Iterated integrals, geometric structures of configuration spaces and applications to quantum topological invariants

• Project/Area Number: 23340014
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: Kohno Toshitake 東京大学, 数理(科)学研究科(研究院), 教授 (80144111)
• Co-Investigator(Kenkyū-buntansha): TERASOMA Tomohide 東京大学, 大学院数理科学研究科, 教授 (50192654)
• Co-Investigator(Renkei-kenkyūsha): SAITO Kyoji 東京大学, カブリ数物連携宇宙研究機構, 教授 (20012445) ; TERAO Hiroaki 北海道大学, 大学院理学研究院, 教授 (90119058) ; MURAKAMI Jun 早稲田大学, 理工学術院基礎理工学部, 教授 (90157751) ; TAMAKI Dai 信州大学, 理学部, 教授 (10252058)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥18,200,000 (Direct Cost: ¥14,000,000、Indirect Cost: ¥4,200,000); Fiscal Year 2014: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2013: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2012: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
• Keywords: 組みひも群 / 反復積分 / 量子群 / 配置空間 / 超幾何関数 / KZ方程式 / 共形場理論 / 高次圏 / 写像類群 / 超幾何積分 / モース理論 / 超平面配置 / 量子表現

Outline of Final Research Achievements

We clarified the relation between quantum representations of braid groups appearing as the monodromy representations of KZ equations and homological representations of braid groups. We gave an expression for the basis of the space of conformal blocks in conformal field theory on Riemann sphere by means of mlti-variable hypergeometric functions by specifying integration cycles. We showed that the KZ connection in conformal field theory can be regarded as a Gauss-Manin
connection. By developing the notion of Chen's formal homology connection and iterated integrals of logarithmic forms, we constructed higher category extensions of quantum representaitons of braid groups as representations of homotopy path groupoids of configuration spaces as higher categories.

