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Geometric structure and integrable systems in mathematical physics

Research Project

Project/Area Number 16340040
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

TAKASAKI Kanehisa  Kyoto University, Graduate School of Human and Environmental Studies, Professor, 人間・環境学研究科, 教授 (40171433)

Co-Investigator(Kenkyū-buntansha) IKEDA Takeshi  Science University of Okayama, Faculty of Science, Lecturer, 理学部, 講師 (40309539)
SASAKI Ryu  Kyoto University, Yukawa Institute, Associate Professor, 基礎物理学研究所, 助教授 (20154007)
SHIMIZU Yuji  International Christian University, Division of Natural Sciences, Associate Professor, 教養学部, 準教授 (80187468)
TAKEBE Takashi  Ochanomizu University, Faculty of Science, Associate Professor, 理学部, 助教授 (60240727)
FUJII Michihiko  Kyoto University, Faculty of Science, Associate Professor, 理学研究科, 助教授 (60254231)
塩田 隆比呂  京都大学, 大学院・理学研究科, 助教授 (20243008)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥8,800,000 (Direct Cost: ¥8,800,000)
Fiscal Year 2006: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2005: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2004: ¥3,300,000 (Direct Cost: ¥3,300,000)
Keywordsintegrable hierarchy / dispersionless integrable system / solvable many body system / gauge theory / free fermion / conformal mapping / Grassmann manifold / hyperbolic structure / 量子可積分系 / 無分散極限 / ランダム行列 / 等モノドロミー変形 / ハミルトン構造 / 同変コホモロジー / Calogero系 / Loewner方程式 / 共形写像 / 広田方程式 / ソリトン方程式 / 共形場理論 / q類似 / 楕円類似 / インスタントン
Research Abstract

1.We considered the instanton sum of four and five dimensional supersymmetric gauge theories as a model of random (plane) partitions, and applied the method of free fermions for integrable hierarchies to derive the Seiberg-Witten curve.
2.We pointed out a relation between a special class of deformation process of conformal mapping and a kind of dispersionless integrable systems. A solution technique (hodograph method) of such integrable systems was also studied.
3.We derived several new dispersionless integrable systems.as quasi-classical limit from integrable hierarchies. An example is related to a q-analogue of the modified KP (and Toda) hierarchy. Another example is obtained from the two-component BKP hierarchy. Moreover, we could identify the so called genus-zero universal Whitham hierarchy to be quasi-classical limit of a multi-component analogue of the KP hierarchy.
4.We elucidated some new features of solvable many body systems (the Calogero-Moser system, the Sutherland systems, and their variants) such as : equilibrium configuration, shape invariance, creation-annihilation operator (as quantum mechanics), direct integration method (as classical mechanics), etc.
5.We obtained several geometric results on Grassmann manifolds, noncommutative algebraic varieties, invariants of low dimensional manifolds, hypergeometric equations related to hyperbolic cones, etc.
6.We did some other researches on random matrices, Seiberg-Witten integrable systems, isomonodromic deformations, integrable systems related to a moduli space of vector bundles, etc.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (42 results)

All 2007 2006 2005 2004

All Journal Article (41 results) Book (1 results)

  • [Journal Article] Hamiltonian structure of PI hierarchy2007

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA (on-line journal) 3

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Hamiltonian structure of PI hierarchy2007

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA 3(on-line journal)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Hamiltonian structure of PI hierarchy2007

    • Author(s)
      K.Takasaki
    • Journal Title

      Symmetry, Integrability and Geometry : Methods and Applications 3

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Explicit solutions of the classical Calogero and Sutherland systems for any root system2006

    • Author(s)
      R.Sasaki, K.Takasaki
    • Journal Title

      Journal of Mathematical Physics 47

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Unified Theory of Annihilation-Creation Operators for Solvable ('Discrete') Quantum Mechanics2006

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Journal of Mathematical Physics 47

    • NAID

      120007114399

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Exact solution in the Heisenberg picture and annihilation-creation operators2006

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Physics Letters B 641

      Pages: 112-117

    • NAID

      120007114398

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Dispersionless Hirota equations of two-component BKP hierarchy2006

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA (on-line journal) 2

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Loewner equations and dispersionless hierarchies2006

    • Author(s)
      T.Takebe, L.-P.Teo, A.Zabrodin
    • Journal Title

      Journal of Physics A : Mathematical and General 39

      Pages: 11479-11501

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Coupled modified KP hierarchy and its dispersionless limit2006

    • Author(s)
      T.Takebe, L.-P.Teo
    • Journal Title

      SIGMA (on-line journal) 2

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Explicit solutions of the classical Calogero and Sutherland systems for any root system2006

    • Author(s)
      R.Sasaki, K.Takasaki
    • Journal Title

      Progress of Theoretical Physics 47

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Exact solution in the Heisenberg picture and annihilation-creation operators2006

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Physics Letters B641

      Pages: 112-117

    • NAID

      120007114398

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dispersionless Hirota equations of two-component BKP hierarchy2006

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA 2(on-line journal)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Coupled modified KP hierarchy and its dispersionless limit2006

    • Author(s)
      T.Takebe, L.-P.Teo
    • Journal Title

      SIGMA 2(on-line journal)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dispersionless Hirota equations of two-component BKP hierarchy2006

    • Author(s)
      K.Takasaki
    • Journal Title

      Symmetry, Integrability and Geometry : Methods and Applications 2

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Loewner equations and dispersionless hierarchies2006

    • Author(s)
      T.Takebe, L.-P.Teo, A.Zabrodin
    • Journal Title

      Journal of Physics A : Mathematical and Geneneral 39

      Pages: 11479-11501

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Coupled modified KP hierarchy and its dispersionless limit2006

