New extensions in topological field theory with instanton effects

In this thesis, we propose two extensions of a topological field theory. One is a construction of new observables. The other is a perturbation theory around a special point of the theory. First, we construct the new observables in the supersymmetric quantum mechanics on a Riemaniann manifold. The observables of this theory correspond to the differential forms on the instanton moduli space. In our case, this space is the space of the gradient trajectories of the Morse function on the manifold, which is a subspace of the space of paths with both endpoints fixed. We consruct such differential forms by the mothod of iterated integrals. We find that the resulting observables are sensitive to the information of the non-commutativity of the fundamental group of the moduli space. Second, we develop a proper method of a perturbation theory around a special limiting point of the topological field theory. It is known that in this special point, one can compute the correlation functions beyond the topological sector of the theory. This limiting point is characterized by the infinite value of a parameter λ of the theory. However, at this point λ = ∞, the theory becomes quite different from the original theory with a finte value of λ. To get desired information, we need to know the value of the correlation functions away from the point λ = ∞. We find that it can be achieved by a kind of perturbation theory around the point λ = ∞. This perturbation theory has properties different from the usual one for a quantum mechanics. We carry out the perturbation theory by the method of the resolvent. We find that the computation on the inifinite dimensional Hilbert space can be reduced to a finite dimensional matrix computation. After reviewing some basic properties of topological field theories, we discuss the two extensions above.報告番号: 甲26642 ; 学位授与年月日: 2011-03-24 ; 学位の種別: 課程博士 ; 学位の種類: 博士(学術) ; 学位記番号: 博総合第1059号 ; 研究科・専攻: 総合文化研究科広域科学専