Topological invariants related to field theory

与场论相关的拓扑不变量

• 批准号: 06640111
• 负责人: KOHNO Toshitake
• 金额: $ 1.28万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06640111/
• 关键词: topological field theory; conformal field theory; braid group; moduli space; Chern-Simons theory; Witten invariants; Vassiliev invariants; Feynman diagrams; バシリエフ不変量; 3次元多様体; ファイマンダイアグラム; topological field theory; conformal field theory; braid group; moduli space; Chern-Simons theory; Witten invariants; Vassiliev invariants; Feynman diagrams; バシリエフ不変量; 3次元多様体; ファイマンダイアグラム

Applying techniques of field theory in mathematical physics, we establishes a general framework to extract topological invariants of manifolds from infinite dimensional data.In late 80's Witten proposed a method to define topological invariants of 3-manifolds as the partition function of the Chern-Simons functional defined over the space of connections on the manifold. We clarified the relation between the Chern-Simons theory for 3-manifolds with boundary and the two-dimensional conformal field theory. We formulated the conformal field theory as the theory of connections on vector bundles over the moduli space of Riemann surfaces and expressed the Witten invariants by the holonomy of the connection. Moreover, from the above point of view we obtained lower estimates for classical invariants for knots and 3-manifolds.The critical points of the Chern-Simons functional are flat connections and it is known that the perturbative expansion at flat connections are described by Feynman diagrams. We investigated topological invariants arising from such perturbative expansion from the viewpoint of integral geometry-integral of Green forms on the configuration space. Motivated by Chern-Simons perturbative theory for 3-manifolds with boundary, we studied the space of chord diagrams on Riemann surfaces and its quantization, together with the symplectic geometry of the moduli space of flat connections. In particular, in the case of the torus, we investigated the holonomy of the elliptic KZ connection and defined Vassiliev type invariants for knots in the product of the torus and the unit inverval.

在数学物理学中应用现场理论的技术，我们建立了一个通用框架，从无限尺寸数据中提取歧管的拓扑不变性。在80年代后期，Witten提出了一种方法，提出了一种定义3个manifolds的拓扑不变性的方法，将其作为Chern-Simons的分区功能，在该连接空间上均可在casticold overold overold of Castionold上定义。我们阐明了Chern-Simons理论的3个模具与边界和二维形式的保形场理论之间的关系。我们将共形场理论提出为矢量束的连接理论，上面的载体束上的模量空间，并通过连接的载体表达了witten不变性。此外，从上面的角度来看，我们获得了针对结和3个manifold的经典不变性的估计值。Chern-Simons功能的临界点是平坦的连接，众所周知，Feynman图描述了平坦连接处的扰动扩展。我们研究了从构型空间上绿色形式的积分几何形式的观点的角度来调查了这种扰动膨胀引起的拓扑不变性。由Chern-simons扰动理论的三个字体带有边界的扰动理论，我们研究了Riemann表面上的和弦图及其量化的空间，以及平面连接模量空间的符号几何形状。特别是，在圆环的情况下，我们研究了椭圆KZ连接的整体，并定义了vassiliev型不变式，用于圆环和单位inverval的产物中的结。