  Matrix string theory is equivalent to type IIA superstring theory in the
light-cone gauge, together with extra degrees of freedom representing D-brane
states. It is therefore the appropriate framework in which to study systems of
multiple fundamental strings expanding into higher-dimensional D-branes.
Starting from Matrix theory in a weakly curved background, we construct the
linear couplings of closed string fields to type IIA Matrix strings. As a
check, we show that at weak coupling the resulting action reproduces light-cone
gauge string theory in a weakly curved background. Further dualities give a
type IIB Matrix string theory and a type IIA theory of Matrix strings with
winding. We comment on the dielectric effect in each of these theories, giving
some explicit solutions describing fundamental strings expanding into various
Dp-branes.
