Weyl group invariant theory, Saito and Frobenius structures

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14531
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Osaka 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
• 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

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.

Free Research Field

幾何学

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．