Topological Structure Of Hopf Invariant And Quantum Radiation Of Black Hole

In this dissertation, we have made in-depth researches on two topics, i.e. topological field theory and general relativity. In the field of topological field theory, in terms of Duan's topological current theory, we study the inner topological structure of Hopf invariant, which is an important topological invariant in differential geometry and topology, and obtain the exact expression for revealing the relationship between Hopf invariant and linking number of knots. In the field of general relativity and black hole physics, we study the Hawking radiation for a series of balck hole background spacetimes and FRW universe by using the methods of quantum tunneling and quantum anomaly, respectively.Firstly, based on Duan's topological current theory, the inner topological structure of Hopf invariant is studied, and we also discuss its applications in gauge field theory. For the simplest case, starting with the spinor representation of Hopf mapping, we derive two important results, which shows that Hopf invariant is actually the degree of Hopf mapping and Hopf invariant is also the linking number of knots. In the case of higher dimensional Hopf invariant, we firstly deduce a complicated expression fot it. Then, according to the definition of linking number for higher dimensional knots in the mathematical literatures, an explicit expression of linking number for higher dimensional knots is derived. Finally, after comparing the two expressions, the relationship between Hopf invariant and the linking number of higher dimensional knots is constructed. Based on these research work, we further study the topological origin of knot-like topological defects in Skyrme-Faddeev model, and explain that Hopf invariant is the proper topological invariant to describe the topology of these knot-like defects.Secondly, we study the scalar and fermonic particle's Hawking radiations in black hole spacetime backgrounds by using the mechanism of quantum tunneling. In terms of the null geodesic method and Hamilton-Jacobi method, we calculate the Hawking temperatures of the spherical symmetric GHS black hole, stationary rotating Kaluza-Klein black hole and Kerr-Sen black hole, respectively. Because the self-gravitational interaction of radiated scalar particles are not taken into account, the radiation spectrums are all pure thermal. We also investigate the fermionic particle's Hawking radiations from the BTZ black hole and Kerr black hole by using the tunneling formalism. Applying WKB approximation to the covariant Dirac equations, we obtain the radiation spectrums for fermions in these spacetime backgrounds.Thirdly, using the the mechanism of quantum anomaly cancellation, we study the Hawking radiation for scalar field in the rotating charged Godel black hole and the Hawk-ing radiation for fermionic field in the general stationary (2+1) dimensional black hole. By performing the dimensional reduction procedure for the action of scalar field in Godel black hole background and the action of fermionic field in (2+1) dimensional black hole back-ground, we find that the original field theories can be approximated by two dimensional effective theories, respectively. By solving the anomaly equation in the two dimensional spacetime background, we obtain the angular flux and energy-momentum flux satisfying the Hawking distribution.Finally, by using the fermion tunneling formalism and the method of gravitational anomaly, we study the Hawking radiation of apparent horizon in FRW universe, and derive the Hawking temperature associated with the apparent horizon of FRW universe, which is extensively applied in investigating the relationship between the first law of thermodynamics and Friedmann equations.