Quantum Group Generators in Conformal Field Theory

TL;DR

This paper explores the construction of quantum group generators within conformal field theory, focusing on Poisson-Lie symmetry and q-deformed Noether charges, with specific analysis of $U_q(sl(2))$ and Liouville theory.

Contribution

It introduces new methods for constructing quantum group generators in conformal field theories, particularly for $U_q(sl(2))$, using Poisson-Lie symmetry and moment maps.

Findings

Developed two approaches to quantum group generator construction.

Applied methods to Liouville theory and Coulomb gas models.

Provided insights into q-deformed symmetries in conformal field theories.

Abstract

These are notes of a seminar given at the 30th International Symposium on the Theory of Elementary Particles, Berlin-Buckow, August 1996. The material is derived from collaborations with E. Cremmer and J.-L. Gervais, and C. Klimcik, and is partially new. Within the general framework of Poisson-Lie symmetry, we discuss two approaches to the problem of constructing moment maps, or q-deformed Noether charges, that generate the quantum group symmetry which appears in many conformal field theories. Concretely, we consider the case of $U_q(sl(2))$ and the operator algebra that describes Liouville theory and other models built from integer powers of screenings in the Coulomb gas picture.