An irregular version of conformal field theory and Painleve equations

• Project/Area Number: 15K17560
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kanazawa University
• Principal Investigator: Nagoya Hajime 金沢大学, 数物科学系, 准教授 (80447367)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: パンルヴェ方程式 / 共形場理論 / モノドロミー保存変形 / conformal field theory / Painleve equations / conformal blocks / Schur polynomials / Whittaker modules / vertex operators / Virasoro algebra

Outline of Final Research Achievements

In this research project, we studied irregular version of the conformal field theory and Painleve equations. Using representation theory of Virasoro algebra, we defined irregular vertex operators and irregular conformal blocks. Using these irregular conformal blocks, we proved Fourier expansion formulas at infinity of the tau functions of the fourth and fifth Painleve equations and gave conjectural Fourier expansion formulas of the tau functions of the second and third Painleve equations.

Research Products

• [Int'l Joint Research] University of Tours(フランス)
• [Int'l Joint Research] トゥール大学(フランス)
• [Int'l Joint Research] Tours University(France)
• [Journal Article] Toward a combinatorial formula for an irregular conformal block of rank one (2017)
  • Author(s): Hajime Nagoya, Yasuhide Numada
  • Journal Title: Josai Mathematical Monographs Volume: 10 Pages: 81-95
  • NAID: 120006028483
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Irregular conformal blocks, with an application to the fifth and fourth Painleve equations (2015)
  • Author(s): Hajime Nagoya
  • Journal Title: Journal of Mathematical Physics Volume: 56 Issue: 12 Pages: 123505-123505
  • DOI: 10.1063/1.4937760
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Fractional calculus of quantum Painleve systems of type A_(1)^l (2015)
  • Author(s): Hajime Nagoya
  • Journal Title: Contemporary Mathe- matics Volume: 651 Pages: 39-64
  • DOI: 10.1090/conm/651/13045
  • Peer Reviewed
• [Presentation] Irregular conformal blocks and Painleve tau functions (2017)
  • Author(s): Hajime Nagoya
  • Organizer: The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
  • Place of Presentation: University of Georgia, USA
  • Year and Date: 2017-03-30
  • Int'l Joint Research / Invited
• [Presentation] 共形場理論とパンルヴェ方程式 (2017)
  • Author(s): 名古屋 創
  • Organizer: Meeting for Study of Number theory, Hopf algebras and related topics
  • Place of Presentation: 富山大学（富山県・富山市）
  • Year and Date: 2017-02-12
  • Invited
• [Presentation] Irregular conformal blocks and Painleve tau functions (2017)
  • Author(s): 名古屋 創
  • Organizer: The XXVth International Conference on Integrable Systems and Quantum symmetries
  • Int'l Joint Research
• [Presentation] On degeneration limits of Viraosoro conformal blocks (2016)
  • Author(s): Hajime Nagoya
  • Organizer: 5th String Theory Meeting in the Greater Tokyo Area
  • Place of Presentation: 首都大学東京（東京都・八王子市）
  • Year and Date: 2016-12-02
  • Int'l Joint Research / Invited
• [Presentation] On confluence of vertex operators of Virasoro algebra (2016)
  • Author(s): Hajime Nagoya
  • Organizer: JMM Workshop on Representation Theory and Differential Equations
  • Place of Presentation: 城西大学（東京都・千代田区）
  • Year and Date: 2016-11-27
  • Invited
• [Presentation] Irregular conformal blocks and Painlev ́e functions (2015)
  • Author(s): Hajime Nagoya
  • Organizer: Differential and Difference Equations: Analytic, Arithmetic and Galoisian Approaches
  • Place of Presentation: Laboratoire Paul Painleve, France
  • Year and Date: 2015-10-19
  • Int'l Joint Research / Invited
• [Presentation] Conformal blocks and Painleve functions (2015)
  • Author(s): 名古屋 創
  • Organizer: RIMS 研究集会「可積分系理論の諸分野への応用」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-08-20
  • Invited
• [Presentation] Irregular conformal blocks and Painleve functions (2015)
  • Author(s): Hajime Nagoya
  • Organizer: TIMS-OCAMI-WASEDA Interna- tional workshop on Painleve equations and related topics
  • Place of Presentation: National Taiwan University, Taiwan
  • Year and Date: 2015-05-11
  • Int'l Joint Research / Invited
• [Presentation] Irregular conformal field theory and Painleve tau functions (2015)
  • Author(s): Hajime Nagoya
  • Organizer: The Ninth IMACS Interna- tional Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
  • Place of Presentation: University of Georgia, USA
  • Year and Date: 2015-04-01
  • Int'l Joint Research / Invited