| 研究課題/領域番号 |
23K19008
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| 研究種目 |
研究活動スタート支援
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| 配分区分 | 基金 |
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| 審査区分 |
0201:代数学、幾何学、解析学、応用数学およびその関連分野
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| 研究機関 | 京都大学 |
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研究代表者 |
DAI Xuanzhong 京都大学, 数理解析研究所, 特定研究員 (70978551)
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| 研究期間 (年度) |
2023-08-31 – 2025-03-31
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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 |
2,860千円 (直接経費: 2,200千円、間接経費: 660千円)
2024年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
2023年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
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| キーワード | CDO / vertex algebras / modular forms / chiral de Rham complex / modular form / Rankin-Cohen bracket |
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| 研究開始時の研究の概要 |
The chiral de Rham complex, as a notable construction of vertex algebra, plays a crucial role in connecting different areas of mathematics and physics. The project uses vertex operator algebra to analyze automorphic forms, demanding diverse expertise and global collaboration.
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| 研究実績の概要 |
In the last fiscal year, I studied the representation theory of the vertex algebras, which provide a quantization of modular forms and give a hint for the generalization of the Kazama-Suzuki duality at the critical level. I also rewrite the character formulas in terms of theta functions and eta functions, which suggests the existence of modular linear differential equations. I also studied the chiral differential operators on the basic affine space and built a lifting formula from functions on cotangent bundles to the global CDO. As an example, I have completely solved the cases for SL_2 and SL_3. I also studied a sequence of non-admissible quasi-lisse vertex algebras coming from 4D/2D duality and classified the irreducible representations and associated varieties of them.
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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
The obtained results are pivotal for further advancement. For instance, the vertex algebra generated by the lifting formulas is expected to be the vertex algebra of global CDO. Moreover, the findings regarding the sequence of non-admissible quasi-lisse vertex algebras are both interesting and unexpected, which can be viewed as a byproduct of the research project.
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| 今後の研究の推進方策 |
I will continue to study the structure and representation theory of the two types of vertex algebras mentioned in the proposal. I plan to use the Kazama-Suzuki duality to understand the representation theory of the vertex algebras labeled by congruence subgroups. As for the CDO over basic affine space, I will continue to study the vertex algebra generated by the lifting formulas and show that the global Virasoro element lies in the nilradical of Zhu's C2 algebra, which prompts to show that the associated variety is isomorphic to the affinization of cotangent bundle.
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