数学・物理学の様々な局面に現れるモジュラー・準モジュラー形式と多重ゼータ値の研究

2015 Fiscal Year Annual Research Report

• Project/Area Number: 23340010
• Research Institution: Kyushu University
• Principal Investigator: 金子 昌信 九州大学, 数理(科)学研究科(研究院), 教授 (70202017)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: モジュラー形式 / 準モジュラー形式 / 多重ゼータ値 / 共系場理論 / 位相不変量

Outline of Annual Research Achievements

坂田との共著論文において，高さが1の多重ゼータ値を，インデックスに1を含まないような多重ゼータ値（”maximal height”）のある対称的な和で表す公式を，いわゆる荒川金子ゼータ関数を用いて証明した． そこでは発散の正規化の議論を用いる．また，正規化版の和公式も証明した．ここにも高さが1の多重ゼータ値が現れる．さらに，同じ重さを持つmaximal height の多重ゼータ値全ての和の母関数が，2のみをインデックスにもつ等号付き多重ゼータ値と3のみをインデックスにもつ多重ゼータ値の母関数によって表されるという公式を発見，証明した．これはホフマン予想（ブラウンの定理）を思い出させる興味深いものであり，深さも動かしたときの2変数の母関数がどうなるか，など，更なる考察の余地があり発展が期待される．
また有家，永友，境との共著論文において，ある種の微分方程式を満たすモジュラー形式の理論が，ある型のアフィン頂点作用素代数の分類に応用が出来ることを示した． 具体的には，正整数レベルをもつアフィン頂点作用素代数で指標の次元が5以下のものをすべてリストアップし，その指標の空間の基底がモジュラー微分方程式の解で張られることを示した．一般には，指標はあるモジュラー微分方程式の解であることが知られているが，解全体が丁度指標の空間に一致する場合とそうでない場合がある．5次元以下の場合は常にそうであることを示したのが本論文であるが，6以上の状況について調べていくのが今後の課題である．
有限多重ゼータ値について，特にその「主予想」について，各地で多くの講演を行った．

