2021 Fiscal Year Final Research Report
Geometric quantum representations, iterated integrals and applications to topological field theory
| Project/Area Number |
16H03931
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Meiji University (2020-2021)
The University of Tokyo (2016-2019) |
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Principal Investigator |
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| 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)
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Keywords | 量子群 / 共形場理論 / 超幾何積分 / 反復積分 / 組みひも群 |
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| 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.
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| Free Research Field |
位相幾何学，数理物理
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| 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．
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