Liouville Exponentials Research Articles

A causal algebra for Liouville exponentials

A causal Poisson bracket algebra for Liouville exponentials on a cylinder is derived using an exchange algebra for free fields describing the in and out asymptotics. The causal algebra involves an even number of spacetime points with a minimum of four. A quantum realization of the algebra is obtained which preserves causality and the local form of non-equal time brackets.

Status of the operator approach to liouville theory

These notes contain a review of work by J.-L. Gervais and the author, as well as recent work in progress by the author. The two main themes are the construction of local observables in Liouville theory — the Liouville exponentials — together with the underlying algebra of chiral vertex operators, and the proper definition of the three point function in the operator framework.

Quantum group structure and local fields in the algebraic approach to 2D gravity

This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.

Two- and three-point functions in Liouville theory

Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two- and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself.

The many faces of the quantum Liouville exponentials

First, it is proven that the three main operator approaches to the quantum Liouville exponentials—that is the one of Gervais-Neveu (more recently developed further by Gervais), Braaten-Curtright-Ghandour-Thorn, and Otto-Weigt—are equivalent since they are related by simple basis transformations in the Fock space of the free field depending upon the zero-mode only. Second, the GN-G expressions for quantum Liouville exponentials, where the U q ( sl(2)) quantum-group structure is manifest, are shown to be given by q-binomial sums over powers of the chiral fields in the J = 1 2 representation. Third, the Liouville exponentials are expressed as operator tau functions, whose chiral expansion exhibits a q Gauss decomposition, which is the direct quantum analogue of the classical solution of Leznov and Saveliev. It involves q exponentials of quantum-group generators with group “parameters” equal to chiral components of the quantum metric. Fourth, we point out that the OPE of the J = 1 2 Liouville exponential provides the quantum version of the Hirota bilinear equation.

The braiding of chiral vertex operators with continous spins in 2D gravity

Chiral vertex-operators are defined for continuous quantum-group spins J from free-field realizations of the Coulomb-gas type. It is shown that these generalized chiral vertex operators satisfy closed braiding relations on the unit circle, which are given by an extension in terms of orthogonal polynomials of the braiding matrix recently derived by Cremmer, Gervais and Roussel. This leads to a natural extension of the Liouville exponentials to continuous powers that remain local.