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Geometry of Quantum Dilogarithm Function

Research Project

Project/Area Number 24654041
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Global analysis
Research InstitutionKyushu University

Principal Investigator

HIKAMI Kazuhiro  九州大学, 数理(科)学研究科(研究院), 准教授 (60262151)

Co-Investigator(Kenkyū-buntansha) MURAKAMI Hitoshi  東北大学, 情報科学研究科, 教授 (70192771)
Co-Investigator(Renkei-kenkyūsha) Rei (INOUE Rei)  千葉大学, 理学研究科, 准教授 (30431901)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords数理物理 / 双曲幾何 / 二重対数関数 / 量子不変量 / 三角形分割 / トポロジー / 結び目 / クラスター代数
Outline of Final Research Achievements

The quantum dilogarithm function was originally introduced by Faddeev to study a discrete analogue of the KdV equation. Sometime ago Kashaev constructed knot invariant by use the quantum dilogarithm function, and he observed that its asymptotic is dominated by the hyperbolic volume of knot complement. Recently it has been realized that the quantum dilogarithm function plays a crucial role in the quantum cluster algebra.
We have used a technique of the quantum cluster algebra to reveal a relationship between the quantum dilogarithm function and the three-dimensional hyperbolic geometry. We constructed the quantum R-operator based on cluster mutation, and derived the Kashaev R-matrix by use of the quantum dilogarithm function.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (23 results)

All 2015 2014 2013 2012 Other

All Journal Article (9 results) (of which Peer Reviewed: 8 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (14 results) (of which Invited: 14 results)

  • [Journal Article] Braids, complex voltme, and cluster algebra2015

    • Author(s)
      Kazuhiro Hikami, Rei Inoue
    • Journal Title

      Algebraic and Geometric Topology

      Volume: 未定

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Torus knots and quantum modular forms2015

    • Author(s)
      K. Hikami and J. Lovejoy
    • Journal Title

      Research in the Mathematical Sciences

      Volume: 2 Issue: 1 Pages: 1-15

    • DOI

      10.1186/s40687-014-0016-3

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a twice-iterated torus knot2015

    • Author(s)
      H. Murakami
    • Journal Title

      Acta Math. Vietnam

      Volume: 39 Issue: 4 Pages: 649-710

    • DOI

      10.1007/s40306-014-0084-x

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Braiding operator via quantum cluster algebra2014

    • Author(s)
      Kazuhiro Hikami, Rei Inoue
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 47 Issue: 47 Pages: 1-21

    • DOI

      10.1088/1751-8113/47/47/474006

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots2014

    • Author(s)
      Kazuhiro Hikami and Rei Inoue
    • Journal Title

      J. Knot Theory and Its Ramifications

      Volume: 23 Issue: 01 Pages: 1450006-1450006

    • DOI

      10.1142/s0218216514500060

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Cluster algebra and complex volume2013

    • Author(s)
      Kazuhiro Hikami, Rei Inoue
    • Journal Title

      RIMS Kokyuroku

      Volume: 1866

    • Related Report
      2013 Research-status Report
  • [Journal Article] The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot2013

    • Author(s)
      Hitoshi Murakami
    • Journal Title

      J.Topology

      Volume: 6 Issue: 1 Pages: 193-216

    • DOI

      10.1112/jtopol/jts036

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] N=2 moonshine2012

    • Author(s)
      T. Eguchi, K. Hikami
    • Journal Title

      Phys.Lett. B

      Volume: 717 Issue: 1-3 Pages: 266-273

    • DOI

      10.1016/j.physletb.2012.09.037

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Twisted Elliptic Genus and Borcherds Product2012

    • Author(s)
      T. Eguchi, K. Hikami
    • Journal Title

      Lett. Math. Phys.

      Volume: 102 Issue: 2 Pages: 203-222

    • DOI

      10.1007/s11005-012-0569-2

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] モックテータ函数の周辺2015

    • Author(s)
      樋上和弘
    • Organizer
      東北大学 情報科学研究科 談話会
    • Place of Presentation
      東北大学
    • Year and Date
      2015-01-16
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Quantum Modular Form from Torus Knots2014

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Low-Dimensional Topology and Number Theory
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach, Germany
    • Year and Date
      2014-08-22
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a cable knot2014

    • Author(s)
      Hitoshi Murakami
    • Organizer
      Low-dimensional Topology and Number Theory
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach, Germany
    • Year and Date
      2014-08-20
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] (mock) modular form and quantum invariant2014

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Mock Modular Forms and Physics: Black Holes, Moonshine, and Conformal Field Theory
    • Place of Presentation
      Institute of Mathematical Sciences, India
    • Year and Date
      2014-04-15
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Hyperbolic geometry and dilogarithm2013

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Quantum Dilogarithm, Modular Double, and Representation Theory
    • Place of Presentation
      大阪市立大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Cluster algebra and complex volume

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Intelligence of Low-Dimensional Topology
    • Place of Presentation
      数理解析研究所、京都大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Cluster algebra and complex volume

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Topology Seminar at KIAS
    • Place of Presentation
      Korea Institute for Advanced Study
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] How much does the colored Jones polynomial of a knot know about representations of the fundamental group to SL(2; C)?

    • Author(s)
      Hitoshi Murakami
    • Organizer
      Quantum Topology and Hyperbolic Geometry
    • Place of Presentation
      Nha Trang, Vietnam
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Survey of the volume conjecture

    • Author(s)
      Hitoshi Murakami
    • Organizer
      Geometric topology in Cortona
    • Place of Presentation
      Cortona, Italy
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] The colored Jones polynomial of a knot and SL(2;C) representations of the fundamental group of its complement

    • Author(s)
      Hitoshi Murakami
    • Organizer
      Knots and Physics Workshop in Amsterdam
    • Place of Presentation
      University of Amsterdam, Netherlands
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] The colored Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of a knot

    • Author(s)
      H. Murakami
    • Organizer
      New Perspectives in Topological Field Theories
    • Place of Presentation
      Center for Mathematical Physics, Hamburg
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Cluster algebra and complex volume of 2-bridge knots

    • Author(s)
      K. Hikami
    • Organizer
      Facets of Integrability: Random Patterns, Stochastic Processes, Hydrodynamics, Gauge Theories and Condensed Matter Systems
    • Place of Presentation
      Simons Center for Geometry and Physics, USA
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] On the complex volume of knots

    • Author(s)
      K. Hikami
    • Organizer
      Bethe Ansatz, Quantum Groups and Beyond
    • Place of Presentation
      数理解析研究所、京都大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Cluster algebra and complex volume of 2-bridge knots

    • Author(s)
      R.Inoue
    • Organizer
      Low Dimensional Topology and Number Theory V
    • Place of Presentation
      Soft Research Park Center, Fukuoka
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2019-07-29  

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