Continous Spins in 2D Gravity: Chiral Vertex Operators and Local Fields

Abstract

We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. {}From the quantum-group viewpoint, they are related to semi-infinite highest or lowest weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations verified, as well as the validity of the quantum Liouville field equations. In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of J.-L. G. and Roussel: for continuous spins the quantum group structure $U_q(sl(2)) \odot U_{\qhat}(sl(2))$ is a non trivial extension of $U_q(sl(2))$ and $U_{\qhat}(sl(2))$. We construct the corresponding generalized exponentials and the generalized Liouville field.

36 pages, LaTex, LPTENS 93/40

References in corpus (4)

• Solving the Strongly Coupled 2D Gravity: 2. Fractional-Spin Operators, and Topological Three-Point Functions
• The Braiding of Chiral Vertex Operators with Continuous Spins in 2D Gravity
• The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap
• The Many Faces of the Quantum Liouville Exponentials

Cited by in corpus (15)

• Liouville theory revisited
• Liouville Field Theory -- A decade after the revolution
• On the Liouville three-point function
• Liouville bootstrap via harmonic analysis on a noncompact quantum group
• Correlation Functions and Vertex Operators of Liouville Theory
• Quantum Group Structure and Local Fields in the Algebraic Approach to 2D Gravity
• Quantum Exchange Algebra and Exact Operator Solution of $A_2$-Toda Field Theory
• On the Canonical Equivalence of Liouville and Free Fields
• Quantum Exchange Algebra and Locality in Liouville Theory
• On the Liouville coupling constants
• A Note on Quantum Liouville Theory via Quantum Group; an Approach to Strong Coupling Liouville Theory
• Negative Screenings in Liouville Theory
• Negative Screenings in Conformal Field Theory and 2D Gravity: The Braiding Matrix
• Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.
• Quantum Group Generators in Conformal Field Theory