A Note on Quantum Liouville Theory via Quantum Group; an Approach to Strong Coupling Liouville Theory

TL;DR

This paper explores quantum Liouville theory at strong coupling using quantum group representations, revealing factorization properties of vertex operators and connections to geometric quantization of Riemann surfaces.

Contribution

It introduces a novel approach to analyze strong coupling Liouville theory through quantum group representations and factorization of vertex operators.

Findings

Vertex operators factorize into classical and quantum parts.

Fusion rules also exhibit factorization.

The model relates to geometric quantization of moduli space.

Abstract

Quantum Liouville theory is analyzed in terms of the infinite dimensional representations of $U_Qsl(2,C)$ with q a root of unity. Making full use of characteristic features of the representations, we show that vertex operators in this Liouville theory are factorized into `classical' vertex operators and those which are constructed from the finite dimensional representations of $U_qsl(2,C)$. We further show explicitly that fusion rules in this model also enjoys 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 c_L satisfies $1<c_L<25$. An interpretation of quantum space-time is also given within this formulation.