Liouville theory and logarithmic solutions to Knizhnik-Zamolodchikov equation

TL;DR

This paper investigates logarithmic solutions to the SL(2,R)_k Knizhnik-Zamolodchikov equation, revealing their structure, singularities, and connections to Liouville theory, with implications for string scattering in AdS3 and the AdS/CFT correspondence.

Contribution

It introduces new logarithmic solutions to the KZ equation, classifies their singularities, and links them to Liouville theory and non-perturbative effects in AdS3 string theory.

Findings

Logarithmic solutions describe string scattering in AdS3.

Classification of singularities and their physical interpretation.

Relation between WZNW correlators and Liouville theory elucidated.

Abstract

We study a class of solutions to the SL(2,R)_k Knizhnik-Zamolodchikov equation. First, logarithmic solutions which represent four-point correlation functions describing string scattering processes on three-dimensional Anti-de Sitter space are discussed. These solutions satisfy the factorization ansatz and include logarithmic dependence on the SL(2,R)-isospin variables. Different types of logarithmic singularities arising are classified and the interpretation of these is discussed. The logarithms found here fit into the usual pattern of the structure of four-point function of other examples of AdS/CFT correspondence. Composite states arising in the intermediate channels can be identified as the phenomena responsible for the appearance of such singularities in the four-point correlation functions. In addition, logarithmic solutions which are related to non perturbative (finite k) effects are found. By means of the relation existing between four-point functions in Wess-Zumino-Novikov-Witten model formulated on SL(2,R) and certain five-point functions in Liouville quantum conformal field theory, we show how the reflection symmetry of Liouville theory induces particular Z_2 symmetry transformations on the WZNW correlators. This observation allows to find relations between different logarithmic solutions. This Liouville description also provides a natural explanation for the appearance of the logarithmic singularities in terms of the operator product expansion between degenerate and puncture fields.