A Note on Quantum Liouville Theory via Quantum Group ——- An Approach to Strong Coupling Liouville Theory ——-

Quantum Liouville theory is annualized in terms of the infinite dimensional representations of Uqsl(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 Uqsl(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 cL satisfies 1 < cL < 25. An interpretation of quantum space-time is also given within this formulation. ———————– ∗ e-mail address: [email protected]