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Applications of quantum toroidal algebras to integrable systems

Research Project

Project/Area Number 16K05183
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionRikkyo University

Principal Investigator

JIMBO Michio  立教大学, 理学部, 特任教授 (80109082)

Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords共形場理論 / 量子トロイダル代数 / 運動の保存量 / ベーテ仮設 / トロイダル量子群 / 可積分系 / 共形ブロック / パンルヴェ方程式 / ベーテ仮説 / ボレル部分代数 / 量子可積分系 / フェルミオン構造
Outline of Final Research Achievements

Integrals of motion (IM) in conformal field theory constitute an important class of quantum integrable systems. We studied its q-deformation and obtained the following results. (1) We constructed non-local IM associated with quantum toroidal gln algebra as Taylor coefficients of transfer matrices (traces of the universal R matrix on the Fock representations). (2) We showed that the commutativity between local and non-local IM can be understood as a (glm,gln) duality. (3)We described the spectrum of IM in the case of gl1 quantum toroidal algebra in terms of Bethe's ansatz.

Academic Significance and Societal Importance of the Research Achievements

共形場理論において重要な運動の保存量の背景に量子トロイダル代数(リー代数に値をとる2変数ローラン多項式の量子変形)があることが構成的に明らかになった。それにより系の解析に可積分系の標準的な手法を適用することが可能となり、スペクトルに対するベーテ仮設による記述が得られた。これによって量子アフィン代数(1変数ローラン多項式の量子変形)を基礎とする従来の量子可積分研究をトロイダル代数へと拡張する新しい研究方向の端緒が開かれた。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (12 results)

All 2019 2017 2016

All Journal Article (5 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 5 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (7 results) (of which Int'l Joint Research: 5 results,  Invited: 7 results)

  • [Journal Article] The (gl_m,gl_n) duality in the quantum toroidal setting2019

    • Author(s)
      B.Feigin, M.Jimbo and E.Mukhin
    • Journal Title

      Commun. Math. Phys.

      Volume: 367 Issue: 2 Pages: 455-481

    • DOI

      10.1007/s00220-019-03405-8

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] CFT approach to the q-Painleve VI equation2017

    • Author(s)
      M. Jimbo, H. Nagoya, and H. Sakai
    • Journal Title

      J. Integrable Syst.

      Volume: 2 Issue: 1

    • DOI

      10.1093/integr/xyx009

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Integrals of motion from quantum toroidal algebras2017

    • Author(s)
      Feigin B、Jimbo M、Mukhin E
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 50 Issue: 46 Pages: 464001-464001

    • DOI

      10.1088/1751-8121/aa8e92

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Finite-type modules and Bethe Ansatz equations2017

    • Author(s)
      B. Feigin, M.Jimbo, T.Miwa and E. Mukhin
    • Journal Title

      Annales Henri Poincare

      Volume: 印刷中 Issue: 8 Pages: 2543-2579

    • DOI

      10.1007/s00023-017-0577-y

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl12017

    • Author(s)
      Feigin B.、Jimbo M.、Miwa T.、Mukhin E.
    • Journal Title

      Communications in Mathematical Physics

      Volume: 356 Issue: 1 Pages: 285-327

    • DOI

      10.1007/s00220-017-2984-9

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] トロイダル量子群と可積分系2019

    • Author(s)
      神保道夫
    • Organizer
      日本数学会年会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Toroidal symmetry in quantum integrable systems2017

    • Author(s)
      Michio Jimbo
    • Organizer
      Correlation Functions of Quantum Integrable Systems and Beyond
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Integrals of motion from quantum toroidal algebras2017

    • Author(s)
      Michio Jimbo
    • Organizer
      The XXVth International Conference on Integrable Systems and Quantum Symmetries
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Integrals of motion from quantum toroidal algebras2017

    • Author(s)
      Michio Jimbo
    • Organizer
      Developments of mathematics at IPMU: in honor of Kyoji SAITO
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Integrals of motion and Bethe ansatz: an approach from quantum toroidal algebras2016

    • Author(s)
      Michio Jimbo
    • Organizer
      String Theory Meeting in Greater Tokyo Area
    • Place of Presentation
      早稲田大学理工学術院(東京都新宿区)
    • Year and Date
      2016-11-28
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] トロイダル量子群とベーテ仮設2016

    • Author(s)
      神保道夫
    • Organizer
      RIMS研究集会「可積分系の数理の現状と展望」
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2016-09-05
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Finite type modules and Bethe ansatz for quantum toroidal gl12016

    • Author(s)
      Michio Jimbo
    • Organizer
      RAQIS16
    • Place of Presentation
      ジュネーヴ (スイス)
    • Year and Date
      2016-08-22
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2016-04-21   Modified: 2020-03-30  

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