Integral Geometry on the Lobachevsky Plane and the Conformal Wess-Zumino-Witten Model of Strings on an ADS3 Background

Abstract

The main purpose of the report is to provide some argumentation that three seemingly distinct approaches of 1. Giveon, Kutasov and Seiberg (hep-th/9806194); 2. Hemming, Keski-Vakkuri (hep-th/0110252); Maldacena, Ooguri (hep-th/0001053) and 3. I. Bars (hep-th/9503205) can be investigated by applying the mathematical methods of integral geometry on the Lobachevsky plane, developed previously by Gel'fand, Graev and Vilenkin. All these methods can be used for finding the transformations, leaving the Kac-Moody and Virasoro algebras invariant. The near-distance limit of the Conformal Field Theory of the SL(2, R) WZW model of strings on an ADS3 background can also be interpreted in terms of the Lobachevsky Geometry : the non - euclidean distance is conserved and the Lobachevsky formulae for the angle of parallelism is recovered. Some preliminary technique from integral geometry for inverting the modified integral representation for the Kac- Moody algebra has been demonstrated.