权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Topological invariants related to field theory

与场论相关的拓扑不变量

基本信息

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

项目摘要

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.

应用数学物理中的场论技巧，建立了从无穷维数据中提取流形拓扑不变量的一般框架，维滕在80年代后期提出了将三维流形的拓扑不变量定义为流形上连通空间上的Chern-Simons泛函的配分函数的方法。阐明了三维有边流形的Chern-Simons理论与二维共形场论之间的关系。我们将共形场论表述为黎曼曲面模空间上向量丛的联络理论，并用联络的完整性来表示维滕不变量。此外，从上述观点出发，我们得到了纽结和三维流形的经典不变量的下界估计。Chern-Simons泛函的临界点是平坦联络，并且已知平坦联络处的微扰展开可用Feynman图描述。我们从位形空间上的积分几何--绿色形式积分的观点研究了这种微扰展开所产生的拓扑不变量。受Chern-Simons微扰理论的启发，我们研究了Riemann曲面上的弦图空间及其量子化，以及平坦联络模空间的辛几何.特别是，在环面的情况下，我们研究了椭圆KZ连接的完整性，并定义了Vassiliev型不变量的环面和单位逆的产品中的结。