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q-analogue of the ADHMN construction and the self-dual Yang-Mills fields

Research Project

Project/Area Number 16540352
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Mathematical physics/Fundamental condensed matter physics
Research InstitutionKisarazu National College of Technology

Principal Investigator

KAMATA Masaru  Kisarazu National College of Technology, Natural Science Education, Professor (10169609)

Co-Investigator(Kenkyū-buntansha) NAKAMURA Atsushi  Kitasato University, Department of Physics, School of Science, Lecturer (90245415)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,740,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsYang-Mills fields / Self-dual / ADHMN construction / q-analogue / Nahm equation / Integrable system / Singularity confinement test / Algebraic entropy / 簡約 / スペクトル曲線
Research Abstract

In mathematical physics, the ADHM (ADHMN) construction is a well-known method to construct most generally the (anti-) selfdual Yang-Mills fields in the four-dimensional Euclidean space R^4 (R×R^3). The head investigator and the investigator presented, in a previous paper, a q-analogue of the ADHMN construction based on the countably infinite dimensional Hilbert space l^2 defined at a discrete q-interval I_q:={±1/2, ±q/2, ±^2/2, ., ±q^n/2, .} with a real parameter q. Further, as a q-analogue of the Nahm equations which describe the multi-monopole configurations in R×R^3, the following q-discretized Nahm equations (or q-Nahm equations in short) are obtained:
D_qT_j(z)=(1/2)ε_{jkl}(T_k(qz)T_l(z)-T_l(qz)T:_k(z)), j,k,l=1,2,3
where D_q is the q-difference operator.
The following have been studied in this project in the above term :
1. The Euler-top type reduction of the q-Nahm equations yields a system of difference equations which pass the singularity confinement test.
2. A Toda-Flaschka type reduction of the q-Nahm equations yields a system of difference equations with at least one conserved quantity.
3. A q-analogue of the caloron solution in S^l×R^3 is obtained through the ADHM construction.
4. Relations between the Nahm equations and the Euler top equations are studied.
5. Singular solutions derived from the known solutions to the CBS equations are investigated.
6. An extended KP hierarchy with the Riemann-Liouville differential-integral operators of the order 1/3 is examined.
7. A preliminary calculation of the algebraic entropy to the q-Nahm equations is given.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (35 results)

All 2007 2006 2005

All Journal Article (14 results) (of which Peer Reviewed: 1 results) Presentation (21 results)

  • [Journal Article] Nahm方程式のq-離散化とその解析2007

    • Author(s)
      中村 厚, 鎌田 勝
    • Journal Title

      九州大学応用力学研究所研究集会報告「非線形波動現象における基礎理論, 数値計算および実験のクロスオーバー」(電子版) No.18ME-S5

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Variant of ADHM construction associated with q-difference analysis2007

    • Author(s)
      Atsushi Nakamula
    • Journal Title

      Journal of Mathematical Physics 48

      Pages: 23505-23513

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary 2006 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Integrability of q-discretized Nahm equations2007

    • Author(s)
      Atsushi Nakamula, Masaru Kainata
    • Journal Title

      Proceedings of Symposium on Crossover among Theoretical, Numerical and Experimental Studies on Nonlinear 'Vaves](electronic version), Research Institute for Appl ied Mathematics, Kushu University(ME-S5, Article) No.18, No.26

      Pages: 1-4

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Variant of ADIIM construction associated with q-difference analysis2007

    • Author(s)
      Atsushi Nakamula
    • Journal Title

      Journal of Mathematical Physics Vol.48

      Pages: 23505-23513

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Nahm方程式のq-離散化とその解析2007

    • Author(s)
      中村 厚, 鎌田 勝
    • Journal Title

      九州大学応用力学研究所研究集会報告「非線形波動現象における基礎理論, 数値計算および実験のクロスオーバー(電子版) (未定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] q-離散Nahm方程式の可積分性2006

    • Author(s)
      中村 厚, 鎌田 勝
    • Journal Title

      九州大学応用力学研究所研究集会報告「非線形波動および非線形力学系の現象と数理」(電子版) No.17ME-S2

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The variant of ADHM construction associated with q-difference operators2006

    • Author(s)
      Atsushi Nakamula
    • Journal Title

      素粒子論研究, 京都大学基礎物理学研究所研究集会報告「場の量子論の基礎的諸問題と応用」 113

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Integrability of q-discretized Nahni equations2006