Research Progress Status

27年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

27年度が最終年度であるため、記入しない。

Research Products

• [Int'l Joint Research] Max Planck Institute for Mathematics(Germany)
  • Country Name: Germany
  • Counterpart Institution: Max Planck Institute for Mathematics
• [Int'l Joint Research] Pohang Univ. of Science and Technology(韓国)
  • Country Name: KOREA (REP. OF KOREA)
  • Counterpart Institution: Pohang Univ. of Science and Technology
• [Journal Article] On multiple zeta values of extremal height (2016)
  • Author(s): Masanobu Kaneko, M. Sakata
  • Journal Title: Bull. Aust. Math. Soc. Volume: 印刷中 Pages: 印刷中
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Affine vertex operator algebras and modular linear differential equations (2016)
  • Author(s): Masanobu Kaneko, Y. Arike, K. Nagatomo, Y. Sakai
  • Journal Title: Letters in Mathematical Physics Volume: 印刷中 Pages: 印刷中
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Classification of vertex operator algebras with central charge 1/2 and -67/8 (2016)
  • Author(s): Kiyokazu Nagatomo
  • Journal Title: Proceeding of Japan Academy Ser. A Math. Sci. Volume: 92 Pages: 33-37
  • DOI: 10.3792/pjja.92.33
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Conformal field theories, modular functors and modular tensor categories (2015)
  • Author(s): Kiyokazu Nagatomo
  • Journal Title: Sugaku Expositions Volume: 28 Pages: 95-123
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Torus knots and quantum modular forms (2015)
  • Author(s): Kazuhiro Hikami, J. Lovejoy
  • Journal Title: Research in the Mathematical Sciences Volume: 2.2 Pages: 1-15
  • DOI: 10.1186/s40687-014-0016-3
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Braids, complex volume and cluster algebras (2015)
  • Author(s): Kazuhiro Hikami, R. Inoue
  • Journal Title: Algebraic & Geometric Topology Volume: 15 Pages: 2175-2194
  • DOI: 10.2140/agt.2015.15.2175
  • Peer Reviewed
• [Journal Article] Moduli spaces of Calabi-Yau complete intersections (2015)
  • Author(s): Shinobu Hosono
  • Journal Title: Nuclear Phys. B Volume: 898 Pages: 661-666
  • DOI: http://dx.doi.org/10.1016/j.nuclphysb.2015.04.001
  • Peer Reviewed / Open Access
• [Journal Article] On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots (2015)
  • Author(s): Toshie Takata, T. Ohtsuki
  • Journal Title: Geometry and Topology Volume: 19 Pages: 853-952
  • DOI: 10.2140/gt.2015.19.853
  • Peer Reviewed
• [Presentation] Quantu modular form (2016)
  • Author(s): Kazuhiro Hikami
  • Organizer: Number Theory Seminar
  • Place of Presentation: Pohang University of Science and Technology, Korea
  • Year and Date: 2016-03-16 – 2016-03-16
  • Int'l Joint Research / Invited
• [Presentation] Quantum dilogarithm and quantum invariant (2016)
  • Author(s): Kazuhiro Hikami
  • Organizer: Quantum Topology Seminar
  • Place of Presentation: Pohang University of Science and Technology, Korea
  • Year and Date: 2016-03-15 – 2016-03-15
  • Int'l Joint Research / Invited
• [Presentation] Finite multiple zeta values (2016)
  • Author(s): Masanobu Kaneko
  • Organizer: Diophantine Analysis and Related Topics
  • Place of Presentation: School of Mathematics and Statistics Wuhan University, China
  • Year and Date: 2016-03-10 – 2016-03-10
  • Int'l Joint Research / Invited
• [Presentation] Conifold transitions in mirror symmetry of CICYs (2016)
  • Author(s): Shinobu Hosono
  • Organizer: Birs workshop "Homological Mirror Geometry"
  • Place of Presentation: Banff International Research Station, Canada
  • Year and Date: 2016-03-08 – 2016-03-08
  • Int'l Joint Research / Invited
• [Presentation] 有限多重ゼータ値について (2016)
  • Author(s): Masanobu Kaneko
  • Organizer: 第9回多重ゼータ研究集会（第29回関西多重ゼータ研究会と共同開催）
  • Place of Presentation: 九州大学IMI オーディトリアム
  • Year and Date: 2016-02-22 – 2016-02-22
  • Invited
• [Presentation] ガロア表現の定義体について (2016)
  • Author(s): Yuichiro Taguchi
  • Organizer: 第134回日本数学会九州支部例会
  • Place of Presentation: 九州大学 IMI オーディトリアム
  • Year and Date: 2016-02-13 – 2016-02-13
  • Invited
• [Presentation] Characterization of the minimal series of Virasoro vertex operator algebras (2016)
  • Author(s): Kiyokazu Nagatomo
  • Organizer: Vertex Algebra and Quantum Groups
  • Place of Presentation: Banff Center, Canada
  • Year and Date: 2016-02-07 – 2016-02-12
  • Int'l Joint Research / Invited
• [Presentation] Quantum modular forms and quantum invariants (2015)
  • Author(s): Kazuhiro Hikami
  • Organizer: 九州大学代数学セミナー
  • Place of Presentation: 九州大学数理学研究院
  • Year and Date: 2015-12-11 – 2015-12-11
  • Invited
• [Presentation] 有限多重ゼータ値 (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: 京都大学数学談話会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-12-09 – 2015-12-09
  • Invited
• [Presentation] On regularizations of multiple zeta values (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: Zeta functions of several variables and applications
  • Place of Presentation: 名古屋大学多元数理科学研究科
  • Year and Date: 2015-11-11 – 2015-11-11
  • Int'l Joint Research / Invited
• [Presentation] Finite multiple zeta values (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: Seminaire de Theorie des Nombres
