New Development Of W Strings And New Physical Effects Of Gauge Theory Of Gravity And Extra Dimensions

This thesis mainly investigate three problems: the spinor field realizations of W strings, conversation charges and conversation laws of Randall-Sundrum (RS) models and telepar-allel theory of gravity, and fermionic zero modes in gauge and self-dual vortex backgrounds on some compact extra dimensions.In first part, we focus our attention on the spinor field realizations of W_2,s strings and W_n strings. Firstly, the spinor field realizations of Virasoro algebra and W_2,s strings are studied, we generalize the grading Becchi-Rouet-Stora-Tyutin (BRST) method of the scalar field realizations of W_2,s strings to the spinor field realizations of W_2,s, strings, and assume that the BRST charges of W_2,s strings in the case of spinor field are also graded as the form of Qb = Q₀ + Q₁ Considering the special character of spinor field, we obtain the explicit constructions of the spinor field BRST charges for W_2,s strings. Secondly, according to the fact that the BRST charges of W_2,s strings are graded and commuted with each other, we generalize the grading BRST method of W_2,s strings to W_N strings. By introducing the ansatz of the spinor field realizations of W_N strings and assuming that the BRST charge of a general W_n string is totally graded, we give the spinor field constructions of W_N strings with the nilpotency conditions. Thirdly, considering the differences between critical W string and non-critical W string, we study the spinor field realizations of non-critical W_2,s strings. Compared with the case of critical W string, these realizations become more richer since the inclusion of matter currents. The BRST charges Q_b for the spinor non-critical W_2,s (s = 3,4) strings are obtained. Finally, using the method that W_2,s algebras can be linearized to W_1,2,s algebras by including a spin-1 current, we give the spinor field and ghost field realizations of the W_2,s strings based on the linear W_1,2,s algebras. This is the new international development of W strings.In second part, we generalize the general covariant conservation laws presented by Prof. Yi-Shi Duan, and investigate the energy-momentum and angular momentum conservationlaws for RS models and the energy-momentum conservation law for the teleparallel gravity theory. By the use of general Noether theorem and local 50(1,4) transformations, we obtain the conservation laws of energy-momentum and angular momentum for the RS models. The energy-momentum and angular momentum currents have superpotential and are therefore identically conserved. The Lagrangian density corresponding to the hidden and visible brane parts do not play a role in the total energy and angular momentum, but have an influence for the energy of gravity. For the solution that gives a flat induced metric on the branes, the momentum and the space-like components of the angular momentum vanish but the total energy of bulk and the non-space-like component of angular momentum are infinite. Most energy of gravity is located near the Planck brane, so the gravity on the Standard Model (SM) brane is very weak. Then, Using the same method, we obtain the energy-momentum conservation law for the teleparallel gravity theory. Basing on this conservation law, we calculate the total energy and momentum of the Friedmann-Robertson-Walker (FRW) universe, which includes contributions of matter and gravitational field. All calculated values vanish, and this is the result that the international theory requires.The third part of this thesis is devoted to discuss the fermionic zero modes in gauge and gravity backgrounds and fermions coupled with self-dual vortex background on some compact extra dimensions by the use of general covariant Dirac equation and the φ-mapping theory. Firstly, we study the fermionic zero modes and the effective Lagrangian for four dimensional Dirac spinor in gauge and gravity backgrounds taking a two dimensional compact manifold as extra dimensions. Two simple cases of these backgrounds are discussed. We get some new simple fermionic zero modes and effective Lagrangian on a torus and a sphere. The effective Lagrangian has a fixed form. Then, using the generalized Abelian Higgs model and φ-mapping theory, we investigate the self-dual vortex equation on an extra two-dimensional curved Riemann surface, and give the relation between gauge potential and Higgs filed. Based on this relation, the fermionic zero modes and the efiFective Lagrangian in self-dual vortex background on a torus and a sphere are discussed.