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

TL;DR

This paper derives the exact string dynamics in conformally invariant AdS backgrounds, showing how the metric and antisymmetric tensor influence string behavior and reducing complex sectors to Liouville and wave equations.

Contribution

It provides a direct derivation of the reduction of WZWN models to Liouville theory based on classical string equations, clarifying the effects of metric and torsion on string dynamics.

Findings

Antisymmetric tensor causes repulsion, canceling dominant attractive metric effects.

Sinh-Gordon and cosh-Gordon sectors reduce to Liouville equations with different signs.

Large classical string size is affected by torsion, medium and small sizes are not.

Abstract

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 {\it 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 find illustrative classes of string solutions in the SL(2,R) background: dynamical closed as well as stationary open spiralling strings, for which the effect of torsion is somewhat like the effect of rotation in the metric. Similarly, the string solutions in the 2+1 BH-AdS background with torsion and angular momentum are fully analyzed.