Topological invariants related to field theory

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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.