Topological Structure Of Gauss-Bonnet-Chern Theorem And φ-mapping Topological Field Theory Of Higher-dimensional Knot-like Branes

In this dissertation, based on the φ-mapping topological current theory and the decomposition theory of SO(N) gauge potential proposed by Prof. Yi-Shi Duan, we investigate some problems in topological field theory. We study the singular topological structure of Gauss-Bonnent-Chern theorem, and constuct the topological field theory of branes and higher dimensional knot like branes. Moreover, we also discuss the topological quantum mechanics of two-component superconductor.Firstly, we discuss the singular topological structure of the Gauss-Bonnet-Chern theorem based on the Chern-Weil theory. It is found that the second Chern-Simions topological classes of Gauss-Bonnent-Chern form between the gauge potential and pure gauge potential are determined by the flat fields under the decomposition result of SO(N) gauge potential. This shows that the Gauss-Bonnet-Chern density may be expressed with the δ-function by making use of φ-mapping topological current theory. Then we offer an intrinsic theoretical framework to reveal the inner relationships among three theories for Eu-ler characteristic number including Gauss-Bonnet-Chern theorem, Hopf-Poincare theorem and Morse theory. Moreover, we consider the Gauss-Bonnet-Chern form imbedded in arbitrary higher-dimensional manifold, which suggests a Hodge dual tensor current. We show the brane structure inherent in the GBC tensor current and obtain the generalized Nambu action for the multi branes with quantized topological charge.Secondly, a φ-mapping topological field theory of higher-dimensional knots is presented. In terms of Chern-Weil theory and a new Chern-Weil-like theory, we construct a series of topological invariants of non-Abelian gauge field theory. They are evaluated to the generalized linking number of higher-dimensional knots. These topological invariants are the non-Abelian and higher-dimensional generalization of the well known helicity integral which measures the linkage of one-dimensional knots.Thirdly, the t' Hooft magnetic monopole theory is generalized. We investigate the generalized t' Hooft magnetic branes in even-dimensional space. Based on the Gauss-Bonnent-Chern theorem, we construct the missing arbitrary odd rank 't Hooft tensor, which universally determines the quantized low energy boundaries of the even-dimensional Georgi-Glashow models under asymptotic conditions. Then the dual magnetic branes theory is built up in terms of -mapping topological current theory. Furthermore, we discuss the nontrivial configuration of closed magnetic branes. We construct a series of topological invariants with non-Abelian symmetry in terms of the generalized t' Hooft tensors and low energy boundaries. Using the decomposition theory of gauge potential, they are evaluated as the generalized linking numbers of higher-dimensional knots with zero width.Finally, the momentum fields oi two-component Ginzburg-Landau superconductor is investigated. We formulize explicitly the inner structure of vortices, especially the co-center vortex lines on ground state and their thermal splitting, which induce the superconductivity carried by charged mode and superfluidity carried by the neutral mode, respectively. Moreover, the topological bifurcation theory of the thermal splitting is built up.