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

2015 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

位相幾何学