2021 Fiscal Year Final Research Report
Isomonodromic tau-functions and representation theory of infinite dimensional algebras
| Project/Area Number |
18K03326
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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 12010:Basic analysis-related
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| Research Institution | Kanazawa University |
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Principal Investigator |
Nagoya Hajime 金沢大学, 数物科学系, 教授 (80447367)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Keywords | conformal field theory / Painleve equations / Virasoro algebra |
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| Outline of Final Research Achievements |
The purpose of this research is construction of Fourier expansions of tau functions of monodromy preserving deformation by representation theory of infinite dimensional algebras, such as Virasoro algebra. Firstly, we proved that the tau functions of the fourth and fifth Painleve equations are expressed by irregular conformal blocks using limiting precedure. Secondly, we introduce irregular vertex operators for a super Virasoro algebra and proved the decomposition of irregular Verma module of a super Virasoro algebra into an infinite sum of irregular Verma module of two Virasoro algebras. Thirdly, we gave Fourier expansions of tau functions of q-difference Painleve equations in terms of q-conformal blocks.
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| Free Research Field |
可積分系
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| Academic Significance and Societal Importance of the Research Achievements |
第４，５パンルヴェ方程式のタウ関数を不確定共形ブロックで表示できることを示したことは, パンルヴェ方程式と共形場理論の間にある不思議な関係の理解を深め, すべてのパンルヴェ方程式のタウ関数が共形ブロックで表示されるという予想の証明に向けて確かな礎となる. super Virasoro 代数に対して不確定頂点作用素を導入できたことによって, 一般の無限次元代数に対する不確定頂点作用素の定義、性質が明らかになりつつある.
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