From the WZWN Model to the Liouville Equation: Exact String Dynamics in Conformally Invariant AdS Background

It has been known for some time that the SL(2,R) WZWN model reduces to Liouville theory. Here we give a direct and physical derivation of this result based on the classical string equations of motion and the proper string size. This allows us to extract precisely the physical effects of the metric and antisymmetric tensor, respectively, on the exact string dynamics in the SL(2,R) background. The general solution to the proper string size is also found. We show that the antisymmetric tensor (corresponding to conformal invariance) generally gives rise to repulsion, and it precisely cancels the dominant attractive term arising from the metric. Both the sinh-Gordon and the cosh-Gordon sectors of the string dynamics in non-conformally invariant AdS spacetime reduce here to the Liouville equation (with different signs of the potential), while the original Liouville sector reduces to the free wave equation. Only the very large classical string size is affected by the torsion. Medium and small size string behaviours are unchanged. We also give illustrative classes of string solutions in the SL(2,R) background: dynamical closed aa well as stationary open spiralling strings, for which the effect of torsion is somewhat like the effect of rotation in the metric. Department of Physics, University of Odense, Campusvej 55, 5230 Odense M, Denmark. Observatoire de Paris, DEMIRM. Laboratoire Associé au CNRS UA 336, Observatoire de Paris et École Normale Supérieure. 61, Avenue de l’Observatoire, 75014 Paris, France.