Geometric quantum representations, iterated integrals and applications to topological field theory

• Project/Area Number: 16H03931
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Meiji University (2020-2021); The University of Tokyo (2016-2019)
• Principal Investigator: Kohno Toshitake 明治大学, 総合数理学部, 専任教授 (80144111)
• Co-Investigator(Kenkyū-buntansha): 加藤 晃史 東京大学, 大学院数理科学研究科, 准教授 (10211848) ; 逆井 卓也 東京大学, 大学院数理科学研究科, 准教授 (60451902)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥17,940,000 (Direct Cost: ¥13,800,000、Indirect Cost: ¥4,140,000); Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2019: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2018: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2017: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2016: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
• Keywords: 量子群 / 共形場理論 / 超幾何積分 / 反復積分 / 組みひも群 / 位相的場の理論 / 高次圏 / 配置空間 / KZ方程式 / 超幾何関数 / 幾何学 / モノロドミー表現

Outline of Final Research Achievements

Homological representations of braid groups are defined as the action of homeomorphisms of a punctured disk on the homology of an abelian covering of its configuration space. These representations were extensively studied by Krammer and Bigelow. I described a relation between homological representations of braid groups and the monodoromy representations of KZ connections based on solutions of the KZ equation expressed by hypergeometric integrals.I also studied the case of resonance at infinity appearing in conformal field theory and investigated the structureof integration cycles. In this case I described the symmetry by quantum groups at roots of unity. I showed that the KZ equation is represented as a differential equation satisfied by period integrals for certain algebraic varieties,and is expressed as a Gauss-Manin connection.

Academic Significance and Societal Importance of the Research Achievements

組みひも群，曲面の写像類群の幾何学的量子表現の理論をより深化させて，幾何学的群論などの問題にフィードバックしていくことが期待される．本研究によって，量子位相不変量についての幾何学的な理解が深まり，曲面結び目の不変量を構成するなどの成果を挙げ，低次元トポロジーの新しい研究手法がもたらされた．また離散群のユニタリ表現の組織的な構成方法が与えられ，これは量子計算の基礎に寄与するものである．