Research Products

• [Int'l Joint Research] Nantes University(France)
• [Int'l Joint Research] University of California, Berkeley(米国)
• [Journal Article] Higher holonomy maps of formal homology connections and braid cobordisms (2016)
  • Author(s): T. Kohno
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] An example of Schwarz map of reducible Appell's hypergoemetric equation E_2 in two variables (2016)
  • Author(s): K. Matsumoto, T.Sasaki, T. Terasoma and M. Yoshida
  • Journal Title: Journal of Mathematical Society of Japan Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Degree formula for Grassmann bundles (2015)
  • Author(s): H. Kaji and T. Terasoma
  • Journal Title: J. Pure Appl. Algebra Volume: 219
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On Burau representations at roots of unity (2014)
  • Author(s): L. Funar and T. Kohno
  • Journal Title: Geometriae Dedicata Volume: 169 Issue: 1 Pages: 145-163
  • DOI: 10.1007/s10711-013-9847-0
  • Peer Reviewed
• [Journal Article] Free subgroups within the images of quantum representations (2014)
  • Author(s): L. Funar and T. Kohno
  • Journal Title: Forum Mathematicum Volume: 印刷中 Issue: 2 Pages: 337-355
  • DOI: 10.1515/form.2011.162
  • Peer Reviewed
• [Journal Article] Novikov homology, jump loci and Massey products (2014)
  • Author(s): T. Kohno and A. Pajitnov
  • Journal Title: Cent. Eur. J. Math. Volume: 12 Issue: 9 Pages: 1285-1304
  • DOI: 10.2478/s11533-014-0413-2
  • Peer Reviewed
• [Journal Article] All order alpha prime-expansion of superstring trees from the Drinfeld associator (2014)
  • Author(s): J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma
  • Journal Title: Physical Review D Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Circle-valued Morse theory for complex hyperplane arrangements (2013)
  • Author(s): T. Kohno, A. Pajitnov
  • Journal Title: Forum Mathematicum Volume: (Published on line) Issue: 4 Pages: 2113-2128
  • DOI: 10.1515/forum-2013-0032
  • Peer Reviewed
• [Journal Article] On Burau representations at roots of unity (2013)
  • Author(s): Louis Funar and Toshitake Kohno
  • Journal Title: Geometriae Dedicata Volume: --
  • Peer Reviewed
• [Journal Article] Homological representations of braid groups and KZ connections (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Journal of Singularities Volume: 5
  • Peer Reviewed
• [Journal Article] Quantum and homological representations of braid groups (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Edizioni della Normale Volume: 14
  • Peer Reviewed
• [Journal Article] Hyperplane arrangements, local system homology and iterated integrals (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 62
  • Peer Reviewed
• [Journal Article] Thomae type formula for K3 surfaces given by double covers of the projective plane branching along six lines (2012)
  • Author(s): Keiji Matsumoto and Tomohide Terasoma
  • Journal Title: J. Reine Angew. Math. Volume: 669
  • Peer Reviewed
• [Journal Article] Varieties of lines on Fermat hypersurfaces (2012)
  • Author(s): Tomohide Terasoma
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 62
  • Peer Reviewed
• [Journal Article] Hyperplane arrangements, local system homology and iterated integrals (2012)
  • Author(s): Toshitake Kohno
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 62 Pages: 157-174
  • Peer Reviewed
• [Journal Article] Free subgroups within the images of quantum representation (2011)
  • Author(s): Louis Funar, Toshitake Kohno
  • Journal Title: Forum Mathematicum Volume: 20(印刷中)
  • Peer Reviewed
• [Presentation] Holonomy of braids and its 2-category extension (2015)
  • Author(s): T. Kohno
  • Organizer: Braids, Configuration Spaces and Quantum Topology
  • Place of Presentation: 東京大学
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] A construction of an algebraic surface with a big higher Chow group (2015)
  • Author(s): T. Terasoma
  • Organizer: Mini-workshop on Calabi-Yau varieties; arithmetic, geometric and physics
  • Place of Presentation: 津田塾大学
  • Year and Date: 2015-08-05
  • Int'l Joint Research / Invited
• [Presentation] Holonomy of braids and its 2-category extension (2015)
  • Author(s): T. Kohno
  • Organizer: Integrability in Algebra, Geometry and Physics: New Trends
  • Place of Presentation: Congressi Stefano Franscini, Switzerland
  • Year and Date: 2015-07-13
  • Int'l Joint Research / Invited
• [Presentation] Conformal block bundles and Gauss-Manin connections (2014)
  • Author(s): T. Kohno
  • Organizer: 第９回代数・解析・幾何学セミナー
  • Place of Presentation: 鹿児島大学
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups (2013)
  • Author(s): T. Kohno
  • Organizer: Quantization of Teichmuller spaces
  • Place of Presentation: Erwin Schrodinger Institute, Vienna
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2013)
  • Author(s): T. Kohno
  • Organizer: Quantum Topology and Hyperbolic Geometry
  • Place of Presentation: Nha Trang, Vietnam
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2013)
  • Author(s): T. Kohno
  • Organizer: Joint International Meeting of American Mathematical Society and Romanian Mathematical Society
  • Place of Presentation: Alba-Iulia, Romania
  • Invited
• [Presentation] Quantum symmetries in homological representations of braid groups and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: The 6th Pacific RIM Conference on Mathematics
