2次元結び目とYang-Millsゲージ理論について

关于二维结和 Yang-Mills 规范理论

• 批准号: 20K22319
• 负责人: 谷口 正樹
• 金额: $ 1.83万
• 依托单位: Institute of Physical and Chemical Research
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Research Activity Start-up
• 财政年份: 2020
• 资助国家: 日本
• 起止时间: 2020-09-11 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20K22319/
• 关键词: 2次元結び目; Yang-Millsゲージ理論; 特異インスタントン; 2次元トーラス結び目; 結び目群; インスタントンFloer理論; 特異接続; トーラス結び目; Chern-Simons汎関数; インスタントンFloerホモロジー; 結び目Floerホモロジー; Khovanovホモロジー; 微分トポロジー; Seifert超曲面

本研究の研究目標は, 2次元結び目をYang-Millsゲージ理論を用いて研究することであった. 本研究課題の主結果であった論文「Seifert hypersurfaces of 2-knots and Chern-Simons functional」においてYang-Millsゲージ理論を用いて2次元結び目のSeifert超曲面と結び目補空間のSU(2)表現を結びつけるに至っていた. また, 前年度は, その議論を2次元トーラスがS^1×S^3にessentialに埋め込まれている場合に拡張できることを考察していた. また, 関連する議論として, 2次元トーラスとS^1×S^3のペアとしての無限巡回被覆空間を考察し, その上のYang-Millsインスタントンモジュライ空間のコンパクト性を考察し, 上記に述べた一部の結果はそのようなモジュライ空間から導かれることを考察した. また, Daemi-佐藤-Scaduto-井森-谷口の研究において特異インスタントン理論を用いて結び目の実数値不変量J(K)を導入し, そこからあるクラスの2次元トーラス結び目に対しても実数値不変量が拡張できることを考察した.これは, 2次元トーラス結び目のスライスが結び目Kである場合に, J(K)は, 2次元トーラス結び目になる, という考察を経て得られる. すなわち, スライスとなる結び目の取り方に依存しないことが分かる. このようにして2次元トーラス結び目の実数値不変量が新しく得られた.

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项目成果

Construction of equivariant singular knot Floer homology I

等变奇异结Floer同调I的构建

• 发表时间: 2023
• 作者: Hokuto Konno;Masaki Taniguchi;谷口正樹
• 通讯作者: 谷口正樹

Intantons, special cycles, and knot concordance I

整数、特殊循环和结索引 I

• 发表时间: 2022
• 作者: 谷口正樹

Instanton Floer theory and local equivalence II

Instanton Florer 理论和局部等价 II

• 发表时间: 2020
• 作者: Kosuke Mitarai ; Yasunari Suzuki ; Wataru Mizukami ; Yuya O. Nakagawa ; Keisuke Fujii;Masaki Taniguchi
• 通讯作者: Masaki Taniguchi

Concordance Invariants from Equivariant Singular Instanton Theory

等变奇异瞬时子理论的一致性不变量

• 发表时间: 2022
• 作者: R. Matsui;M. Ota;Y. Fukuda;Y. Sakawa and Y. Kishimoto;谷口正樹
• 通讯作者: 谷口正樹

Gauge theory and knotted 2-spheres in the 4-space

规范理论和 4 空间中的结 2 球体

• 发表时间: 2020
• 作者: Gao Yanlin;Okada Susumu;Masaki Taniguchi
• 通讯作者: Masaki Taniguchi