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Conformal Field Theory and Witten Invariant

Research Project

Project/Area Number 15540064
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Institute of Technology

Principal Investigator

YOSHIDA Tomoyoshi  Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (60055324)

Co-Investigator(Kenkyū-buntansha) FUTAKI Akito  Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (90143247)
SHIGA Hiroshige  Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (10154189)
MURAKAMI Hitoshi  Tokyo Institute of Technology, Graduate School of Science and Engineering, Assistant Professor, 大学院・理工学研究科, 助教授 (70192771)
KOJIMA Sadayoshi  Tokyo Institute of Technology, Graduate School of Information Science, Professor, 大学院・情報理工学研究科, 教授 (90117705)
ENDO Hisaaki  Osaka University, Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (20323777)
北野 晃朗  東京工業大学, 大学院・情報理工学研究科, 助教授 (90272658)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2004: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2003: ¥2,000,000 (Direct Cost: ¥2,000,000)
KeywordsConformal field theory / 3-dimensional manfold / Witten invariant / Gauge theory / WZWモデル / 量子不変量 / ベクトル束のモジュライ空間
Research Abstract

We obtained an explicit expression of a base of the conformal block of $SU(2)$ WZW model in terms of the classical Riemann theta functions and theta constants as well as their transformation law. From these expression we can obtain the main results in the conformal field theory systematically, including the existence of a projectively flat connection on the bundle of the conformal blocks and a Hermitian product on the space of the conformal block which is compatible with the connection. With those results we carried out the abelianization of $SU(2)$ WZW model proposed by Atiyah and Hitchin around 1980. Using these result we defined vacuum vector for 3-dimensional handlebodies in the conformal block of $SU(2)$ WZW model and a series of topological invariants for oriented closed 3-manifolds with canonical framing in terms of the Hermitian product of the vacuum vectors. We can calcculate the asymptotic behavior of the invariants and we proved that the leading term of the asymptotic expansion can be expressed in terms of Chern-Simons invariant and Reidemeister torsion of 3-manifolds associated with $SU(2)$ representations of the fundamental groups.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (33 results)

All 2005 2004 2003 2001 Other

All Journal Article (27 results) Publications (6 results)

  • [Journal Article] Stability, integral invariants and canonical Kahler metrics2005

    • Author(s)
      A.Futaki
    • Journal Title

      DGA Proc.Prague (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] Asymptotic Chow semi-stability and integral invariants2004

    • Author(s)
      A.Futaki
    • Journal Title

      Internat.J.Math. Vol.15

      Pages: 967-979

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Dirichlet solutions on bordered Riemann surfaces and quasiconformal mappings2004

    • Author(s)
      H.Shiga
    • Journal Title

      J.D'Analyse Math. 92

      Pages: 117-135

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Asymptotic behaviors of the colored Jones polynomials of a torus knot2004

    • Author(s)
      H.Murakami
    • Journal Title

      Internat.J.Math. 15

      Pages: 547-555

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The dehn filling space of a certain hyperbolic 3-orbifold2004

    • Author(s)
      S.Kojima, S.Mizushima
    • Journal Title

      Contemporary Math. 347

      Pages: 131-140

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Asymptotic Chow semi-stability and integral invariants2004

    • Author(s)
      A.Futaki
    • Journal Title

      Internat.J.Math. Vol.15, No.9

      Pages: 967-979

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Dirichlet solutions on bordered Riemann surfaces and quasiconformal mappings.2004

    • Author(s)
      Shiga, H.
    • Journal Title

      J.D'Analyse Math. 92

      Pages: 117-135

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On holomorphic mappings of complex manifolds with ball model.2004

    • Author(s)
      Shiga, H.
    • Journal Title

      J.Math.Soc.Japan 56

      Pages: 1087-1108

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Asymptotic behaviors of the colored Jones polynomials of a torus knot2004

    • Author(s)
      H.Murakami
    • Journal Title

      Internat.J.Math. 15 No.6

      Pages: 547-555

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Some limits of the colored Jones polynomials of the figure-eight knot2004

