2023 Fiscal Year Final Research Report
New developments in spin geometry
Project/Area Number |
19K03480
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Review Section |
Basic Section 11020:Geometry-related
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Research Institution | Waseda University |
Principal Investigator |
HOMMA Yasushi 早稲田大学, 理工学術院, 教授 (50329108)
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Project Period (FY) |
2019-04-01 – 2024-03-31
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Keywords | スピン幾何学 / クリフォード解析 / スピノール場 / ラリタ-シュインガー場 / 高階スピン / 実グラスマン多様体 / ヒッグス代数 |
Outline of Final Research Achievements |
(1) We attempted to pioneer spin geometry with higher spin. The first result was to give a method for calculating the eigenvalues for the Rarita-Schwinger operator on spin 3/2 spinors on symmetric spaces. And we got all the eigenvalues on the sphere, the complex projective space, and the quaternion projective space. The second result was to clarify the behavior of spinor fields with higher spin and symmetric tensor fields on spaces of constant curvature. As an application, we got all the eigenvalues of the higher spin Dirac operator on the sphere. These results were obtained in collaboration with T. Tomihisa. (2) We generalized the Pizzetti formula in spherical harmonic analysis to the real Grassmannian manifold of oriented 2 plances, Gr(2,n). In the process, we clarified that invariant differential operators on Gr(2,n) consist of a deformation algebra of sl(2,R) called the Higgs algebra. This was done in collaboration with D. Eelbode.
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Free Research Field |
微分幾何学
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Academic Significance and Societal Importance of the Research Achievements |
(1) 幾何学ではラリタ-シュインガー場の研究が行われ,物理学では量子重力や高次スピンのゲージ理論の研究が行われ,最近は高階スピンのスピノール場の研究が活発である.本課題の成果は,定曲率空間や対称空間という条件のもと,高階スピンのスピノール解析を行ったものであり,スピンが異なる場のツイスター作用素を通した関係が把握できる幾何学・物理学分野にインパクトある成果である.実際,ド-ジッター空間上の調和解析という物理学分野へ応用されている. (2) 実グラスマン多様体上の不変微分作用素がヒッグス代数を成すことを発見したことは意義があり,Gr(k,n)へ一般化した場合の代数の解明が今後の課題である.
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