Research Products

• [Int'l Joint Research] University of California, Berkeley(米国)
• [Int'l Joint Research] University of Sydney(オーストラリア)
• [Int'l Joint Research] University of California, Berkeley(米国)
• [Int'l Joint Research] IRMA, Strasbourg(フランス)
• [Int'l Joint Research] ENS de Lyon(France)
• [Int'l Joint Research] Universty of California, Berkeley(米国)
• [Int'l Joint Research] KAIST/KIAS(韓国)
• [Journal Article] Higher holonomy and iterated integrals (2021)
  • Author(s): Toshitake Kohno
  • Journal Title: IRMA Lectures in Mathematics and Theoretical Physics Volume: 33 Pages: 307-325
  • Peer Reviewed
• [Journal Article] Formal connections, higher holonomy functors and iterated integrals (2021)
  • Author(s): Toshitake Kohno
  • Journal Title: Topology and Its Applications Volume: - Pages: 107985-107985
  • DOI: 10.1016/j.topol.2021.107985
  • Peer Reviewed
• [Journal Article] Torelli group, Johnson kernel and invariants of homology spheres (2020)
  • Author(s): Shigeyuki Morita, Takuya Sakasai and Masaaki Suzuki
  • Journal Title: Quantum Topology Volume: 11 Issue: 2 Pages: 379-410
  • DOI: 10.4171/qt/138
  • Peer Reviewed
• [Journal Article] Morita's trace maps on the group of homology cobordisms (2020)
  • Author(s): T. Sakasai, S. Morita and M. Suzuki
  • Journal Title: Quantum Topology Volume: 未定
  • Peer Reviewed
• [Journal Article] Morita's trace maps on the group of homology cobordisms (2019)
  • Author(s): Gwenael Massuyeau, Takuya Sakasai
  • Journal Title: Journal of Topology and Analysis Volume: 未定 Issue: 03 Pages: 775-818
  • DOI: 10.1142/s179352531950064x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Higher holonomies for hyperplane arrangements (2019)
  • Author(s): Toshitake Kohno
  • Journal Title: European Journal of Mathematics Volume: 1 Issue: 3 Pages: 1-23
  • DOI: 10.1007/s40879-019-00382-z
  • Peer Reviewed
• [Journal Article] Homology of Artin groups: A combinatorial group theoretic approach (2018)
  • Author(s): Ye Liu
  • Journal Title: Oberwolfach Report Volume: 印刷中 Issue: 1 Pages: 43-123
  • DOI: 10.4171/owr/2018/2
  • Peer Reviewed
• [Journal Article] 曲面の写像類群と自由群の外部自己同型群のコホモロジーに関するこれまでの歩みと最近の進展 (2018)
  • Author(s): 逆井卓也
  • Journal Title: 数学（日本数学会） Volume: 70-3
  • NAID: 130007880390
  • Peer Reviewed
• [Journal Article] Quantum representations of braid groups and holonomy Lie algebras (2017)
  • Author(s): Toshitake Kohno
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 72 Pages: 117-144
  • Peer Reviewed
• [Journal Article] Homological Representations of Braid Groups and the Space of Conformal Blocks (2017)
  • Author(s): Toshitake Kohno
  • Journal Title: Springer INdAM Series Volume: 19 Pages: 427-442
  • DOI: 10.1007/978-3-319-58971-8_16
  • ISBN: 9783319589701, 9783319589718
  • Peer Reviewed
• [Journal Article] Symmetry of symplectic derivation Lie algebras of free Lie algebras (2017)
  • Author(s): Takuya Sakasai, Shigeyuki Morita, Masaaki Suzuki
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B66 Pages: 185-193
  • Peer Reviewed
• [Journal Article] Johnson-Morita theory in mapping class groups and monoids of homology cobordisms of surfaces (2017)
  • Author(s): Takuya Sakasai
  • Journal Title: Winter Braids Lecture Notes Volume: 3 Pages: 1-25
  • Peer Reviewed
• [Journal Article] Quiver Mutation Sequences and $q$-Binomial Identities (2017)
  • Author(s): Kato Akishi、Mizuno Yuma、Terashima Yuji
  • Journal Title: International Mathematics Research Notices Volume: 108 Issue: 23 Pages: 1-24
  • DOI: 10.1093/imrn/rnx108
  • Peer Reviewed
• [Journal Article] An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra (2017)
  • Author(s): T. Sakasai, S. Morita and M. Suzuki
  • Journal Title: Experimental Mathematics Volume: 未定
  • Peer Reviewed
• [Journal Article] Higher holonomy maps of formal homology connections and braid cobordisms (2016)
  • Author(s): Toshitake Kohno
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 26 Issue: 12 Pages: 1642007-1642007
  • DOI: 10.1142/s0218216516420074
