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2006 Fiscal Year Final Research Report Summary

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)
Project Period (FY) 2004 – 2006
Keywordsintegrable hierarchy / dispersionless integrable system / solvable many body system / gauge theory / free fermion / conformal mapping / Grassmann manifold / hyperbolic structure
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.

  • Research Products

    (28 results)

All 2007 2006 2005 2004

All Journal Article (28 results)

  • [Journal Article] Hamiltonian structure of PI hierarchy2007

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA (on-line journal) 3

      Pages: Paper 042

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

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA 3(on-line journal)

      Pages: paper 042

    • Description
      「研究成果報告書概要(欧文)」より
  • [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

      Pages: 012701(13)

    • Description
      「研究成果報告書概要(和文)」より
  • [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

      Pages: 102102(33)

    • Description
      「研究成果報告書概要(和文)」より
  • [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

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

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA (on-line journal) 2

      Pages: paper 057

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

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

      SIGMA (on-line journal) 2

      Pages: paper 072

    • Description
      「研究成果報告書概要(和文)」より
  • [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

      Pages: 012701 (13)

    • Description
      「研究成果報告書概要(欧文)」より
  • [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

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

    • Author(s)
      K.Takasaki
    • Journal Title

      SIGMA 2(on-line journal)

      Pages: paper 057

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

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

      SIGMA 2(on-line journal)

      Pages: paper 072

    • Description
      「研究成果報告書概要(欧文)」より
  • [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
  • [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
  • [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
  • [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

    • Description
      「研究成果報告書概要(和文)」より
  • [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
  • [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

      Pages: paper 056

    • Description
      「研究成果報告書概要(和文)」より
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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)

      Pages: paper 056

    • Description
      「研究成果報告書概要(欧文)」より
  • [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
  • [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
  • [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
  • [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

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Published: 2008-05-27  

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