  We construct a string field Hamiltonian describing the dynamics of open and
closed strings with effective target-space dimension $c\le 1 $. In order to do
so, we first derive the Dyson-Schwinger equations for the underlying large $N$
vector+matrix model and formulate them as a set of constraints satisfying
decoupled Virasoro and U(1) current algebras. The Hamiltonian consists of a
bulk and a boundary term having different scaling dimensions. The time
parameters corresponding to the two terms are interpreted from the the point of
view of the fractal geometry of the world surface.