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Quantum computation and homological representations of braid groups (2021)
  • Author(s): Toshitake Kohno
  • Organizer: Workshop on "Computational Knot Theory" KAIST
  • Int'l Joint Research / Invited
• [Presentation] 数理模型から空間の幾何化定理を読み解く (2021)
  • Author(s): 河野俊丈
  • Organizer: 分野協働のための図学, 日本図学会
  • Invited
• [Presentation] Higher dimensional extensions of Johnson homomorphisms via bordism groups (2021)
  • Author(s): Takuya Sakasai
  • Organizer: Geometry of discrete groups and hyperbolic spaces
  • Int'l Joint Research / Invited
• [Presentation] On a group of Kim-Manturov (2021)
  • Author(s): Takuya Sakasai
  • Organizer: Workshop on Groups in Geometry
  • Int'l Joint Research / Invited
• [Presentation] Higher holonomy functors and iterated integrals (2020)
  • Author(s): Toshitake Kohno
  • Organizer: Homotopy Theory Symposium 2020
  • Int'l Joint Research / Invited
• [Presentation] Higher holonomy maps and iterated integrals (2019)
  • Author(s): Toshitake Kohno
  • Organizer: New trends in geometry and mathematical physics
  • Int'l Joint Research / Invited
• [Presentation] Mathematical forms - geometric models, lattices and crystals (2019)
  • Author(s): Toshitake Kohno
  • Organizer: Design Innovation from Nature Symposium, UC Berkeley
  • Int'l Joint Research / Invited
• [Presentation] Higher holonomy maps and iterated integrals (2019)
  • Author(s): Toshitake Kohno
  • Organizer: Japan-Australia Symposium "Hyperplane Arrangements and Singularities"
  • Int'l Joint Research / Invited
• [Presentation] Computations on Johnson homomorphisms and their applications (2019)
  • Author(s): Takuya Sakasai
  • Organizer: Computational Problems in Low-dimensional Topology II
  • Int'l Joint Research / Invited
• [Presentation] Computations in symplectic derivation Lie algebras and their applications (2019)
  • Author(s): Takuya Sakasai
  • Organizer: Graph complexes in algebraic geometry and topology, The University of Manchester
  • Int'l Joint Research / Invited
• [Presentation] Higher category extensions of holonomy maps and iterated integrals (2019)
  • Author(s): Toshitake Kohno
  • Organizer: Higher structures in algebra, geometry and quantum field theory, University of Hamburg
  • Int'l Joint Research / Invited
• [Presentation] Mathematical forms - collection of geometric models at the University of Tokyo (2019)
  • Author(s): Toshitake Kohno
  • Organizer: Tokyo-Berkeley Symposium "Forms in Nature and Art"
  • Int'l Joint Research / Invited
• [Presentation] Quantum symmetry in homological representations of braid groups (2019)
  • Author(s): Toshitake Kohno
  • Organizer: East Asian Conference on Geometric Topology
  • Int'l Joint Research / Invited
• [Presentation] 力学の変遷 ―古典・量子・弦― (2019)
  • Author(s): 加藤晃史
  • Organizer: 日本数学会 市民講演会
  • Invited
• [Presentation] Higher category extensions of KZ connections and representations of braid cobordisms (2018)
  • Author(s): Toshitake Kohno
  • Organizer: 101e rencontre entre mathematiciens et physiciens theoriciens, IRMA, Strasbourg
  • Int'l Joint Research / Invited
• [Presentation] グラフホモロジーとモジュライ空間のホモロジー群 (2018)
  • Author(s): 逆井卓也
  • Organizer: 種々の幾何学的構造と基本群に現れる様々な特性類とその不変量への応用
  • Invited
• [Presentation] Computations on Johnson homomorphisms (2018)
  • Author(s): Takuya Sakasai
  • Organizer: Cohomological study of mapping class groups and related topics, IRMA
  • Int'l Joint Research
• [Presentation] Higher category extensions of holonomy maps and representations of braid cobordims (2018)
  • Author(s): Toshitake Kohno
  • Organizer: The 13th Algebra-Analysis-Geometry Seminar
  • Int'l Joint Research / Invited
• [Presentation] Quiver mutation loops and partition q-series-Tropical geometry and related topics (2017)
  • Author(s): Akashi Kato
  • Organizer: 日本数学会年会無限可積分セッション特別講演
  • Place of Presentation: 首都大学東京
  • Year and Date: 2017-03-24
  • Invited
• [Presentation] Higher category extensions of holonomy mapsfor hyperplane arrangements (2017)
