  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 ${\bf k}$ along
the domain wall. They are characterized by a conserved current $J_\mu = (Q,{\bf
J})$, and are classified into magnetic ($J^2 < 0$), 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 ${\bf E}$. We also
find that $(Q, {\bf J}, \omega, {\bf k})$ is dual to $(B, {\bf E}, H, {\bf D})$
with $H$ and ${\bf D}$ being a magnetizing field and a displacement field,
respectively.

High Energy Physics - Theory