Geometry of Quantum Invariants of Knots and 3-Manifilds, and Its Applications to Physics

• Project/Area Number: 18740227
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: Naruto University of Education (2008); The University of Tokyo (2006-2007)
• Principal Investigator: HIKAMI Kazuhiro Naruto University of Education, 大学院・学校教育研究科, 准教授 (60262151)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥270,000); Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2006: ¥1,600,000 (Direct Cost: ¥1,600,000)
• Keywords: 数理物理 / 量子不変量 / 結び目 / 量子情報 / エントロピー / 量子ホール効果 / ジョーンズ多項式 / 擬テータ関数 / 双曲幾何 / 3次元多様体 / 保型形式 / 擬データ関数

Research Abstract

結び目・3次元多様体の量子不変量と双曲幾何学・保型形式との関連を考察した。保型性を用いて漸近展開を解析するとともに、量子ダイログ函数を用いて新しい不変量を構成し、その幾何学的な性質を解析した。また、量子不変量の手法を量子ホール系に適用して、エンタングルメント・エントロピーを厳密に計算した。その結果、エントロピーにおける粒子の量子次元の役割を明らかにした。

Research Products

• [Journal Article] Colored Jones Polynomial with Polynomial Growth (2008)
  • Author(s): K. Hikami and H. Murakami
  • Journal Title: Communications in Contemporary Mathematics 10 Suppl. 1 Pages: 815-834
• [Journal Article] Skein theory and topological quantum registers : braiding matrices and topological entanglement entropy of non-abelian quantum Hall states (2008)
  • Author(s): K. Hikami
  • Journal Title: Annals of Physics 323 Pages: 1829-1769
• [Journal Article] Duality of linking pairing in Arnold's singularities (2008)
  • Author(s): K. Hikami
  • Journal Title: Proceedings of Japan Academy 84 Series A Pages: 81-86
  • NAID: 40016292872
• [Journal Article] Duality of Linking pairing in Arnold's singularities (2008)
  • Author(s): K. Hikami
  • Journal Title: Proceedings of Japan Academy Series A 84 Pages: 81-86
  • NAID: 40016292872
  • Peer Reviewed
• [Journal Article] Skein theory and topological quantum registers: braidin g matrices and topological entanglement entropy (2008)
  • Author(s): K. Hikami
  • Journal Title: Annals of Physics 323 Pages: 1729-1769
  • Peer Reviewed
• [Journal Article] Colored Jones polynomial with polynomial growth (2008)
  • Author(s): K. Hikami, H. Murakami
  • Journal Title: Communications in Contemporary Mathematirs 10 Suppl ement 1 Pages: 815-834
  • Peer Reviewed
• [Journal Article] Generalized volume conjecture and the A-polynomial-the Neumann-Zagier potential function as a classical limit of partition function (2007)
  • Author(s): K. Hikami
  • Journal Title: Journal of Geometry and Physics 57 Pages: 1895-1940
• [Journal Article] Hecke type formula for unified Witten-Reshetikhin-Turaev invariant as higher order mock theta functions (2007)
  • Author(s): K. Hikami
  • Journal Title: International Mathematical Research Notices 2007 rnm022 Pages: 32-32
• [Journal Article] Asymptotics of the colored Jones polynomial and the A-polynomial (2007)
  • Author(s): K. Hikami
  • Journal Title: Nuclear Physics B 773 Pages: 184-202
• [Journal Article] Generalized Volume Conjecture and the A-Polynomials: the Neumann-Zagier Potential Function as a Classical Limit of Partition Function (2007)
  • Author(s): K. Hikami
  • Journal Title: Journal of Geometry and Physics 57 Pages: 1895-1940
  • Peer Reviewed
• [Journal Article] Hecke Type Formula for Unified Witten-Reshetikhin-Turaev Invariant as Higher Order Mock Theta Functions (2007)
  • Author(s): K. Hikami
  • Journal Title: International Mathematical Research Notices 2007
  • Peer Reviewed
• [Journal Article] Boson-Fermion Duality in SU (m|n) Supersymmetric Haldane-Shastry S$ in Chain (2007)
  • Author(s): B. Basu-Mallick, N. Bondyopadhaya, K. Hikami, D. Sen
  • Journal Title: Nuclear Physics B 782 Pages: 276-295
  • Peer Reviewed
• [Journal Article] Quantum Invariants, Modular Forms, and Lattice Points II (2006)
  • Author(s): K.Hikami
  • Journal Title: Journal of Mathematical Physics 47
• [Journal Article] On the Quantum Invariants for the Spherical Seifert Manifolds (2006)
  • Author(s): K.Hikami
  • Journal Title: Communications in Mathematical Physics 268・2 Pages: 285-319
• [Journal Article] q-Series and L-functions related to half-derivatives of the Andrews-Gordon identity (2006)
  • Author(s): K.Hikami
  • Journal Title: The Ramanujan Journal 11 Pages: 175-197
• [Presentation] Around the Volume Conjedture (2009)
  • Author(s): K. Hikami
  • Organizer: Quantum Discrete Integrable Systems
  • Place of Presentation: Newton Institute, Cambridge
  • Year and Date: 2009-03-26
• [Presentation] Around the Volume Conjecture (2009)
  • Author(s): K. Hikami
  • Organizer: Workshop on Quantum Discrete Integrable Systems
  • Place of Presentation: Newton Institute
• [Presentation] From the Quantum Dilogarithm Function to the A-Polynomial (2008)
  • Author(s): K. Hikami
  • Organizer: Workshop on Geometry and Integrability
  • Place of Presentation: メルボルン(オーストラリア)
  • Year and Date: 2008-02-13
• [Presentation] From the quantum dilogarithm function to the A-polynomial, Workshop on Geometry and Integrability (2008)
  • Author(s): K. Hikami
  • Place of Presentation: Melbourne University
• [Presentation] 量子不変量と保型形式に関連した話題 (2006)
  • Author(s): 樋上和弘
  • Organizer: 日本数学会秋季総合分科会招待講演
  • Place of Presentation: 大阪市立大学