  We continue the studies of localization of the U(1) gauge fields on domain
walls. Depending on dynamics of the bulk theory the gauge field localized on
the domain wall can be either in the Coulomb phase or squeezed into flux tubes
implying (Abelian) confinement of probe charges on the wall along the wall
surface. First, we consider a simple toy model with one flavor in the bulk at
weak coupling (a minimal model) realizing the latter scenario. We then suggest
a model presenting an extension of the Seiberg--Witten theory which is at
strong coupling, but all theoretical constructions are under full control if we
base our analysis on a dual effective action. Finally, we compare our findings
with the wall in a "nonminimal" theory with two distinct quark flavors that had
been studied previously. In this case the U(1) gauge field trapped on the wall
is exactly massless because it is the Goldstone boson of a U(1) symmetry in the
bulk spontaneously broken on the wall. The theory on the wall is in the Coulomb
phase. We explain why the mechanism of confinement discussed in the first part
of the paper does not work in this case, and strings are not formed on the
walls.