  • Author(s): Toshitake Kohno
  • Organizer: Arrangements and beyond: Combinatorics, geometry, topology and applications
  • Int'l Joint Research / Invited
• [Presentation] Representations of the category of braid cobordisms (2017)
  • Author(s): Toshitake Kohno
  • Organizer: Self-distributive system and quandle (co)homology theory in algebra and low-dimensional topology
  • Int'l Joint Research / Invited
• [Presentation] Local systems on configuration spaces, KZ connections and conformal blocks (2017)
  • Author(s): Toshitake Kohno
  • Organizer: Topology of Arrangements and Representation Stability, Oberwolfach
  • Int'l Joint Research / Invited
• [Presentation] Quiver mutation loops and partition q-series (2017)
  • Author(s): Akishi Kato
  • Organizer: リーマン面に関連する位相幾何学
  • Invited
• [Presentation] Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces (2017)
  • Author(s): Takuya Sakasai
  • Organizer: Geometric Topology Fair 2017, KAIST
  • Int'l Joint Research / Invited
• [Presentation] Johnson homomorphisms and symplectic representation theory (2017)
  • Author(s): Takuya Sakasai
  • Organizer: ジョンソン準同型とその周辺, 東京大学
  • Invited
• [Presentation] Invariants of homology cobordisms of surfaces (2017)
  • Author(s): Takuya Sakasai
  • Organizer: Tsuda University Topology Workshop
  • Invited
• [Presentation] 自由群の acyclic closure とその自己同型群について (2017)
  • Author(s): 逆井卓也
  • Organizer: ストリングトポロジーとその周辺
  • Invited
• [Presentation] Higher category extensions of holonomy maps for hyperplane arrangements (2016)
  • Author(s): Toshitake Kohno
  • Organizer: Summer Conference on Hyperplane Arrangements (SCHA) in Sapporo
  • Place of Presentation: Hokkaido University
  • Year and Date: 2016-08-08
  • Int'l Joint Research / Invited
• [Presentation] Higher category extensions of the holonomy maps of KZ connections (2016)
  • Author(s): Toshitake Kohno
  • Organizer: Workshop on Grothendieck-Teichmuller Theories
  • Place of Presentation: Chern Institute of Mathematics, Nankai University, Tianjin, China
  • Year and Date: 2016-07-24
  • Int'l Joint Research / Invited
• [Presentation] Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces (2016)
  • Author(s): Takuya Sakasai
  • Organizer: 離散群と双曲空間のトポロジーと解析
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Topological approaches to Mumford-Morita-Miller classes (2016)
  • Author(s): Takuya Sakasai
  • Organizer: MCM 2016
  • Place of Presentation: 沖縄科学技術大学院大学
  • Int'l Joint Research / Invited
• [Presentation] An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra (2016)
  • Author(s): Takuya Sakasai
  • Organizer: トポロジーとコンピュータ 2016
  • Place of Presentation: カレッジプラザ
  • Invited
• [Presentation] On the dihedral invariant Lie subalgebra of the associative symplectic derivation Lie algebra (2016)
  • Author(s): Takuya Sakasai
  • Organizer: GD2016---微分同相群と離散群
  • Place of Presentation: 文部科学省共済組合箱根宿泊所
  • Invited
• [Book] 曲率とトポロジー - 曲面の幾何から宇宙のかたちへ (2021)
  • Author(s): 河野俊丈
  • Total Pages: 315
  • Publisher: 東京大学出版会
  • ISBN: 9784130629287
• [Remarks] Toshitake Kohno's Homepage
  • URL: https://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks] Toshitake Kohno's Homepage
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Remarks] Toshitake Kohno's Home Page
  • URL: http://www.ms.u-tokyo.ac.jp/~kohno/
• [Funded Workshop] Japan-Australia Symposium "Hyperplan Arrangements and Singularities" (2019)
• [Funded Workshop] Interdisciplinary Workshop "Mathematics and Pharmaceutical Sciences" (2019)
• [Funded Workshop] Tokyo-Berkeley Data Science Boot Up Camp (2018)
• [Funded Workshop] Tokyo-Berkeley Symposium "Forms in Nature and Art" (2018)
• [Funded Workshop] The 12th East Asian School of Knots and Related Topics (2017)
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2017-02-13
• [Funded Workshop] Tokyo-Lyon Conference in Mathematics (2017)
• [Funded Workshop] Berkeley-Tokyo Summer School "Geometry, Representation Theory, and Mathematical Physics" (2017)