    • Author(s)
      H.Murakami
    • Journal Title

      Kyungpook Math.J. 44 No.13

      Pages: 369-383

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Dehn filling space of a certain hyperbolic 3-orbifold2004

    • Author(s)
      S.Kojima, S.Mizushima
    • Journal Title

      Contemporary Math. 347

      Pages: 131-140

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Asymptotic Chow semi-stability and integral invariants2004

    • Author(s)
      A.Futaki
    • Journal Title

      Internat.J.Math. Vol15,No9

      Pages: 967-979

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Oehn filling space of a certain hyperbolic 3-orbifold2004

    • Author(s)
      S.Kojima, S.Mizushima
    • Journal Title

      Contemporary Math. 347

      Pages: 131-140

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Asymptotic behaviors of the colored Jones polynomials of a torus knot2004

    • Author(s)
      H.Murakami
    • Journal Title

      Internat.J.Math. Vol15,No.6

      Pages: 547-555

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Dirichlet solutions on bordered Riemann surfaces and q.e.mappings2004

    • Author(s)
      H.Shiga
    • Journal Title

      J.D'Analyse Math. 92

      Pages: 117-135

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a distance defined by the length spectrum on Teichmuller space.2003

    • Author(s)
      Shiga, H.
    • Journal Title

      Ann.Acad.Sci.Fenn.Math. 28

      Pages: 315-326

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Circle packings on surfaces with projective structures2003

    • Author(s)
      S.Kojima, S.Mizushima, S.P.Tan
    • Journal Title

      J.Differential Geom. 63

      Pages: 349-397

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On holomorphic families of rational maps : Finiteness, Rigidity and Stability2001

    • Author(s)
      Shiga, H.
    • Journal Title

      Kodai Math.J. 24

      Pages: 48-65

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An abelianization of SU(2) WZW model

    • Author(s)
      T.Yoshida
    • Journal Title

      Annals of Mathematics (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

    • Author(s)
      H.Endo, S.Nagami
    • Journal Title

      Transactions of the A.M.S. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An abelianization of SU(2) WZW model

    • Author(s)
      T.Yoshida
    • Journal Title

      Annals of Mathematics (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] Circle Packings on Surfaces with Projective Structures and Uniformization

    • Author(s)
      S.Kojima, S.Mizushima, S.P.Tan
    • Journal Title

      Pacific J.Math. (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] Circle Packings on Surfaces with Projective Structures : A survey

    • Author(s)
      S.Kojima, S.Mizushima, S.P.Tan
    • Journal Title

      LMS Lecture Notes (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] Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

    • Author(s)
      H.Endo, S.Nagami
    • Journal Title

      Transactions of the American Mathematical Society (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] Various aspects of degenerate families of Riemann surfaces, to appear in Sugaku Expositions, and fibered knots

    • Author(s)
      T.Ashikaga, H.Endo
    • Journal Title

      Comment.Math.Helv. (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] An abelianization of SU(2) WZW model

    • Author(s)
      T.Yoshida
    • Journal Title

      Annals of Mathematics (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Dirisibility of twisted Alexander polynomial and fibered knots

    • Author(s)
      T.Kitano, T.Morifuji
    • Journal Title

      Annali della Scoula normale superiore di Pisa (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] A.Futaki, T.Mabuchi: "Moment maps and bilinear forms associated with Symplectic classes"Asian J.Math. 6. 349-372 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Kojima, S.Mizushima, S.P.Tan: "Circle packings on surface with projective structures"J.Differential Geom.6. 63. 349-397 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Kojima, S.Mizushima: "The Dehn filling space of a certain hyperbolic 3-orbifold"Contemporary Math.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Murakami, J.Murakami: "Kashaev's conjecture and the Chern-Simons invariants of knots and links"Experiment Math.. 11. 427-435 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Kitano, T.Morifuji, M.Takasawa: "L^2-forsion invariants of a surface bundles"J.Math.Soc.Japan. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Endo, M.Korkmaz, D.Kotschick: "Commutators, Lefschetz fibrations and signature of surface bundles"Topology. 41. 961-977 (2002)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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