Localization of a spin-$1/2$ fermion on the braneworld is an important and interesting problem. It is well known that a five-dimensional free massless fermion $\mathrm{\Psi }$ minimally coupled to gravity cannot be localized on the Randall-Sundrum braneworld. In order to trap such a fermion, the coupling between the fermion and bulk scalar fields should be introduced. In this paper, localization and quasilocalization of a bulk fermion on the thick braneworld generated by two scalar fields (a kink scalar $\varphi$ and a dilaton scalar $\pi$) are investigated. Two types of couplings between the fermion and two scalars are considered. One coupling is the usual Yukawa coupling $-\eta \overline{\mathrm{\Psi }}\varphi \mathrm{\Psi }$ between the fermion and kink scalar, another one is $\lambda \overline{\mathrm{\Psi }}{\mathrm{\Gamma }}^M{\partial }_M\pi {\gamma }^5\mathrm{\Psi }$ between the fermion and dilaton scalar. The left-chiral fermion zero mode can be localized on the brane, and both the left- and right-chiral fermion massive Kaluza-Klein modes may be localized or quasilocalized. Hence the four-dimensional massless left-chiral fermion and massive Dirac fermions, whose lifetime is infinite or finite, can be obtained on the brane.

In this paper, we obtain a gauge invariant effective action for a bulk massless U(1) gauge vector field on a brane with codimension two by using a general Kaluza-Klein (KK) decomposition for the field. It suggests that there exist two types of scalar KK modes to keep the gauge invariance of the action for the massive vector KK modes. Both the vector and scalar KK modes can be massive. The masses of the vector KK modes m ⁽ⁿ⁾ contain two parts, m ₁ ^{( n)} and m ₂ ^{( n)} , due to the existence of the two extra dimensions. The masses of the two types of scalar KK modes m _ϕ ^{( n)} and m _φ ^{( n)} are related to the vector ones, i.e., m _ϕ ^{( n)} = m ₁ ^{( n)} and m _φ ^{( n)} = m ₂ ^{( n)} . Moreover, we derive two Schrödinger-like equations for the vector KK modes, for which the effective potentials are just the functions of the warp factor.

In this paper, writing the most general form $f(R)={\sum }_{i=1}^n{a}_i{R}^i+\mathrm{\Lambda }$ the solutions of the pure geometric thick $f(R)$ brane are investigated. For the certain value of $n$, the analytical thick brane solution can be calculated, and when $n=3$, $n=4$, and $n=10$, the thick brane solutions are presented. The solutions are stable against linear tensor perturbations. The zero mode of gravity and scalar field can be localized on thick $f(R)$ branes naturally. The zero mode of a vector field and left-chiral fermion can be localized on thick $f(R)$ branes by introducing the coupling with scalar curvature $R$ of spacetime, and the massive resonant modes can be quasilocalized on the brane with the large coupling coefficients.

We investigate the gauge invariance and localization of vector KK modes for a bulk U(1) gauge field under three kinds of localization mechanism on a brane with one extra dimension. By a general KK decomposition for the bulk U(1) gauge field, there are both vector and scalar KK modes on the brane, which couple with each other. We demonstrate that for a localization mechanism with a gauge invariant bulk action of the U(1) gauge field, the effective action of the KK modes on the brane can be formalized to gauge invariant form. However, only the massive vector KK modes and their accompanying scalar ones can be both localized on the brane, which depends on the solution of the brane, the gauge invariance of the massive vector field is finally preserved. For a localization mechanism with a broken gauge invariant bulk action of the field, it is impossible to rebuild the gauge invariance on the brane.

It is known that the scalar fields can be trapped on branes of different types, and the U ( 1 ) gauge vector fields can be localized on the thick de Sitter brane, or the thick Minkowski brane via coupling with gravity. The Kalb-Ramond fields can be localized on the Minkowski brane and the thick de-Sitter brane, with certain couplings. In this paper, with considering a coupling mechanism between the kinetic term of the q -form fields and the background spacetime, we investigate the localization of Kaluza-Klein modes for the q -form fields in D -dimensional spacetime. Concrete q -form fields are discussed within five-dimensional brane models with typical spacetime geometries: Minkowski, de Sitter, and anti–de Sitter. In the Minkowski brane case, the zero modes of various q -form fields can be localized on the brane. In the de Sitter brane case, the zero mode of the U ( 1 ) gauge vector fields can be localized on the brane. Lastly in the anti–de Sitter brane case, the zero mode of the Kalb-Ramond fields can be localized on the brane. For the massive Kaluza-Klein modes of these q -form fields, they could be localized or quasi-localized on the brane of different types. Besides, subtle and detailed behaviors of the Kaluza-Klein modes for q -form fields are observed: the zero modes could be localized on both sides of the brane, and the massive modes could be localized or quasilocalized at the brane position.