Towards a Strong Coupling Liouville Gravity

A possibility of strong coupling quantum Liouville gravity is investigated via infinite dimensional representations of Uqsl(2,C) with q at a root of unity. It is explicitly shown that vertex operator in this model can be written by a tensor product of a vertex operator of the classical Liouville theory and that of weak coupling quantum Liouville theory. Some discussions about the strong coupling Liouville gravity within this formulation are given. In the recent developments of 2D quantum Liouville gravity theory[1] as a noncritical string theory[2]-[6], one of the important features is that it has quantum group structure[7, 8]. Precisely, the vertex operators in the theory can be expressed in terms of the representations of Uqsl(2,C). Upon remembering that there are two kinds of representations of Uqsl(2,C), namely, of finite dimension and of infinite one, and that they are completely different from each other, we can expect that there are two phases in the Liouville gravity. These phases correspond to, so-called, weak coupling regime and strong coupling regime. A number of works have revealed remarkable features of the weak coupling Liouville gravity where finite dimensional representations appear. However, in the weak coupling regime, the well-known restriction, that is, the D = 1 barrier has bothered us. On the other hand, a successful work for the strong coupling Liouville gravity has been done by Gervais and his collaborators[9]. In these works, they made use of the infinite dimensional representations of Uqsl(2,C) with generic q and have shown that consistent theories can be built provided the central charge of the Liouville gravity Present address, YITP-Uji, University of Kyoto, Uji 611, Japan