  The theory of a massless two-dimensional scalar field with a periodic
boundary interaction is considered. At a critical value of the period this
system defines a conformal field theory and can be re-expressed in terms of
free fermions, which provide a simple realization of a hidden $SU(2)$ symmetry
of the original theory. The partition function and the boundary $S$-matrix can
be computed exactly as a function of the strength of the boundary interaction.
We first consider open strings with one interacting and one Dirichlet boundary,
and then with two interacting boundaries. The latter corresponds to motion in a
periodic tachyon background, and the spectrum exhibits an interesting band
structure which interpolates between free propagation and tight binding as the
interaction strength is varied.
