Deepening and application to integrable systems of index theory via perturbation of Dirac operator

• Project/Area Number: 26800045
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• 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): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: Dirac作用素 / 同変指数 / 指数の局所化 / トーリック多様体 / 特異ファイバー / Origami多様体 / Delzant多面体 / 幾何学的量子化 / origami多様体 / ループ群 / Dirac型作用素 / 指数 / 同境

Outline of Final Research Achievements

The results in this research are followings. 1. We defined the natural notion of cobordism in index theory via perturbation by Dirac operators along fibers, and we showed cobordism invariance of our index. 2. We gave a geometric proof of localization of equivariant Riemann-Roch number of toric origami manifolds. 3. We revised a paper about an S1-equivariant index of non-compact symplectic manifold with Hamiltonian S1-action. We also studied a localization of index in recent development of loop group equivariant index theory.

Research Products

• [Journal Article] A Danilov-type formula for toric origami manifolds via localization of index, (2018)
  • Author(s): Hajime Fujita
  • Journal Title: Osaka Journal of Mathematics Volume: 55
  • NAID: 120006542316
  • Peer Reviewed / Open Access
• [Journal Article] $S^1$-equivariant local index and transverse index for non-compact symplectic manifolds (2016)
  • Author(s): Hajime Fujita
  • Journal Title: Mathematical Research Letters Volume: 未定
  • Peer Reviewed
• [Journal Article] Cobordism invariance and the well-definedness of local index (2015)
  • Author(s): Hajime Fujita
  • Journal Title: Annals of Global Analysis and Geometry Volume: 47 Issue: 4 Pages: 399-414
  • DOI: 10.1007/s10455-015-9452-6
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Delzant 多面体のモジュライ空間上の距離関数について, (2018)
  • Author(s): 藤田玄
  • Organizer: 淡路島幾何学研究集会2018
• [Presentation] Equivariant Riemann-Roch number for non-compact symplectic manifolds (2018)
  • Author(s): Hajime Fujita
  • Organizer: Gauge theory in Fukuoka
  • Int'l Joint Research / Invited
• [Presentation] 一般化されたジェンガゲームとその最大種数 (2017)
  • Author(s): 藤田玄
  • Organizer: 2017年度日本数学会年会・トポロジー分科会
  • Place of Presentation: 首都大学東京
  • Year and Date: 2017-03-24
• [Presentation] Equivariant Riemann-Roch number of non-compact symplectic manifolds (2017)
  • Author(s): Hajime Fujita
  • Organizer: 変換群を核とする 代数的位相幾何学
  • Invited
• [Presentation] On a metric on the moduli space of Delzant polytopes (2017)
  • Author(s): Hajime Fujita
  • Organizer: Toric Topology 2017 in Osaka
  • Invited
• [Presentation] Localization of index via torus fibrations and a Danilov-type formula for toric origami manifolds (2016)
  • Author(s): Hajime Fujita
  • Organizer: The First Japan-Taiwan Joint Conference on Differential Geometry & the 8th TIMS-OCAMI-WASEDA Joint International Workshop on Differential Geometry and Geometric Analysis
  • Place of Presentation: Waseda University
  • Year and Date: 2016-12-13
  • Int'l Joint Research / Invited
• [Presentation] A Danilov-type formula for toric origami manifolds via localization of index (2016)
  • Author(s): Hajime Fujita
  • Organizer: Toric topology 2016
  • Place of Presentation: Kagoshima University
  • Year and Date: 2016-04-19
  • Int'l Joint Research / Invited
• [Presentation] A Danilov-type formula for toric origami manifolds via localization of index (2016)
  • Author(s): Hajime Fujita
  • Organizer: Toric topology 2016
  • Place of Presentation: 鹿児島大学 理学部 (鹿児島県鹿児島市)
  • Year and Date: 2016-04-19
  • Int'l Joint Research / Invited
• [Presentation] 幾何的量子化周辺のあれこれ‐局所化の観点から- (2016)
  • Author(s): 藤田玄
  • Organizer: 新しい幾何学に向かって2
  • Place of Presentation: 東京理科大学 森戸記念館(東京都,新宿区)
  • Year and Date: 2016-03-27
  • Invited
• [Presentation] Toric origami多様体に対するDanilov型定理 (2015)
  • Author(s): 藤田玄
  • Organizer: 松山TGSAセミナー
  • Place of Presentation: 愛媛大学理学部数学科 (愛媛県,松山市)
  • Year and Date: 2015-12-02
  • Invited
• [Presentation] Danilov type formula for toric origami manifolds via localization of index (2015)
  • Author(s): Hajime Fujita
  • Organizer: 6th International Conference on Geometry and Quantization (GEOQUANT)
  • Place of Presentation: ICMAT, Campus de Cantoblanco, Madrid(Spain)
  • Year and Date: 2015-09-14
  • Int'l Joint Research
• [Presentation] Danilov type theorem for toric origami manifolds via localization of index (2015)
  • Author(s): 藤田玄
  • Organizer: 非可換幾何学と数理物理学2015
  • Place of Presentation: 慶應義塾大学日吉キャンパス来往舎 (神奈川県,横浜市)
  • Year and Date: 2015-07-24
  • Invited
• [Presentation] 非コンパクトシンプレクティック多様体に対する[Q,R]=0型定理 (2015)
  • Author(s): 藤田玄
  • Organizer: 淡路島幾何学研究集会2015
  • Place of Presentation: 兵庫県南あわじ市 国民宿舎慶野松原荘
  • Year and Date: 2015-01-23 – 2015-01-25
• [Presentation] On well-definedness of the local index (2014)
  • Author(s): Hajime Fujita
  • Organizer: Topology of Torus actions and Applications to Geometry and Combinatorics (ICM2014 satellite conference)
  • Place of Presentation: Daejeon, Korea
  • Year and Date: 2014-08-07 – 2014-08-11