Reduction of the six-dimensional $q$-form fields to the four-dimensional fields by coupling with gravity

In this paper, we investigate the localization of various $q$-form fields on a codimension-two brane. In particular, the $0$-form scalar field, the $1$-form $U(1)$ gauge vector field, and the $2$-form Kalb-Ramond field are considered with gravitational coupling, where a coupling function $F(R)$ is introduced into the six-dimensional actions of these fields. The function $F(R)$ depends on the scalar curvature of the bulk. Within this framework, we find that the massless modes of different $q$-form fields can be localized on the thick brane for positive values of the coupling parameters $t_1$ and $t_2$. For the massive modes, the different $q$-form fields exhibit similar localization properties determined by the coupling parameter $t_2$. When $0<t_2<v^2\mathrm{/}24$, the massive modes of these fields cannot be localized on the brane, while they may exist as the resonant modes. In the case $t_2=v^2\mathrm{/}24$, a finite number of massive modes can be localized on the thick brane, and the number of localized modes increases with the coupling parameter $t_1$. Finally, when $t_2>v^2\mathrm{/}24$, the effective potentials associated with the Kaluza-Klein modes of these $q$-form fields form infinitely deep potential wells, so that all massive modes can be localized on the brane. Moreover, the tachyonic massive modes can always be excluded for different $q$-form fields.