group and fixed-point free automorphisms of the Leech lattice that satisfy a "no massless states" condition, conjectured by Tuite in 1993. Using this duality, I show that a class of "non-Fricke monstrous" Lie algebras with string-theoretic origin satisfy the Borcherds-Kac-Moody property. From this, I showed that the ambiguous constants that appear in the Generalized Moonshine conjecture for Hauptmoduln are necessarily roots of unity.
2) Integral forms (submitted): I constructed a self-dual integral form of the Moonshine vertex operator algebra with monster symmetry. The existence of this object resolves Ryba's 1994 Modular Moonshine conjecture.
3) Vertex algebras and non-monstrous functions (submitted): In joint work with T. Komuro and S. Urano, I work out conditions on completely replicable functions that are necessary for such functions to come from holomorphic vertex operator algebras. Using orbifold conformal field theory, we eliminate all but 3 of the 157 candidate non-monstrous functions. This provides additional evidence for the uniqueness of the Moonshine vertex operator algebra, as conjectured by Frenkel, Lepowsky, and Meurman in 1988.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
My research path moved in a different direction than originally intended, but progress has been rapid. I had planned a homological refinement of Generalized Moonshine for the end of next year, but I already have a result that reduces the ambiguous constants to roots of unity. It remains to show that they are controlled by a cocycle. My construction of a monster-symmetric self-dual integral form of the Moonshine module was unanticipated, and it gives a final resolution to Ryba's 1994 Modular Moonshine conjecture. My constructions lead to new questions of unifying Modular Moonshine and Generalized Moonshine.
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| 今後の研究の推進方策 |
I am continuing my work on twisted conformal blocks on log-smooth curves and non-abelian fusion, and expect to finish by the end of the academic year, following the plan outlined in the grant application. However, I am also considering integrality questions as part of a new project to unify Modular Moonshine and Generalized Moonshine.
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| 次年度使用額が生じた理由 |
I have been invited to talk at three international conferences within the next five months. At these conferences, I plan to communicate my results and exchange research-related information with the other participants. I am also planning to attend two domestic conferences in the Kyoto area during the next three months. I have been invited to speak at one of them. I plan to use the Kakenhi for my travel expenses for these three conferences.
I expect there will be further travel later in this academic year, but I do not know where or when.
I also plan to buy one or two USB drives for file transfer and backup, and a few textbooks for reference.
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