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Studies in gauge theoretic deformation invariants and their generating functions

Research Project

Project/Area Number 21K03246
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

田中 祐二  名古屋大学, 多元数理科学研究科, 博士研究員 (00647993)

Project Period (FY) 2021-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2024: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2023: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordsゲージ理論 / 仮想基本類 / 壁越え公式 / 壁超え公式
Outline of Research at the Start

近年,Gromov-Witten不変量やDonaldson-Thomas不変量の理論などにおける仮想基本類の理論の発展に伴い射影曲面上の数え上げ不変量,特に望月拓郎氏によるDonaldson型不変量,Vafa-Witten,不変量および,類似の不変量の研究が新たな展開を見せている.この計画ではより多くの具体例でこれらの不変量の計算を行う.また,理論物理学で予想されているこれらの不変量の生成関数が持つ保型性などの性質や頂点代数との関係などをいくつかの具体的な場合に調べる.より一般の場合を調べるためには厳密な意味で半安定な対象が存在する場合も含めて議論する必要があるためそのための理論的整備も行う.

Outline of Annual Research Achievements

射影曲面上のVafa-Witten不変量および関連する数え上げ不変量に関する研究を進めた.Vafa-Witten不変量に関しては具体例の計算を行い,物理学者による関連する研究との連携を進めた.さらに,射影曲面上の関連する数え上げ不変量に関する基礎理論の構築も進めた.

また,近年,Borisov-JoyceおよびOh-Thomasらによって,Donaldson-Thomas不変量の4次元Calabi-Yau多様体での類似,あるいは,実4次元多様上で展開されているDonaldson不変量の理論の「複素類似」の研究が著しく発展しているが,そのゲージ理論的起源であるSpin(7)インスタントンの例を構成するという研究をGaldeano氏,Platt氏,Wang氏と共同で行った.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

共同研究者とVafa-Witten不変量に関連する変形不変量の具体的計算を進めることができた.また関連する数え上げ不変量の基礎研究についても新しい構造を発見することができた.これらの研究成果は次年度以降順次発表していく予定である.また,Spin(7)インスタントンの構成は難解なものとして知られているが,上述の通りGaldeano氏,Platt氏,Wang氏と共同で研究を行いその研究成果をまとめた論文をarXivで公表することができた.

さらに,様々な研究打合せを通して今後の研究の進展に有益な多くのフィードバックを得ることができた.

Strategy for Future Research Activity

2024年度では, 射影曲面上の変形不変量の代数幾何学的研究をさらに進める予定である.いずれも国内外の関連する研究者と議論を重ね進めていく予定である.

Report

(3 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • Research Products

    (17 results)

All 2023 2022 Other

All Int'l Joint Research (9 results) Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (7 results) (of which Int'l Joint Research: 7 results,  Invited: 7 results)

  • [Int'l Joint Research] Imperial College London(英国)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] Universitaet Hamburg(ドイツ)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] Stanford University/University of Maryland(米国)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] University of Waterloo(カナダ)

    • Related Report
      2023 Research-status Report
  • [Int'l Joint Research] マックス・プランク数学研究所(ドイツ)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] オスロ―大学(ノルウェー)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] ウプサラ大学(スウェーデン)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Stanford University(米国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Oxford University(英国)

    • Related Report
      2021 Research-status Report
  • [Journal Article] Universal Structures in C-Linear Enumerative Invariant Theories2022

    • Author(s)
      Gross Jacob, Joyce Dominic, Tanaka Yuuji
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 18

    • DOI

      10.3842/sigma.2022.068

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] On a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications2023

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Workshop on Algebraic Geometry,
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications2023

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Geometry of moduli spaces of Higgs bundles
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the moduli spaces of semistable Higgs sheaves on projective surfaces2023

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Geometry of moduli spaces of Higgs bundles
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications2023

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Physics and Special Holonomy
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces and its applications2023

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Gauge Theory, Moduli Spaces and Representation Theory, Kashiwa 2023, In honor of the 60th birthday of Hiraku Nakajima
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces2022

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Workshop on Mirror symmetry and Related Topics, Kyoto 2022
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On a blowup formula for sheaf-theoretic virtual enumerative invariants on projective surfaces2022

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Pacific Rim Complex & Symplectic Geometry Conference
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2021-04-28   Modified: 2024-12-25  

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