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Algebraic and geometric aspects of string dualities

Research Project

Project/Area Number 13640264
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 素粒子・核・宇宙線
Research InstitutionThe University of Tokyo

Principal Investigator

KATO Akishi  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (10211848)

Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsquantum field theory / string theory / supersymmetric gauge theory / duality / renormalization group / derived category / rational elliptic surface / moduli space / E-string / 弦双対性 / 位相的場の理論
Research Abstract

The renormalization of quantum field theories are commonly phased as "the procedure to extract finite results from divergent quntities by subtracting infinity." This obscures why one can get a reliable answer avoiding arbitrariness. Cannes and Kreimer proposed a Hopf algebra point of view to the renormalization. In their paper, however, it was not completely clear which part is assumption and which part is logical consequence. As a part of our project of understanding dualities algebraically, I started to clarify the relationship between Hopf algebra and theory of renormalization. In particular, I studied the renormalizability and the Birkhoff decomposition in detail using a toy model.
Discovery of string dualities enabled us to "geometrize" the various phenomena in physics, and provided a tool to analyze the dynamics beyond the reach of perturbative method. In particular, new type of fixed points were discovered in six dimensional N=(1,0) supersymmetric gauge theories with E8 global symmetry. F-theoretic interpretation of this critical point is the vanishing locus of codimension one rational elliptic surface in Calabi-Yau threefold. Conjecturally elementary excitations called "E-strings" will play an important role in the critical point. As has been successful in Seiberg-Witten theory, renormalization group flow can be seen as the deformation family of elliptic curves fibered over a moduli space. The duality group of the quantum field theory acts not only as covering transformations associated with the fibration, but also the autoequivalence of the derived categories on the surface. I investigated the relationship between the geometry of rational elliptic surfaces and the dynamics of the supersymmetric gauge theories, especially the partition functions of the topological gauge theories (E-strings). This work was done with Professors H.Awata (Nagoya), S.Kondo (Nagoya), Y.Saito (Tokyo), Y.Shimizu (ICU) and A.Tsuchiya (Nagoya).

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (15 results)

All 2004 2003 Other

All Journal Article (11 results) Publications (4 results)

  • [Journal Article] D-braneの安定性について2004

    • Author(s)
      加藤晃史
    • Journal Title

      日本物理学会 第59回年次大会 アブストラクト

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On stabilities of D-branes2004

    • Author(s)
      Akishi Kato
    • Journal Title

      The 59^<th> annual meetings of Japanese Physical Society

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] D-braneの安定性について2004

    • Author(s)
      加藤晃史
    • Journal Title

      日本物理学会 第59回年次大会

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 場の理論とトポロジー2003

    • Author(s)
      加藤晃史
    • Journal Title

      数理科学 480・6

      Pages: 49-55

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] 繰り込み・Hopf代数・行列模型2003

    • Author(s)
      加藤晃史
    • Journal Title

      非可換幾何 2003 秩父研究集会(代表 森吉仁志)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Topology and quantum field theories2003

    • Author(s)
      Akishi Kato
    • Journal Title

      Mathematical Sciences vol.480 no.6

      Pages: 48-53

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization, Hopf Algebra, and matrix models2003

    • Author(s)
      Akishi Kato
    • Journal Title

      Workshop on noncommutative geometry in Chichibu

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] 繰り込み・Hopf代数・行列模型2003

    • Author(s)
      加藤晃史
    • Journal Title

      非可換幾何2003秩父研究集会(代表 森吉仁志)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Renormalization and Hopf Algebra

    • Author(s)
      加藤晃史
    • Journal Title

      Proceedings of Kinosaki Workshop on Mirror Symmetry (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization and Hopf Algebra

    • Author(s)
      Akishi Kato
    • Journal Title

      Proceedings of Kinosaki Workshop on Mirror Symmetry (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Renormalization and Hopf Algebra

    • Author(s)
      加藤晃史
    • Journal Title

      Proceedings of Kinosaki Workshop on Mirror Symmetry (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] 加藤晃史: "場の理論とトポロジー"数理科学. 480・6. 49-55 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 加藤晃史: "CFT and Topological string amplitudes"日本物理学会 第59回年次大会. (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 加藤晃史: "繰り込み・Hopf代数・行列模型"非可換幾何2003秩父研究集会(代表 森吉仁志). (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 加藤晃史: "場の理論とトポロジー"数理科学. 480・6. (2003)

    • Related Report
      2002 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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