Applications of quantum toroidal algebras to integrable systems

• Project/Area Number: 16K05183
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Rikkyo 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: ¥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。

Research Products

• [Journal Article] The (gl_m,gl_n) duality in the quantum toroidal setting (2019)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] CFT approach to the q-Painleve VI equation (2017)
  • Author(s): M. Jimbo, H. Nagoya, and H. Sakai
  • Journal Title: J. Integrable Syst. Volume: 2 Issue: 1
  • DOI: 10.1093/integr/xyx009
  • Peer Reviewed / Open Access
• [Journal Article] Integrals of motion from quantum toroidal algebras (2017)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Finite-type modules and Bethe Ansatz equations (2017)
  • 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
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl1 (2017)
  • 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
  • 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
  • Invited
• [Presentation] Toroidal symmetry in quantum integrable systems (2017)
  • Author(s): Michio Jimbo
  • Organizer: Correlation Functions of Quantum Integrable Systems and Beyond
  • Int'l Joint Research / Invited
• [Presentation] Integrals of motion from quantum toroidal algebras (2017)
  • Author(s): Michio Jimbo
  • Organizer: The XXVth International Conference on Integrable Systems and Quantum Symmetries
  • Int'l Joint Research / Invited
• [Presentation] Integrals of motion from quantum toroidal algebras (2017)
  • Author(s): Michio Jimbo
  • Organizer: Developments of mathematics at IPMU: in honor of Kyoji SAITO
  • Int'l Joint Research / Invited
• [Presentation] Integrals of motion and Bethe ansatz: an approach from quantum toroidal algebras (2016)
  • 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
  • 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
  • Invited
• [Presentation] Finite type modules and Bethe ansatz for quantum toroidal gl1 (2016)
  • Author(s): Michio Jimbo
  • Organizer: RAQIS16
  • Place of Presentation: ジュネーヴ (スイス）
  • Year and Date: 2016-08-22
  • Int'l Joint Research / Invited