  • Place of Presentation: Institut Fourier, France
  • Year and Date: 2015-09-17 – 2015-09-17
  • Int'l Joint Research / Invited
• [Presentation] Looking geometry from the moduli spaces of CICYs (2015)
  • Author(s): Shinobu Hosono
  • Organizer: Russia-Japanese conference "Categorical and analytic invariants in Algebraic geometry 1
  • Place of Presentation: Steklov Mathematical Institute in Moscow, Russia
  • Year and Date: 2015-09-16 – 2015-09-16
  • Int'l Joint Research / Invited
• [Presentation] Poly-Bernoulli numbers, multiple zeta values, and related zeta functions (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: French-Japanese Workshop on multiple zeta functions and applications
  • Place of Presentation: Universite Jean Monnet, Saint-Etienne, France
  • Year and Date: 2015-09-08 – 2015-09-08
  • Int'l Joint Research / Invited
• [Presentation] 4th order modular linear differential equations and minimal models (2015)
  • Author(s): Yusuke Arike
  • Organizer: Vertex Operator Algebras and related Topics
  • Place of Presentation: Sichuan University, China
  • Year and Date: 2015-09-07 – 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] Logarithmic invariant of knots and its applications (2015)
  • Author(s): Jun Murakami
  • Organizer: Braids, Configuration Spaces, and Quantum Topology
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2015-09-07 – 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] On finite multiple zeta values (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: Seminaire de Theorie des Nombres de Caen
  • Place of Presentation: Laboratoire de Mathmatiques, Universite de Caen、France
  • Year and Date: 2015-09-04 – 2015-09-04
  • Int'l Joint Research / Invited
• [Presentation] On finite multiple zeta values (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: Seminaire de Groupe d’Etude sur les Problemes Diophantiens
  • Place of Presentation: Universite Paris 6 Pierre et Marie Curie-Jussieu, France
  • Year and Date: 2015-09-03 – 2015-09-03
  • Int'l Joint Research / Invited
• [Presentation] Galois 表現の Hecke 体について (2015)
  • Author(s): Yuichiro Taguchi
  • Organizer: 2015大分整数論研究集会
  • Place of Presentation: ホルトホール大分2階 サテライトキャンパスおおいた講義室
  • Year and Date: 2015-09-01 – 2015-09-01
  • Invited
• [Presentation] The slope conjecture for graph knots (2015)
  • Author(s): Toshie Takata
  • Organizer: 拡大KOOKセミナー2015
  • Place of Presentation: 神戸大学
  • Year and Date: 2015-08-19 – 2015-08-19
  • Invited
• [Presentation] Classification of vertex operator algebras with central charge 1/2 and -67/8 (2015)
  • Author(s): Kiyokazu Nagatomo
  • Organizer: Lie algebras, vertex operator algebras, and related topics
  • Place of Presentation: University of Notre Dame, USA
  • Year and Date: 2015-08-15 – 2015-08-15
  • Int'l Joint Research / Invited
• [Presentation] On the elliptic modular function j (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: Seminar on Algebra, Geometry and Physics
  • Place of Presentation: Max Planck Institute for Mathematics, Germany
  • Year and Date: 2015-08-11 – 2015-08-11
  • Int'l Joint Research / Invited
• [Presentation] On the Hecke Fields of Galois Representations (2015)
  • Author(s): Yuichiro Taguchi
  • Organizer: Minisymposium “Class Groups and Zeta Functions of Function Fields”, SIAM Conference on Applied Algebraic Geometry
  • Place of Presentation: NIMS, Daejeon, Korea
  • Year and Date: 2015-08-04 – 2015-08-04
  • Int'l Joint Research / Invited
• [Presentation] Logarithmic invariants of knots in a three manifold (2015)
  • Author(s): Jun Murakami
  • Organizer: New Developments in TQFT
  • Place of Presentation: Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Denmark
  • Year and Date: 2015-07-31 – 2015-07-31
  • Int'l Joint Research / Invited
• [Presentation] WAFOM with parameter for higher QMC: Revenge of the algebraic code, Part II (2015)
  • Author(s): Makoto Matsumoto
  • Organizer: 10th IMACS Seminar on Monte Carlo Methods
  • Place of Presentation: Johannes Kepler Universitat, Austria
  • Year and Date: 2015-07-06 – 2015-07-10
  • Int'l Joint Research
• [Presentation] Modular forms, 3rd order ordinary linear differential equations and affine vertex operator algebras (2015)
  • Author(s): Kiyokazu Nagatomo
  • Organizer: Representation XI
  • Place of Presentation: Dubrovnik, Croatia
  • Year and Date: 2015-06-21 – 2015-06-27
  • Int'l Joint Research / Invited
• [Presentation] On the elliptic modular j-function (2015)
  • Author(s): Masanobu Kaneko
  • Organizer: Modular forms workshop
  • Place of Presentation: University of Hawaii, USA
  • Year and Date: 2015-06-19 – 2015-06-19
  • Int'l Joint Research / Invited
• [Presentation] On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots (2015)
  • Author(s): Toshie Takata
  • Organizer: Intelligence of Low-dimensional Topology
  • Place of Presentation: 京都大学 数理解析研究所
  • Year and Date: 2015-05-22 – 2015-05-22
  • Int'l Joint Research / Invited
• [Remarks] 金子昌信 論文
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/papers.html