  We consider the field theory of $N$ massless bosons which are free except for
an interaction localized on the boundary of their 1+1 dimensional world. The
boundary action is the sum of two pieces: a periodic potential and a coupling
to a uniform abelian gauge field. Such models arise in open string theory and
dissipative quantum mechanics, and possibly in edge state tunneling in the
fractional quantized Hall effect. We explicitly show that conformal invariance
is unbroken for certain special choices of the gauge field and the periodic
potential. These special cases are naturally indexed by semi-simple, simply
laced Lie algebras. For each such algebra, we have a discrete series of
conformally invariant theories where the potential and gauge field are
conveniently given in terms of the weight lattice of the algebra. We compute
the exact boundary state for these theories, which explicitly shows the group
structure. The partition function and correlation functions are easily computed
using the boundary state result.
