Classical and quantal supersymmetric Liouville theory

The classical supersymmetric Liouville theory is shown to be invariant under the supersymmetric extension of the conformal group. Lax pair and Bäcklund transformations are derived and the general classical solution is obtained. The isotropy group for every solution is constant on the solution manifold and equal to an $N=1\mathrm{}$ conformal supersymmetry. For the quantum theory, the effective potential is computed exactly. The spectrum of the theory is continuous, bounded from below by zero, but no translationally invariant ground state exists. Translation invariance may be broken without the appearance of a Goldstone boson, and a consistent perturbation theory in the coupling constant is obtained. The constant $N=1\mathrm{}$ supersymmetry is also broken to a constant $N=\frac{1}{2}$ supersymmetry, and no Goldstone fermion arises. Space is spontaneously reduced to the half-line. The $N=1\mathrm{}$ conformal supersymmetry remains exact to all orders of perturbation theory.