| Project/Area Number |
23K19008
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Kyoto University |
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Principal Investigator |
DAI Xuanzhong 京都大学, 数理解析研究所, 特定研究員 (70978551)
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| Project Period (FY) |
2023-08-31 – 2025-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2024: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2023: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | CDO / vertex algebras / modular forms / chiral de Rham complex / modular form / Rankin-Cohen bracket |
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| Outline of Research at the Start |
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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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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