Operator product expansion in SL(2) conformal field theory

In the conformal field theories having affine SL(2) symmetry, we study the operator product expansion (OPE) involving primary fields in highest weight representations. For this purpose, we analyze properties of primary fields with definite SL(2) weights, and calculate their twoand three-point functions. Using these correlators, we show that the correct OPE is obtained when one of the primary fields belongs to the degenerate highest weight representation. We briefly comment on the OPE in the SL(2, R) WZNW model. May 2001 [email protected] [email protected] 1. The conformal field theories having affine SL(2) symmetry have been an interesting topic in recent string theory, since the SL(2) symmetry expresses the isometry of the AdS3 target space or its Euclidean counterpart known as simplest examples exhibiting the holography. The CFT on the Euclidean AdS3, namely, the H + 3 WZNW model is now well controlled [1, 2, 3] and, starting from this, one may extract useful results for other models with the affine SL(2) symmetry [4, 5]. In this note, we continue this line of studies. Our point here is to focus on the primary fields with definite SL(2) weights. These are important in dealing with highest weight representations, since highest weight conditions are expressed by certain relations between the SL(2) spin and weight. In particular, we present pieces of properties of the primary fields mentioned above, and calculate their twoand three-point functions. Using these correlators, we discuss the operator product expansion including primary fields in highest weight representations. When one of the primary fields belongs to the degenerate highest weight representation, we show that the correct OPE is obtained. We briefly comment on the OPE in the SL(2, R) WZNW model. Our analyses may serve also as preparatory steps for further studies. 2. The H 3 WZNW model [1, 2, 3] has the action S = k π ∫