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New developments in spin geometry

Research Project

Project/Area Number 19K03480
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionWaseda University

Principal Investigator

HOMMA Yasushi  早稲田大学, 理工学術院, 教授 (50329108)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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 / ディラック作用素 / ラリタ・シュインガー場 / ケーリー・ラプラス作用素 / higher spin / Higgs代数 / ラリタ-シュインガー作用素 / 高次スピン / 定曲率空間 / 対称空間 / Howe双対性 / 幾何学 / 四元数ケーラー幾何学
Outline of Research at the Start

スピン1/2のスピノール場を用いたスピン幾何学は,数学・物理学の様々な話題が関連する重要な分野である.本研究課題の目的は,スピンを3/2へ上げた新しい幾何学を開発することである.アインシュタイン多様体や8次元特殊幾何学との関連性などを解明し,新しい方向性を探る.このため,ドイツの研究者と国際共同研究を行う.もう一つの目的は,グラスマン多様体上の調和解析学をwedge-ディラック作用素という新しい道具を用いて開発することである.このため,ベルギーの研究者と国際共同研究を行う.研究経費は主に研究打ち合せ旅費として使用する.

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.

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.

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (15 results)

All 2023 2021 2020 2019 Other

All Int'l Joint Research (3 results) Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Open Access: 1 results) Presentation (7 results) (of which Int'l Joint Research: 2 results,  Invited: 4 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Sttutgart University (ドイツ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] アントワープ大学(ベルギー)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] アントワープ大学(ベルギー)

    • Related Report
      2019 Research-status Report
  • [Journal Article] The spinor and tensor fields with higher spin on spaces of constant curvature2021

    • Author(s)
      Yasushi HOMMA, Takuma TOMIHISA
    • Journal Title

      Annals of Global Analysis and Geometry

      Volume: 60 Issue: 4 Pages: 829-861

    • DOI

      10.1007/s10455-021-09791-4

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Spectra of the Rarita-Schwinger Operator on Some Symmetric Spaces2021

    • Author(s)
      Homma Y.、Tomihisa T.
    • Journal Title

      Journal of Lie Theory

      Volume: 31 Pages: 249-264

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Pizzetti formula on the Grassmannian of 2-planes2020

    • Author(s)
      Eelbode D.、Homma Y.
    • Journal Title

      Annals of Global Analysis and Geometry

      Volume: 58 Issue: 3 Pages: 325-350

    • DOI

      10.1007/s10455-020-09731-8

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Rarita-Schwinger fields on Einstein manifolds2023

    • Author(s)
      Yasushi HOMMA
    • Organizer
      Semiclassic seminar at University of Cologne
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Higgs algebra in harmonic analysis on the Grassmannian of 2-planes2023

    • Author(s)
      本間泰史
    • Organizer
      mini-workshop, Global analysis and geometrry
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Higgs algebra in harmonic analysis on the Grassmannian of 2-planes2021

    • Author(s)
      本間泰史
    • Organizer
      第68回幾何学シンポジウム
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Pizzetti formula on the Grassmannian of 2-planes2021

    • Author(s)
      本間泰史, David Eelbode
    • Organizer
      日本数学会 年会
    • Related Report
      2020 Research-status Report
  • [Presentation] 定曲率空間上のスピノール解析2021

    • Author(s)
      富久拓磨, 本間泰史
    • Organizer
      日本数学会 年会
    • Related Report
      2020 Research-status Report
  • [Presentation] 対称空間上のRarita-Schwinger作用素の固有値について2020

    • Author(s)
      富久拓磨, 本間泰史
    • Organizer
      日本数学会 秋季総合分科会
    • Related Report
      2020 Research-status Report
  • [Presentation] The Rarita-Schwinger Operator and Spin Geometry2019

    • Author(s)
      Yasushi HOMMA
    • Organizer
      The second Taiwan-Japan Joint Conference on Differential Geometry
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Global Analysis and Geometry 20232023

    • Related Report
      2023 Annual Research Report
  • [Funded Workshop] mini-workshop, Global analysis and geometry2023

    • Related Report
      2022 Research-status Report

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Published: 2019-04-18   Modified: 2025-01-30  

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