    • Author(s)
      T.Takebe, L.-P.Teo
    • Journal Title

      Symmetry, Integrability and Geometry : Methods and Applications 2

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Explicit solutions of the classical Calogero & Sutherland systems for any root system2006

    • Author(s)
      R, Sasaki, K.Takasaki
    • Journal Title

      Journal of Mathematical Physics 47(掲載予定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The Neumann-Siebenmann invariant and Seifert surgery2005

    • Author(s)
      M.Ue
    • Journal Title

      Mathematische Zeitschrift 250

      Pages: 475-493

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] An integral lift of the Rochlin invariant of spherical 3-manifolds and finite surgery2005

    • Author(s)
      M.Ue
    • Journal Title

      Journal of Mathematics of Kyoto University 45

      Pages: 21-37

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Shape Invariant Potentials in Discrete Quantum Mechanics2005

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      J. Nonlinear Mathematical Physics, Supplement 1

      Pages: 507-521

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems, and Orthogonal Polynomials2005

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Progress of Theoretical Physics 114

      Pages: 1245-1260

    • NAID

      110002975384

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Schur function expansion for normal matrix models and associated discrete matrix model2005

    • Author(s)
      A.Yu.Orlov, T.Shiota
    • Journal Title

      Physics Letters A 343

      Pages: 384-396

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions2005

    • Author(s)
      T.Maeda, T.Nakatsu, K.Takasaki, T.Tamakoshi
    • Journal Title

      JHEP (on-line journal) 03

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Free Fermion and Seiberg-Witten Differential in Random Plane Partitions2005

    • Author(s)
      T.Maeda, T.Nakatsu, K.Takasaki, T.Tamakoshi
    • Journal Title

      Nuclear Physics B 715

      Pages: 275-303

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] q-analogue of modified KP hierarchy and its quasi-classical limit2005

    • Author(s)
      K.Takasaki
    • Journal Title

      Letters in Mathematical Physics 72

      Pages: 165-181

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Degeneration of hyperbolic structures on the figure-eight knot complement and points of finite order on an elliptic curve2005

    • Author(s)
      M.Fujii
    • Journal Title

      Journal of Mathematics of Kyoto University 45

      Pages: 343-354

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] An integral lift of the Rochlin invariant of spherical 3-manifolds and finite surgery2005

    • Author(s)
      M.Ue
    • Journal Title

      J onrnal of Mathematics of Kyoto Univ. 45

      Pages: 21-37

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Schur function expansion for normal matrix models and associated discrete matrix models2005

    • Author(s)
      A.Yu.Orlov, T.Shiota
    • Journal Title

      Physics Letters A343

      Pages: 384-396

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions2005

    • Author(s)
      T.Maeda, T.Nakatsu, K.Takasaki, T.Tamakoshi
    • Journal Title

      JHEP 03(on-line journal)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Free Fermion and Seiberg-Witten Differential in Random Plane Partitions2005

    • Author(s)
      T.Maeda, T.Nakatsu, K.Takasaki, T.Tamakoshi
    • Journal Title

      Nuclear Physics B715

      Pages: 275-303

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Shape Invariant potentials in Discrete Quantum Mechanics2005

    • Author(s)
      S.Odake, E.Sasaki
    • Journal Title

      J・Nonlinear Mathematical Physics, supplement Vol.1

      Pages: 507-521

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] q-analogue of modified KP hierarchy and its quasi-classical limit2005

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Letters in Mathematical Physics 72・3

      Pages: 165-181

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions2005

    • Author(s)
      T.Maeda, T.Nakatsu, K.Takasaki, T.Tamakoshi
    • Journal Title

      JHEP 03・056(オンラインジャーナル)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Calogero-Sutherland-Moser systems, Ruijsenaars-Schneider-van Diejen systems, and orthogonal polynomials2005

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Progress of Theoretical Physics 114

      Pages: 1245-1260

    • NAID

      110002975384

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Degeneration of hyperbolic structures on the figure-eight knot complement and points of finite order on an elliptic curve2005

    • Author(s)
      M.Fujii
    • Journal Title

      Journal of Mathematcs of Kyoto University 45

      Pages: 343-354

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Elliptic spectral parameter and infinite dimensional Grassmann variety2005

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Progress in Mathematics(単行本,Birkhauser) 237

      Pages: 169-197

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Trigonometric degeneration and obifold Wess-Zumino-Witten model. II2005

    • Author(s)
      Takashi Takebe
    • Journal Title

      Progress in Mathematics(単行本,Birkhauser) 237

      Pages: 199-218

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Shape invariant potentials in 'discrete' quantum mechanics2005

    • Author(s)
      Satoru Odake, Ryu Sasaki
    • Journal Title

      Jounral of Nonlinear Mathematical Physics 12,Suppl.1

      Pages: 507-521

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Equilibria of 'discrete' integrable systems and deformations of classical orthogonal polynomials2004

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Journal of Physics A : Mathematical and General 37

      Pages: 11841-11876

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Polynomials Associated with Equilibria of Affine Toda-Sutherland Systems2004

    • Author(s)
      S.Odake, R.Sasaki
    • Journal Title

      Journal of Physics A : Mathematical and General 37

      Pages: 11401-11405

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Equilibria of 'discrete' integrable systems and deformations of classical orthogonal polynomials2004

    • Author(s)
      Satoru Odake, Ryu Sasaki
    • Journal Title

      Journal of Physics A 37

      Pages: 11401-11405

    • Related Report
      2004 Annual Research Report
  • [Book] Elliptic Integrable Systems (Rokko Lectures in Mathematics, vol.18)2005

    • Author(s)
      M.Noumi, K.Takasaki(編)
    • Total Pages
      363
    • Publisher
      神戸大学理学部数学教室
    • Related Report
      2005 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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