    • Author(s)
      Atsushi Nakamula, Masaru Kamata
    • Journal Title

      Proceedings of Symposium on Phenomena and Mathematical Theory of Nonlinear Waves and Nonlinear Dynamical Systems(electronic version), Research Institute for Applied Mathematics, Kushu University(Article) No.17, ME-S2, No, 17

      Pages: 1-4

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The variant of ADHM construction associated with q-difference operators2006

    • Author(s)
      Atsushi Nakamula
    • Journal Title

      Proceedings of Yorkshop on Fundamental Problems of Quantum Field Theory and its Application, Soryusiron-Kenkyuu, Research Institute for Fundamental Physics, Koto University. Vol.113

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] q-離散Nahm方程式の可積分性2006

    • Author(s)
      中村 厚, 鎌田 勝
    • Journal Title

      九州大学応用力学研究所研究集会「非線形波動および非線形力学系の現象と数理」プロシーディングズ 未定(電子版)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The variant of ADHM construction associated with q-difference operators2006

    • Author(s)
      Atsushi Nakamula
    • Journal Title

      素粒子論研究,京都大学基礎物理学研究所研究集会「場の量子論の基礎的諸問題と応用」プロシーディングズ 未定

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 分数階Riemann-Liouville積分と拡張KP階層2005

    • Author(s)
      鎌田 勝, 中村 厚
    • Journal Title

      木更津工業高等専門学校紀要 38

      Pages: 97-106

    • NAID

      110004302423

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Fractional Rieniann-Liouville Integrals and extended KP hierarchy2005

    • Author(s)
      Masaru Kamata, Atsushi Nakaniula
    • Journal Title

      Bulletin of Kisarazu National College of Technology No.38

      Pages: 97-106

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] 分数階Riemann-Liouville積分と拡張KP階層2005

    • Author(s)
      鎌田 勝, 中村 厚
    • Journal Title

      木更津工業高等専門学校紀要 第38号

      Pages: 97-106

    • NAID

      110004302423

    • Related Report
      2004 Annual Research Report
  • [Presentation] Aspects of q-discretized Nahm equations2007

    • Author(s)
      Atsushi Nakamula
    • Organizer
      The Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena:Computation and Theory, Oral presentation
    • Place of Presentation
      University of Georgia,Athens,GA,USA
    • Year and Date
      2007-04-18
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Aspects of q-discretized Nahm equations2007

    • Author(s)
      Atsushi Nakamula
    • Organizer
      The Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena : Computation and Theory, Oral presentation
    • Place of Presentation
      University of Georgia, Athens, USA
    • Year and Date
      2007-04-18
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Aspects of q-discretized Nahm equations2007

    • Author(s)
      Atsushi Nakamula
    • Organizer
      The Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
    • Place of Presentation
      Athens, GA University of Georgia
    • Year and Date
      2007-04-18
    • Related Report
      2007 Annual Research Report
  • [Presentation] CBS方程式の不連続解について2007

    • Author(s)
      中村 厚, 安藤 洋祐
    • Organizer
      日本物理学会2007年春季大会・口頭発表
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2007-03-18
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Discontinuous solutions to the CBS euuation2007

    • Author(s)
      Atsushi Nakamula, Yoske Ando
    • Organizer
      Spring Meeting of Physical Society of Japan, Oral presentation
    • Place of Presentation
      Kagoshima University
    • Year and Date
      2007-03-18
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Nahm方程式のq-離散化とその解析2006

    • Author(s)
      中村 厚, 鎌田 勝
    • Organizer
      九州大学応用力学研究所研究集会「非線形波動現象における基礎理論・数値計算および実験のクロスオーバー」・ポスター発表
    • Place of Presentation
      九州大学応用力学研究所
    • Year and Date
      2006-11-07
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] q-discretization of Nahm equations and its analysis2006

    • Author(s)
      Atsushi Nakamula, Masaru Kamata
    • Organizer
      Proceedings of Symposium on Crossover among Theoretical, Numerical and Experimental Studies on Nonlinear Waves, Poster presentation, Research Institute for Applied Mathematics
    • Place of Presentation
      Kvushvn Uniwrci
    • Year and Date
      2006-11-07
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On q-discretized Nahm equations2006

