A note on quantum Liouville theory via the quantum group An approach to strong coupling Liouville theory

Abstract

Quantum Liouville theory is analysed in terms of the infinite dimensional representations of U q sl(2, C) with q a root of unity. Making full use of the characteristic features of the representations, we show that the vertex operators in this Liouville theory are factorized into classical vertex operators and those which are constructed from the finite dimensional representations of U q sl(2, C) We further show explicitly that the fusion rules in this model also enjoy such a factorization. Upon the conjecture that the Liouville action effectively decouples into the classical Liouville action and that of a quantum theory, correlation functions and transition amplitudes are discussed, especially an intimate relation between our model and geometric quantization of the moduli space of Riemann surfaces is suggested. The most important result is that our Liouville theory is in the strong coupling region, i.e. the central charge CL satisfies 1 < C L < 25. An interpretation of the quantum space-time is also given within this formulation.