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Studies in the Vafa-Witten invariants

Research Project

Project/Area Number 16K05125
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionNagoya University

Principal Investigator

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

Project Period (FY) 2016-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords完全障害理論 / 仮想基本類 / 壁越え公式 / ウーレンベック・コンパクト化 / ゲージ理論 / (対称)完全障害理論
Outline of Final Research Achievements

Thomas and myself formulated Vafa-Witten invariants for projective surfaces and confirmed vorious conjectures. Kuhn and myself proved a blowup formula for enumerative invariants on projective surfaces. By using this blowup formula, Kuhn, Leigh, and myself obtained a modular expression of the blowup formula for the virtual chi_y genera of the moduli spaces. Gross, Joyce, and myself proved a wall-crossing formula for the moduli of representations of a quiver by using Joyce's theory of vertex algebras. Joyce, Upmeier, and myself developed new methods to study orientations for gauge-theoretic moduli spaces. I also investigated analytic aspects of the Vafa-Witten theory and the Kapustin-Witten one, furthermore, Liu, Rayan, and myself clarified a relationship between the latter and the non-abelian Hodge theory.

Academic Significance and Societal Importance of the Research Achievements

射影曲面上のVafa-Witten不変量の数学的定式化は20年以上なされていなかったものであり,この研究は多くの後続研究を生んでいる.仮想chi_y種数の爆発公式はこの不変量の生成関数の計算に決定的な役割を果たし現在様々な研究が進行中である.箙の表現のモジュライの壁越え公式は最近Joyceによって射影曲面上でも同種の公式が成立する事が示されている.モジュライ空間の向きに関する研究は,その後のゲージ理論的モジュライ空間の向きの問題に関する様々な研究で本質的に使われているものである.Vafa-WittenおよびKapustin-Witten理論における解析的研究は今後の研究の基礎となるものである.

Report

(8 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (37 results)

All 2023 2022 2021 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (15 results) Journal Article (8 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 8 results,  Open Access: 2 results,  Acknowledgement Compliant: 1 results) Presentation (11 results) (of which Int'l Joint Research: 8 results,  Invited: 10 results) Remarks (3 results)

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

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

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

    • Related Report
      2022 Annual Research 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
  • [Int'l Joint Research] Saskatchewan University(カナダ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] National Cheng Kung University(台湾)

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

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Saskatchewan University(カナダ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] National Cheng Kung University(台湾)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University of Oxford(英国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Imperial College London/University of Oxford(英国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] スウェーデン王立工科大学(スウェーデン)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Imperial College London/University of Oxford(英国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Imperial College London(英国)

    • Related Report
      2016 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 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The Kapustin-Witten equations and nonabelian Hodge theory2022

    • Author(s)
      Liu Chih-Chung、Rayan Steven、Tanaka Yuuji
    • Journal Title

      European Journal of Mathematics

      Volume: 印刷中 Issue: S1 Pages: 23-41

    • DOI

      10.1007/s40879-022-00538-4

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On orientations for gauge-theoretic moduli spaces2020

    • Author(s)
      Joyce Dominic、Tanaka Yuuji、Upmeier Markus
    • Journal Title

      Advances in Mathematics

      Volume: 362 Pages: 106957-106957

    • DOI

      10.1016/j.aim.2019.106957

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Vafa-Witten invariants for projective surfaces I: stable case2019

    • Author(s)
      Tanaka, Yuuji, Thomas, Richard P.
    • Journal Title

      Journal of Algebraic Geometry

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds2018

    • Author(s)
      Tanaka, Yuuji
    • Journal Title

      Glasgow Mathematical Journal

      Volume: 61 Issue: 2 Pages: 471-486

    • DOI

      10.1017/s0017089518000307

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Vafa-Witten invariants for projective surfaces II: semistable case2018

    • Author(s)
      Tanaka Yuuji、Thomas Richard P.
    • Journal Title

      Pure and Applied Mathematics Quarterly

      Volume: 13 Issue: 3 Pages: 517-562

    • DOI

      10.4310/pamq.2017.v13.n3.a6

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kahler surfaces2018

    • Author(s)
      Yuuji Tanaka
    • Journal Title

      Geometriae Dedicata

      Volume: 印刷中 Issue: 1 Pages: 177-187

    • DOI

      10.1007/s10711-018-0344-3

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Some bounded properties of solutions to the Vafa-Witten equations on closed 4-manifolds2017

    • Author(s)
      Yuuji Tanaka
    • Journal Title

      The Quarterly Journal of Mathematics

      Volume: 印刷中 Issue: 4 Pages: 1203-1225

    • DOI

      10.1093/qmath/hax015

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [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 Annual Research 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 Annual Research 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 Annual Research 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 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the virtual Euler characteristics of the moduli spaces of semistable sheaves on a complex projective surface2021

    • Author(s)
      田中祐二
    • Organizer
      第16回代数・解析・幾何学セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] On the Vafa-Witten theory on closed four-manifolds2018

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Special Holonomy: Progress and Open Problems 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Vafa-Witten invariants on projective surfaces2017

    • Author(s)
      Yuuji Tanaka
    • Organizer
      Gauge theory and categorification
    • Place of Presentation
      Institute for Pure and Applied Mathematics, UCLA, the U.S.A.
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Seiberg-Witten type equations on compact symplectic 6-manifolds2016

    • Author(s)
      Yuuji Tanaka
    • Organizer
      13th Taiwan geometry symposium
    • Place of Presentation
      National Cheng Kung University, Tainan, Taiwan
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A construction of Spin(7)-instantons2016

    • Author(s)
      田中祐二
    • Organizer
      日本数学会2016年度秋季総合分科会
    • Place of Presentation
      関西大学
    • Related Report
      2016 Research-status Report
  • [Presentation] A construction of Spin(7)-instantons2016

    • Author(s)
      Yuuji Tanaka
    • Organizer
      The second China-Japan geometry conference
    • Place of Presentation
      Fujian Normal University, Fuzhou, China
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A construction of Spin(7)-instantons2016

    • Author(s)
      田中祐二
    • Organizer
      第63回幾何学シンポジウム
    • Place of Presentation
      岡山大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Remarks] Yuuji Tanaka's homepage

    • URL

      https://sites.google.com/site/yu2tanaka/

    • Related Report
      2017 Research-status Report 2016 Research-status Report
  • [Remarks] スタッフ | 大阪大学大学院理学研究科数学専攻・理学部数学科

    • URL

      http://www.math.sci.osaka-u.ac.jp/staff/tanaka.html

    • Related Report
      2016 Research-status Report
  • [Remarks] People

    • URL

      http://www.math.sci.osaka-u.ac.jp/eng/staff_tanaka.html

    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2024-01-30  

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