  • Place of Presentation: 札幌コンベンションセンター
  • Invited
• [Presentation] Monodromy groups of conformal field theory (2013)
  • Author(s): T. Kohno
  • Organizer: The 6th Pacific RIM Conference on Mathematics
  • Place of Presentation: 札幌コンベンションセンター
  • Invited
• [Presentation] Geometric representation theory of braid groups related quantum groups and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: Geometry and Dynamics
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Invited
• [Presentation] Braids, quantum symmetry and hypergeometric integrals (2013)
  • Author(s): T. Kohno
  • Organizer: 日本数学会秋季総合分科会，特別講演
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] Discriminantal arrangements and fusion rules in WZW model (2013)
  • Author(s): T. Kohno
  • Organizer: Hyperplane arrangements and characteristic classes
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Brown-Zagier relation for associators (2013)
  • Author(s): Tomohide Terasoma
  • Organizer: Arithmetic and Algebraic Geometry
  • Place of Presentation: The University of Tokyo
  • Invited
• [Presentation] Brown-Zagier relation for associators (2013)
  • Author(s): Tomohide Terasoma
  • Organizer: Research Project, Grothendieck-Teichmuller group, Deformation and Operad
  • Place of Presentation: Newton institute Cambridge
  • Invited
• [Presentation] Quantum and homological representations of braid groups (2012)
  • Author(s): 河野俊丈
  • Organizer: Low Dimensional Topology and Number Theory
  • Place of Presentation: 九州大学(福岡県)(招待講演)
  • Year and Date: 2012-03-12
• [Presentation] Deligne bar complexに関するcomparison copath (2012)
  • Author(s): 寺杣友秀
  • Organizer: 代数幾何とモチーフ理論ワークショップ
  • Place of Presentation: 東北大学(宮城県)(招待講演)
  • Year and Date: 2012-03-09
• [Presentation] Motif associated to fundamental group of universal elliptic curves and its regulator (2012)
  • Author(s): 寺杣友秀
  • Organizer: Arithmetic and Algebraic Geometry 2012
  • Place of Presentation: 東京大学(東京都)
  • Year and Date: 2012-02-17
• [Presentation] Quantum and homological representations of braid groups (2012)
  • Author(s): 河野俊丈
  • Organizer: East Asian School of Knots and Related Topics
  • Place of Presentation: KAIST(韓国)(招待講演)
  • Year and Date: 2012-01-09
• [Presentation] Quantum symmetries in homological representations of braid groups and applications (2012)
  • Author(s): Toshitake Kohno
  • Organizer: Geometric Group Theory
  • Place of Presentation: RIMS Kyoto University
  • Invited
• [Presentation] Homological representations of braid groups and KZ connections (2012)
  • Author(s): Toshitake Kohno
  • Organizer: 6th European Congress of Mathematics
  • Place of Presentation: Krakow, Poland
  • Invited
• [Presentation] Hodge realization of Bloch-Kriz mixed Tate motives (2012)
  • Author(s): Tomohide Terasoma
  • Organizer: 1st Vietnam-France Math Congress
  • Place of Presentation: Hue University, Vietnam
  • Invited
• [Presentation] Bar comodules associated to fundamental group of punctured elliptic curves (2011)
  • Author(s): 寺杣友秀
  • Organizer: Taiwan-Japan workshop on Arithmetic Algebraic Geometry and Related Topics
  • Place of Presentation: Academia Sinica (台湾)(招待講演)
  • Year and Date: 2011-11-19
• [Presentation] Relative regulator and automorphic L functions (2011)
  • Author(s): 寺杣友秀
  • Organizer: 代数幾何シンポジウム
  • Place of Presentation: 城崎大会議場(兵庫県)(招待講演)
  • Year and Date: 2011-10-27
• [Presentation] Homological representations of braid groups and KZ equations (2011)
  • Author(s): 河野俊丈
  • Organizer: 仙台シンポジウム
  • Place of Presentation: 東北大学(宮城県)(招待講演)
  • Year and Date: 2011-08-23
• [Presentation] Morse-Novikov theory for hyperplane arrangements (2011)
  • Author(s): 河野俊丈
  • Organizer: Hyperplane arrangements and applications
  • Place of Presentation: PIMS(カナダ)(招待講演)
  • Year and Date: 2011-08-12
• [Presentation] {Monodromy groups of conformal field theory (2011)
  • Author(s): 河野俊丈
  • Organizer: Singularity theory and its applications
  • Place of Presentation: 中国科学技術大学(招待講演)
  • Year and Date: 2011-07-27
• [Presentation] Reltaive regulator for Kuga fiber space and Beilinson conjecture (2011)
  • Author(s): 寺杣友秀
  • Organizer: Workshop on multiple zeta values, modular forms and mixed elliptic motives
  • Place of Presentation: University of Bristol(英国)(招待講演)
  • Year and Date: 2011-05-06
• [Presentation] Cohomological representation of Grothendieck Teichmuller group
  • Author(s): T. Terasoma
  • Organizer: Recent advances in Hodge theory: Period Domains, Algebraic cycles, and Arithmetic
  • Place of Presentation: UBC, Vancouver
  • Invited
• [Presentation] Hypergeometric identities and Brown-Zagier identities
  • Author(s): T. Terasoma
  • Organizer: 多重ゼータ値の諸相
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] A construction of surfaces with big Higher Chow group
  • Author(s): T. Terasoma
  • Organizer: Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Invited
• [Book] 結晶群 (2015)
  • Author(s): 河野俊丈
  • Total Pages: 200
  • Publisher: 共立出版
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/teacher/kohno.html
• [Remarks] Toshitake Kohno's Home Page
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks]
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Funded Workshop] Braids, Configuration Spaces and Quantum Topology (2015)
  • Place of Presentation: 東京大学大学院数理科学研究科大講義室
  • Year and Date: 2015-09-07