The grading Becchi-Rouet-Stora-Tyutin (BRST) method gives a way to construct the integer W2,s strings, where the BRST charge is written as QB= Q0 +Q1. Using this method, we reconstruct the nilpotent BRST charges Q0 for the integer W2,s strings and the half-integer W2,s strings. Then we construct the exact grading BRST charge with spinor fields and give the new realizations of the half-integer W2,s strings for the cases of s= 3/2, 5/2, and 7/2.

In this paper, we mainly investigate the WM2,s WL2,s system, in which the matter and the Liouville subsystems generate WM2,s and WL2,s algebras respectively. We first give a brief discussion of the physical states for corresponding W stings. The lower states are given by freezing the spin-2 and spin-s currents.Then, introducing two pairs of ghost-like fields, we give the realizations of W1,2,s algebras. Based on these linear realizations, BRST operators for W2,s algebras are obtained. Finally, we construct new BRST charges of Liouville system for WL2,s strings at the specific values of central charges c: c = −225/5 for WL2,3 algebra,c = −24 for WL2,4 algebra and c = −2,−286 /3 for WL2,6 algebra, at which the corresponding WL 2,s algebras are singular.

Motivated by the recent work on the fermion resonances on scalar-constructed thick branes (arXiv:0901.3543 and arXiv:0904.1785), we extend the idea to multi-scalar generated thick branes and complete previous work. The fermion localization and resonances on the three-field and two-field thick branes are investigated. With the Numerov method, our numerical results show that the resonance states also exist in the brane besides the single-field thick branes and the two-field thick branes in the former cases. This interesting phenomenon is related to the internal structure of the brane and the coupling of fermions and scalars. We find that the Kaluza-Klein chiral decomposition of massive fermion resonances is the parity-chiral decomposition. For the couplings k and k for three-field and two-field models with odd positive k,respectively, the number of the resonant states decreases with k. This result is opposite to the one obtained in the single-field de Sitter thick brane (arXiv:0904.1785).

The first order perturbations of the energy levels of a stationary hydrogen atom in static external gravitational field, with Schwarzschild metric, are investigated. The energy shifts are calculated for the relativistic 1S, 2S, 2P, 3S, 3P, 3D, 4S, 4P, 4D and 4F levels. The results show that the energy-level shifts of the states with total angular momentum quantum number 1/2 are all zero, and the ratio of absolute energy shifts with total angular momentum quantum number 5/2 is 1:4:5. This feature can be used to help us to distinguish the gravitational effect from other effect.

A three-parameter (positive odd integers, thickness factor, and asymmetry factora) family of asymmetric thick brane solutions in five dimensions were constructed from a two-parameter (s and) family of symmetric ones in [R. Guerrero, R.O. Rodriguez, and R. Torrealba, Phys. Rev. D72, 124012 (2005).]. The values s = 1 and s 3 correspond to single branes and double branes,respectively. These branes have very rich inner structure. In this paper, by presenting the massindependent potentials of Kaluza–Klein (KK) modes in the corresponding Schrodinger equations,we investigate the localization and mass spectra of fermions on the symmetric and asymmetric thick branes in an AdS background. In order to analyze the effect of gravity-fermion interaction (i.e., the effect of the inner structure of the branes) and scalar-fermion interaction to the spectrum of fermion KK modes, we consider three kinds of typical kink-fermion couplings. The spectra of left chiral fermions for these couplings are consisted of a bound zero mode and a series of gapless continuous massive KK modes, some discrete bound KK modes including zero mode (exist mass gaps) and a series of continuous massive KK modes, infinite discrete bound KK modes, respectively. The structures of the spectra are investigated in detail.