Zamolodchikov relations and Liouville hierarchy in SL(2,R)_k WZNW model

TL;DR

This paper explores the deep mathematical relationship between Zamolodchikov relations in Liouville theory and the SL(2,R)_k WZNW model, revealing a classical hierarchy and connections to the AdS_3/CFT_2 correspondence.

Contribution

It provides a new formulation of classical relations in SL(2,R)_k as a Liouville hierarchy using uniformizing Schwarzian operators and links solutions to representation theory.

Findings

Classical relations in SL(2,R)_k form a Liouville hierarchy.

Solutions span spin j representations of SL(2,R).

Connection established with Hamiltonian reduction to Liouville theory.

Abstract

We study the connection between Zamolodchikov operator-valued relations in Liouville field theory and in the SL(2,R)_k WZNW model. In particular, the classical relations in SL(2,R)_k can be formulated as a classical Liouville hierarchy in terms of the isotopic coordinates, and their covariance is easily understood in the framework of the AdS_3/CFT_2 correspondence. Conversely, we find a closed expression for the classical Liouville decoupling operators in terms of the so called uniformizing Schwarzian operators and show that the associated uniformizing parameter plays the same role as the isotopic coordinates in SL(2,R)_k. The solutions of the j-th classical decoupling equation in the WZNW model span a spin j reducible representation of SL(2,R). Likewise, we show that in Liouville theory solutions of the classical decoupling equations span spin j representations of SL(2,R), which is interpreted as the isometry group of the hyperbolic upper half-plane. We also discuss the connection with the Hamiltonian reduction of SL(2,R)_k WZNW model to Liouville theory.