Analytical Solution of the Sl(2,r)/u(1) Wznw Black Hole Model

The gauged SL(2,R)/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. We have found a Lax pair representation for the non-linear equations of motion, and a Bäcklund transformation. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory, and its fundamental solutions describe the general solution of the WZNW model as well. The physical and free fields are related by non-local transformations. The (anti-)chiral component of the energymomentum tensor which factorizes into conserved quantities satisfying a non-linear algebra characteristic for parafermions assumes the canonical free field form. So the black hole model is expected to be prepared for an exact canonical quantization. Witten has shown [1] that the conformal SL(2,R)/U(1) gauged WZNW action [2] describes string propagation in the background of a two-dimensional space-time black hole. He conjectured that “the conformal field theory governing the black hole is more or less exactly soluble.” However, this theory is not conformal in the ordinary sense. It is true that it has classically a traceless energy-momentum tensor, but the β-function is non-zero [3, 1, 4, 5] as the gauged WZNW action corresponds to a σ-model with curvature. The inherent non-local structure of the theory might settle this question. But that requires, obviously, a complete analytical solution of the theory, and ∗E-mail: [email protected] ∗∗E-mail: [email protected]