Conformal Bootstrap in Liouville Field Theory

Abstract

Attention to the two-dimensional Liouville Field Theory (LFT) is drawn basically for two reasons. First it was recognized [1] as an effective field theory of the 2d. quantum gravity. In particular it is very relevant in the string theory [1–4]. Second, it is an example of non-rational conformai field theory (CFT) which is very likely exactly solvable (e.g., the classical equations of motion are integrable).The interest to LFT was renewed recently with the development of the matrix model approach to 2d gravity [5, 6]. It was shown that LFT is able to reproduce some of the predictions of the matrix model approach, in particular the scaling behavior [7–9], the genus one partition functions [10] and some of the integrated correlation functions [11–15]. It is very plausible therefore that LFT describes the same 2d quantum gravity as the matrix models do (at least in the “weak coupling region” C _L > 25).