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

TL;DR

This paper explores the application of integral geometry on the Lobachevsky plane to analyze various approaches in string theory on an ADS3 background, revealing geometric invariants and providing new mathematical tools for conformal field theory.

Contribution

It introduces a novel application of Lobachevsky geometry to investigate string theory models and their algebraic invariants, connecting integral geometry with conformal field theory.

Findings

Transformations leaving Kac-Moody and Virasoro algebras invariant identified

Lobachevsky geometry explains near-distance limits in SL(2,R) WZW models

Preliminary techniques for inverting integral representations of Kac-Moody algebra developed

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.