Loop group equivariant index theory and its localization

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03288
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Japan Women's University
• Principal Investigator: FUJITA Hajime 日本女子大学, 理学部, 准教授 (50512159)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: Dirac作用素 / 局所化 / トーリック多様体 / Gromov-Hausdorff収束 / 双対平坦構造 / ダイバージェンス / 情報幾何学

Outline of Final Research Achievements

We developed an analytic index theory on non-compact manifolds with torus action via deformation by using differential operator along orbits. In addition we proved that a KK-homology cycle giving the analytic index is naturally defined.
As a joint work with Yu Kitabeppu (Kumamoto univ.) and Ayato Mitsuishi (Fukuoka univ.) we had a continuity of the Delzant construction between toric manifolds and Delzant polytopes under a relatively strong assumption.
We had an extension of the generalized Pythagorean theorem to the boundary for the divergence of the dually flat structure on a Delzant polytope via toric geometry.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

先行研究を群作用がある状況で発展および応用させ, 既存の結果により見通しのよい理解を与え, さらに非コンパクトな設定への一般化を得た.
研究で得たトーリック幾何の知見を活かし, Delzant対応の連続性や, トーリック幾何を用いた双対平坦構造の研究とその情報幾何的応用という新たな研究課題へとつながった.