Liouville Theory Revisited

Liouville theory seems to be a kind of universal building block for a variety of models for two-dimensional gravity and non-trivial backgrounds in string theory. Some aspects of it were important for understanding what the matrix models of 2D gravity actually describe (see e.g. [GM]), and it keeps popping up in sometimes unexpected circumstances such as the the physics of membranes in string theory (e.g. [SW]). In the context of string theory, Liouville theory and close relatives such as the SL(2) or SL(2)/U(1) WZNW models seem to be the simplest examples where the new qualitative features of nontrivial (possibly curved) non-compact backgrounds can be studied. For all this it is crucial that Liouville theory, as indeed supported by many investigations of this issue, can be quantized as a conformal field theory (CFT), implying in particular that the space of states forms a representation of the Virasoro algebra. What makes the analysis of the quantized theory much more difficult as compared to other conformal field theories is the fact that the set of Virasoro representations that make up the space of states is continuous. This can be viewed as a reflection of the noncompactness of the space in which the Liouville zero mode Õ Ê ¾ ¼ ¨´µ takes values. Liouville theory may furthermore be seen as probably the simplest prototypical example for a class of conformal field theories called non-compact CFT which have continuous spectrum of representations of the Virasoro algebra. It may well be expected to play a role in the development of a general theory of such CFT's that is analogous to the role of the minimal models as prototype for rational CFT. This is in fact one of our main motivations: We believe that other non-compact CFT will share many features with Liouville theory that distinguish non-compact from rational CFT. Moreover, once the technical tools for the proper investigation of Liouville theory are established, it should not be too difficult to generalize them to other non-compact CFT. For example, many results from Liouville theory can be carried over fairly directly to the À · ¿-WZNW model, as will be explained elsewhere.-02-1.2. Aims and scope This paper focuses on the understanding of Liouville theory on a (space-time) cylinder with circumference ¾, time-coordinate Ø and (periodic) space-coordinate as a two dimensional quantum field theory in its own right. (Semi-)classically the theory is defined by the action (1) Ë ½ …