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Integrable hierarchies related to Gromov-Witten invariants

Research Project

Project/Area Number 18K03350
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionKindai University

Principal Investigator

TAKASAKI Kanehisa  近畿大学, 理工学部, 教授 (40171433)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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)
Keywordsグロモフ-ウィッテン不変量 / 可積分階層 / Dubrovin-Zhang理論 / Givental理論 / 格子KP階層 / 戸田階層 / フルヴィッツ数 / ホッジ積分 / 格子KdV階層 / 格子GD階層 / 一般化ILW階層 / 同変戸田階層 / 拡張戸田階層 / 対数的時間発展 / スケール極限 / 行列模型 / リーマン球面 / 同変グロモフ-ウィッテン不変量 / フォック空間 / ギヴェンタール群 / 頂点作用素 / 双線形方程式 / 位相的頂点 / ヴォルテラ型階層 / 一般化KdV階層 / ゲリファント-ディキー階層 / グロモフ・ウィッテン不変量 / ヴォルテラ型可積分階層 / τ函数 / 対数的ラックス作用素 / Gromov-Witten不変量
Outline of Final Research Achievements

The Gromov-Witten invariants are a rich source of studies on integrable hierarchies. Major progress therein has been achieved by the Dubrovin-Zhang theory and the Givental theory. The present research is focused on the cases that are related to the lattice P and Toda hierarchies and various reductions thereof. To be more precise, we have considered the Hurwitz numbers and the Gromov-Witten invariants of the Riemann sphere and the Hodge integrals on the moduli space of stable curves, and found that the Volterra-type hierarchies, the equivariant Toda hierarchy, the lattice Gelfand-Dickey hierarchy and the generalized ILW hierarchy show up as the integrable structures of these geometric objects. Moreover, these integrable hierarchies urn out to possess many novel features.

Academic Significance and Societal Importance of the Research Achievements

本研究は代数解析的な可積分系研究の一環である.その要となるのはτ函数の概念であり,無限次元グラスマン多様体,無限次元リー群とその表現,自由フェルミ場とそのフォック空間などを駆使してτ函数の構造や性質を記述する.グロモフ-ウィッテン不変量に関するDubrovin-Zhang理論やGivental理論もτ函数の概念を共有しているが,方法論的には代数解析的方法とかなり異質である.本研究はDubrovin-Zhang理論やGivental理論をヒントにして代数解析的な可積分階層の理論の拡張を試みたことに学術的意義がある.この試みはまだ道半ばであり,今後も継続して行く価値がある.

Report

(6 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
  • Research Products

    (17 results)

All 2023 2022 2021 2020 2019 2018

All Journal Article (9 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 9 results,  Open Access: 2 results) Presentation (8 results) (of which Int'l Joint Research: 4 results,  Invited: 7 results)

  • [Journal Article] Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy2023

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      J. Phys. A: Math. Theor.

      Volume: 56 Issue: 16 Pages: 165201-165201

    • DOI

      10.1088/1751-8121/acc495

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Extended lattice Gelfand-Dickey hierarchy2022

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      J. Phys. A: Math. Theor.

      Volume: 55 Issue: 30 Pages: 305203-305203

    • DOI

      10.1088/1751-8121/ac7ca2

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Dressing operators in equivariant Gromov-Witten theory of CP12021

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 54 Issue: 35 Pages: 35LT02-35LT02

    • DOI

      10.1088/1751-8121/ac1828

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Integrable structures of specialized hypergeometric tau functions2021

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B87 Pages: 57-78

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Cubic Hodge integrals and integrable hierarchies of Volterra type2021

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Proceedings of Symposia in Pure Mathematics

      Volume: 103 Pages: 481-502

    • DOI

      10.1090/pspum/103.1/01844

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Three-Partition Hodge Integrals and the Topological Vertex2020

    • Author(s)
      T. Nakatsu and K. Takasaki
    • Journal Title

      Communications in Mathematical Physics

      Volume: 376 Issue: 1 Pages: 201-234

    • DOI

      10.1007/s00220-019-03648-5

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] 4D limit of melting crystal model and its integrable structure2019

    • Author(s)
      Takasaki Kanehisa
    • Journal Title

      Journal of Geometry and Physics

      Volume: 137 Pages: 184-203

    • DOI

      10.1016/j.geomphys.2018.12.012

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Toda hierarchies and their applications2018

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 20 Pages: 203001-203001

    • DOI

      10.1088/1751-8121/aabc14

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Hurwitz numbers and integrable hierarchy of Volterra type2018

    • Author(s)
      Takasaki Kanehisa
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 43 Pages: 43LT01-43LT01

    • DOI

      10.1088/1751-8121/aae10b

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] 弦理論・ゲージ理論における戸田階層2020

    • Author(s)
      高崎金久
    • Organizer
      Quantum Geometry in Gauge Theory and Strings
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] CP1の同変Gromov-Witten理論と同変戸田階層2020

    • Author(s)
      高崎金久
    • Organizer
      日本数学会2020年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] Integrable structures of cubic Hodge integrals2019

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      2nd IBS-CGP Workshop on integrable systems and applications
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Volterra-type hierarchies for specialized hypergeometric tau functions2019

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      China-Japan Joint Workshop on Integrable Systems 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3次ホッジ積分の可積分構造2019

    • Author(s)
      高崎金久
    • Organizer
      数理解析研究所共同研究「可積分系数理の進化と展望」
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 位相的弦理論の量子ミラー曲線2019

    • Author(s)
      高崎金久
    • Organizer
      第72回Encounter with Mathematics,中央大学2019年1月11日~12日
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Hurwitz numbers and integrable hierarchy of Volterra type2018

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      AIMS Conferencer 2018, Taipei, Taiwan, July 5-9, 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Toda and q-Toda equations for Nekrasov partition functions2018

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      SIDE13, JR博多シティ2018年11月12日ー16日
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2018-04-23   Modified: 2024-01-30  

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