  We study a $U(N)$-invariant vector+matrix chain with the color structure of a
lattice gauge theory with quarks and interpret it as a theory of open andclosed
strings with target space $\Z$. The string field theory is constructed as a
quasiclassical expansion for the Wilson loops and lines in this model. In a
particular parametrization this is a theory of two scalar massless fields
defined in the half-space $\{x\in \Z , \tau >0\} $. The extra dimension $\tau$
is related to the longitudinal mode of the strings. The topology-changing
string interactions are described by a local potential. The closed string
interaction is nonzero only at boundary $\tau =0$ while the open string
interaction falls exponentially with $\tau$.