    • Author(s)
      Atsushi Nakamula
    • Organizer
      The 6th International Conference on Soliton and Integrable Systems, Oral presentation
    • Place of Presentation
      西北大学,西安,中華人民共和国
    • Year and Date
      2006-10-05
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On q-discretized Nahm equations2006

    • Author(s)
      Atsushi Nakamula
    • Organizer
      The 6th International Conference on Soliton and Integrable Systems, Oral presentation
    • Place of Presentation
      Northwest University, Xian, People's Republic of China
    • Year and Date
      2006-10-05
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] q-離散Nahm方程式の戸田型簡約II2006

    • Author(s)
      鎌田 勝, 中村 厚
    • Organizer
      日本物理学会2006年秋季大会・口頭発表
    • Place of Presentation
      千葉大学
    • Year and Date
      2006-09-24
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Reduction of q-discretized Nahm equations to a Toda-type system II2006

    • Author(s)
      Masaru Kamata, Atsushi Nakamula
    • Organizer
      Autumnal Meeting of Physical Society of Japan, Oral presentation
    • Place of Presentation
      Chiba University
    • Year and Date
      2006-09-24
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Integrability of q-discretized Nahm Equations2006

    • Author(s)
      M.Kamata and A.Nakamula
    • Organizer
      SIDE VII (The 7th International Meeting of Symmetries and Integrability of Difference Equations), Poster presentation
    • Place of Presentation
      Melbourne University,Melbourne,Australia
    • Year and Date
      2006-07-11
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] q-離散Nahm方程式の戸田型簡約2006

    • Author(s)
      中村 厚, 鎌田 勝
    • Organizer
      日本物理学会第61回年次大会・口頭発表
    • Place of Presentation
      愛媛大学
    • Year and Date
      2006-03-27
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The variant of ADHM construction associated with q-difference operators2005

    • Author(s)
      Atsushi Nakamula
    • Organizer
      京都大学基礎物理学研究所研究集会「場の量子論の基礎的諸問題と応用」・ポスター発表
    • Place of Presentation
      京都大学基礎物理学研究所
    • Year and Date
      2005-12-19
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The variant of ADHM construction associated with q-difference operators2005

    • Author(s)
      Atsushi Nakamula
    • Organizer
      Workshop on Fundamental Problems of Quantum Field Theory and its Application, oster presentation
    • Place of Presentation
      Research Institute for Fundamental Physics, Kyoto University
    • Year and Date
      2005-12-19
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] q-離散Nahm方程式の可積分性2005

    • Author(s)
      中村 厚, 鎌田 勝
    • Organizer
      九州大学応用力学研究所研究集会「非線形波動および非線形力学系の現象と数理」・ポスター発表
    • Place of Presentation
      九州大学応用力学研究所
    • Year and Date
      2005-11-11
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Integrability of q-discretized Nahm equations2005

    • Author(s)
      Atsushi Nakamula, Masaru Kamata
    • Organizer
      Symposium on Fhenomena and Mathematics of Nonlinear Waves and Dynamical Systems, Posterpresentation, Research Institute for Applied Mathematics
    • Place of Presentation
      Kyushyu University
    • Year and Date
      2005-11-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] q-離散Nahm方程式の構成2005

    • Author(s)
      中村 厚, 鎌田 勝
    • Organizer
      日本物理学会2005年秋季大会・口頭発表
    • Place of Presentation
      同志社大学
    • Year and Date
      2005-09-19
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Construction ofq-discretized Nahm equations2005

    • Author(s)
      Masaru Kamata, Masaru Kamata
    • Organizer
      Autumnal Meeting of Physical Society of Japan, Oral presentation
    • Place of Presentation
      Doshishya University
    • Year and Date
      2005-09-09
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Integrability of q-discretized Nahm Equations2005

    • Author(s)
      Masaru Kamata, Atsushi Nakamula
    • Organizer
      SIDE VII(The 7' International Meeting of Symmetries and Integrability of Difference Equations), Poster presentation
    • Place of Presentation
      Melbourne University, Melbourne, Australia
    • Year and Date
      2005-07-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Reduction of q-discretized Nahm equations to a Toda-type system2005

    • Author(s)
      Atsushi Nakamula, Masaru Kamata
    • Organizer
      The 61'h Annual Meeting of Physical Society of Japan, Oral presentation
    • Place of Presentation
      Ehime University
    • Year and Date
      2005-03-27
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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

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