Two Dimensional Anti-de Sitter Space and Discrete Light Cone Quantization

We realize the two dimensional anti-de Sitter (AdS2) space as a KaluzaKlein reduction of the AdS3 space in the framework of the discrete light cone quantization (DLCQ). Introducing DLCQ coordinates which interpolate the original (unboosted) coordinates and the light cone coordinates, we discuss that AdS2/CFT correspondence can be deduced from the AdS3/CFT . In particular, we elaborate on the deformation of WZW model to obtain the boundary theory for the AdS2 black hole. This enables us to derive the entropy of the AdS2 black hole from that of the AdS3 black hole. Typeset using REVTEX [email protected] [email protected] [email protected] 1 One of the main progresses achieved recently in the string theory is the AdS/CFT duality [1,2], which connects the gravity in the D-dimensional anti-de Sitter (AdSD) space and the (D − 1) dimensional conformal field theory (CFT ) on its boundary. Among the AdS/CFT dualities in the various dimensions are the AdS3/CFT and AdS2/CFT dualities relevant for the black hole physics, since most of the black holes in the string theories are known to contain either AdS3 space or AdS2 space in their near horizon geometries [1,3]. Thus, the AdS/CFT dualities in low dimensions would play a key role in understanding the quantum aspects of the black holes. However, compared with the case of the AdS3/CFT duality [4–6] the AdS2/CFT duality is less well discussed in the literature. Observing that the near horizon geometry of the three dimensional BTZ (Bañados-Teitelboim-Zanelli) black hole [7] becomes effectively AdS2 in the low energy regime, one may attempt to derive the AdS2/CFT duality from the known AdS3/CFT . This approach was taken by Strominger in his recent work on AdS2/CFT duality [8]. Here in this paper we will employ a different strategy to derive the AdS2/CFT duality, namely the DLCQ (discrete light cone quantization) [9], which reveals the relationship between two dualities more transparently. If the BTZ black hole is viewed in the light cone frame along the circle direction, the metric components in the light like directions are constant and can be scaled by boosting the frame. Thus, if the light cone coordinate, x is taken to be periodic, the Kaluza-Klein compactification can be easily performed. In order to have a periodic light cone coordinate, we employ the DLCQ procedure, which has been discussed recently [10] in the context of the Matrix M-theory [11]. It is found useful to introduce DLCQ coordinates, which interpolate the original (unboosted) coordinates and the light cone ones when we apply the DLCQ procedure to the AdS3 black hole. One advantage of this approach is that we do not need to confine ourselves to the near horizon region. Let us begin with the well-known D1-D5 black hole in ten dimensions, which has its near horizon geometry as MBTZ × S × T 4 [12] ds α′ = U l2 (−dt + dx5) + U0 2 l2 (cosh σ dt+ sinh σ dx5) 2