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

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 /CF T 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.

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…where the function W (x) encoding the Schwarzian equation (136) is a simple expression of the first two coefficients of the linear differential operator (see (56)). For order-two linear differential operators this Schwarzian condition turns out to be sufficient.…”

“…When dealing with linear differential operators, we have seen the emergence of Schwarzian derivatives, consequence of the fact that the Schwarzian derivative is appropriate for the composition of functions [19] (see the chain rule of the Schwarzian derivative of the composition of function). Do higher order Schwarzian derivatives [55,56,57,58] occur for pullback-symmetries of non-linear ODE's, or, more generally, for functional equations?…”

“…Remark 1: There is nothing specific with order-four linear differential operators, one has the same result for two operators of arbitrary orders N equal up to pullback and conjugation (see (G.25)): the expressions of W (M N , x) and W (L N , x) being the ones given in (56), (57). One also has W (M N , x) − W (L N , y(x)) · y ′ (x) 2 + {y(x), x} = 0.…”

“…Remark: the Schwarzian condition (55) and the associated function W (x) given by (56), correspond to an elimination of the conjugation function v(x) in (53). If one changes the order-N linear differential operator L N by conjugation, L N →…”

Classical geometry from the quantum Liouville theory

“…On the other hand there exists so called geometric approach originally proposed by Polyakov [11] and further developed by Takhtajan [12][13][14][15] (see also [16][17][18][19]). In contrast to the operator formulation of the DOZZ theory the correlators of primary fields with elliptic and parabolic weights are expressed in terms path integral over conformal class of Riemannian metrics with prescribed singularities at the punctures.…”