The Many Faces of the Quantum Liouville Exponentials

Abstract

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.

(23 pages, Latex file) preprint LPTENS-93/30

References in corpus (5)

• W-Geometries
• Classical A_n--W-Geometry
• Gravity-Matter Couplings from Liouville Theory
• The Braiding of Chiral Vertex Operators with Continuous Spins in 2D Gravity
• On the Exact Operator Formalism of Two-Dimensional Liouville Quantum Gravity in Minkowski Spacetime

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• Classical Liouville action on the sphere with three hyperbolic singularities
• The Multivalued Free-Field Maps of Liouville and Toda Gravities
• Non-critical strings at high energy
• Quantum Exchange Algebra and Locality in Liouville Theory
• Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes
• Negative Screenings in Liouville Theory
• String theory and classical integrable systems
• Quantum deformations of $sl(2)$: The hard core of two dimensional gravity
• One-dimensional dynamics of QCD2 string
• Zero modes for the quantum Liouville model