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Weyl group invariant theory, Saito and Frobenius structures

Research Project

Project/Area Number 19K14531
Research Category

Grant-in-Aid for Early-Career Scientists

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

Principal Investigator

Shiraishi Yuuki  大阪大学, インターナショナルカレッジ, 講師 (40773990)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords原始形式 / ワイル群不変式論 / 平坦・フロベニウス構造 / 安定性条件の空間 / 導来圏 / 平坦構造 / フルビッツ・フロベニウス構造 / 拡大ワイル群不変式論 / 非可換特異点解消 / 境界点付き曲面 / 写像類群 / ワイル群 / 一般化ルート系 / 不変式 / 不変式論 / フロベニウス多様体 / フルビッツ・フロベニウス多様体 / 齋藤構造 / ミラー対称性 / フロベニウス構造・齋藤構造
Outline of Research at the Start

フロベニウス構造とは,その接層に平坦な,非退化対称双線形式とそれに両立する代数構造を持つ複素多様体です.この構造は,とある非線形偏微分方程式を満たす正則函数と対応しています.また様々な幾何学的・表現論的・数論的に素性のよい数列の母函数は,その偏微分方程式を満たすことが知られています.本研究は,一般化したルート系から系統的にフロベニウス構造を構成し,その様々な性質を調べることを目標とするものです.

Outline of Final Research Achievements

A Frobenius manifold is a complex manifold reflecting important numerical invariants from symplectic geometry, complex geometry and representation theory, on its tangent sheaf. A conjectural isomorphism among Frobenius manifolds implies non-trivial relations among these geometries and representation theory, and is called mirror symmetry conjecture. Derived categories of objects in these geometries and representation theory are constructed from their homological algebraic natures. This research found a clue for constructing a Frobenius manifold from their categorical structure though we only checked this for some concrete examples.

Academic Significance and Societal Importance of the Research Achievements

これまで,フロベニウス多様体の同型は一意性定理等の間接的な方法で示されてきました.導来圏の同値からこの同型を導くことによって,それぞれの幾何学や表現論の重要な数値的不変量の間のより内在的な理解に繋がります.この計画はKontsevich氏により提起されました.導来圏からどのようにフロベニウス多様体を構成するかには有力な候補と様々な進展があるものの未だ謎が多く,その解決の一歩に本研究は貢献しました.

Report

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

    (4 results)

All 2023 2022 2019

All Journal Article (1 results) (of which Peer Reviewed: 1 results) Presentation (3 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results)

  • [Journal Article] A Frobenius manifold for l-Kronecker quiver2022

    • Author(s)
      Ikeda Akishi、Otani Takumi、Shiraishi Yuuki、Takahashi Atsushi
    • Journal Title

      Letters in Mathematical Physics

      Volume: 112 Issue: 1

    • DOI

      10.1007/s11005-022-01506-5

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Presentation] A Frobenius manifold for l-Kronecker quiver2023

    • Author(s)
      白石勇貴
    • Organizer
      Combinatorics, geometry and commutative algebra of hyperplane arrangements
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] $\ell$-クロネッカー箙から構成されるフロベニウス多様体2022

    • Author(s)
      白石勇貴(登壇者),高橋篤史,池田暁志,大谷拓己
    • Organizer
      日本数学会秋季総合分科会一般講演
    • Related Report
      2022 Annual Research Report
  • [Presentation] Natural Saito structures on orbit spaces of duality groups2019

    • Author(s)
      Yuuki Shiraishi
    • Organizer
      Interaction Between Algebraic Geometry and QFT
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2019-04-18   Modified: 2024-01-30  

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