MATHEMATICAL PHYSICS,TOPOLOGY AND RELATED ALGEBRAIC STRUCTURE

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1.Construction of 3-manifold invariants derived from conformal field theory and its applicationsBased on Chern-Simons gauge theory, Witten proposed topological invariants of 3-manifolds. Several works have been done afterwards from geometric or combinatorial viewpoints. We constructed 3-manifold invariants based on representations of mapping class groups appearing in conformal field theory and Heegaard splitting of 3-manifolds.As an application, using the unitarity of the monodromy of conformal field theory, we obtained lower estimates for classical invariants, such as Heegaard genus and tunnel numbers of knots etc. Investigating the symmetry derived from Dynkin diagram automorphisms, we refined Witten invariant and established the level-rank duality.2.Graph complex and differential forms on knot spaceThe object of this research is differential forms on the space of all knots, which is an infinite dimensional space. We constructed a morphism from the graph complex to the de Rham complex on the knot space. This might be considered to be a generalization of the bar complex for the loop space. Especially, as the zero dimensional cohomology of the graph complex, the Vassiliev invariants can be represented by integrals appearing in Chern-Simons perturbation theory.Applying the de Rham homotopy theory to the pure braid group, we showed that the filtration derived from the Vassiliev invariants for pure braids coinsides with the lower central series. It turns out that the Vassiliev invariants are strong enough to distinguish any pure braid.