INTEGRABILITY AND UNIFORMIZATION IN LIOUVILLE THEORY - THE GEOMETRICAL ORIGIN OF QUANTIZED SYMMETRIES

We prove, in the classical limit of the Liouville theory, that the BPZ null vector decoupling equations admit a geometrical interpretation as uniformization equations. Using the Feigin-Fuchs representation of the Liouville theory we obtain a uniformization which gives rise to a propagator-vertex picture of the underlying Riemann surface. The space of solutions of these uniformization equations turns out to be the classical limit of the screened vertex operators and defines the spin-1/2 representation of the classical limit of SU(2)q+. This is the perturbative part of the full quantum group Q of the Liouville theory. In the strong coupling regime 1 < c < 25 the non perturbative quantum group is shown to be isomorphic to a quantum deformation of the four-dimensional Lorentz group.