J-kink domain walls and the DBI action

We study $J$-kink domain walls in $D=4$ massive $\mathbb{C}{P}^1$ sigma model. The domain walls are not static but stationary, since they rotate in an internal $S^1$ space with a frequency $\omega$ and a momentum $\mathbf{k}$ along the domain wall. They are characterized by a conserved current $J_\mu = (Q,{\bf J})$,andareclassifiedintomagnetic($J^2 < 0$),null($J^2=0$), and electric ($J^2>0$) types. Under a natural assumption that a low energy effective action of the domain wall is dual to the $D=4$ DBI action for a membrane, we are lead to a coincidence between the $J$-kink domain wall and the membrane with constant magnetic field $B$ and electric field $\mathbf{E}$. We also find that $(Q,\mathbf{J},\omega ,\mathbf{k})$ is dual to $(B,\mathbf{E},H,\mathbf{D})$ with $H$ and $\mathbf{D}$ being a magnetizing field and a displacement field, respectively.