  In this paper we study the holographic dual, in several spacetime dimensions,
of the Higgs branch of gauge theories with fundamental matter. These theories
contain defects of various codimensionalities, where the matter fields are
located. In the holographic description the matter is added by considering
flavor brane probes in the supergravity backgrounds generated by color branes,
while the Higgs branch is obtained when the color and flavor branes recombine
with each other. We show that, generically, the holographic dual of the Higgs
phase is realized by means of the addition of extra flux on the flavor branes
and by choosing their appropriate embedding in the background geometry. This
suggests a dielectric interpretation in terms of the color branes, whose vacuum
solutions precisely match the F- and D-flatness conditions obtained on the
field theory side. We further compute the meson mass spectra in several cases
and show that when the defect added has codimension greater than zero it
becomes continuous and gapless.
