Notes on AdS 3

We use the conjectured strong-weak coupling worldsheet duality between the SL(2)/U (1) and Sine-Liouville conformal field theories to study some properties of degenerate operators and to compute correlation functions in CFT on AdS 3. The same quantities have been computed in the past by other means. The agreement between the different approaches provides new evidence for the duality. We also discuss the supersymmetric analog of this duality, the correspondence between SCFT on the cigar and N = 2 Liouville. We show that in the spacetime CFT dual to string theory on AdS 3 via the AdS/CFT correspondence, the central term in the Virasoro algebra takes different values in different sectors of the theory. In a companion paper we use the results described here to study D-branes in AdS 3. In this paper we study string theory on AdS 3 (the infinite cover of the SL(2) group manifold). This system received a lot of attention over the years (see e.g. [1,2,3,4,5] for some recent discussions and additional references), for a variety of reasons. Some of the reasons for our interest in this model are: 1. We regard it as a warmup exercise for the study of string theory in time dependent backgrounds. For instance, the coset [SL(2) × SU (2)]/R 2 is a cosmological background corresponding to a closed universe which begins and ends with a singularity [6]. AdS 3 CFT is an important ingredient in analyzing the physics of this model. 2. It is relevant for the study of asymptotically linear dilaton theories, such as the Liou-ville model and the cigar CFT [7], which describes a Euclidean two dimensional black hole [8,9]. Liouville can be obtained from SL(2) (after a certain twist) by gauging a Borel subgroup. The cigar corresponds to the coset SL(2)/U (1) where the U (1) is associated with the compact timelike direction in SL(2). Gauging the non-compact spacelike U (1) gives a Lorentzian two dimensional black hole. 3. Linear dilaton models have many applications. Liouville theory plays an important role in two dimensional string theory, which is holographically dual to a certain matrix quantum mechanics in the large N limit (see e.g. [10] for a review). The cigar appears both in two dimensional string theory, where it describes the high energy thermodynamics (see [11] for a recent discussion) and in Little String Theory (LST), where it again plays a role in the thermodynamics (see e.g. [12]), …