Topological invariants related to field theory

应用数学物理中的场论技术，建立了从无限维数据中提取流形拓扑不变量的一般框架。80年代末，Witten提出了一种将3-流形的拓扑不变量定义为流形上连接空间上的chen - simons泛函的配分函数的方法。阐明了具有边界的3-流形的chen - simons理论与二维共形场理论的关系。我们将共形场理论表述为黎曼曲面模空间上向量束上的连接理论，并用连接的完整性表示了Witten不变量。此外，从上述观点出发，我们得到了结点和3流形经典不变量的较低估计。chen - simons泛函的临界点是平面连接，已知平面连接处的微扰展开用费曼图来描述。从构型空间上格林形式的积分几何的角度研究了这种微扰展开引起的拓扑不变量。基于具有边界的3流形的chen - simons微扰理论，研究了Riemann曲面上弦图的空间及其量化，以及平面连接模空间的辛几何。特别地，在环面情况下，我们研究了椭圆KZ连接的完整性，并定义了环面与单位逆积中的结点的Vassiliev型不变量。

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.