Symmetry Breaking, Central Charges and the AdS2/CFT1 Correspondence

When two-dimensional Anti-de Sitter space (AdS2 ) is endowed with a non-constant dilaton the origin of the central charge in the Virasoro algebra generating the asymptotic symmetries of AdS2 can be traced back to the breaking of the SL(2, R) isometry group of AdS2 . We use this fact to clarify some controversial results appeared in the literature about the value of the central charge in these models. E-Mail: [email protected] E-Mail: [email protected] The Anti-de Sitter(AdS)/conformal field theory (CFT) correspondence in two spacetime dimensions [1, 2, 3] seems to contradict the general belief that low-dimensional physics is simpler than the higher-dimensional one. Compared with the higher-dimensional cases, the AdS2/CFT1 duality has many puzzling and controversial features. Even though general arguments suggest that gravity on AdS2 should be related to some sort of conformal mechanics living on the one-dimensional boundary of AdS2 , the realization of this correspondence is rather involved. Although it has been shown that, analogously to the three-dimensional (3D) case [4], the conformal symmetry involved is infinite dimensional [2], the search for a physical system realizing this symmetry has not been very successful. On the other hand, the fact that AdS2 has a disconnected boundary makes the AdS2/CFT1 correspondence problematic even at a more fundamental level. At a technical level there are features of gravity on AdS2 that represent a further difficulty for sheding light on the subject, e.g. the existence of several two-dimensional (2D) gravity models admitting AdS2 as solution ( models with or without dilatons) or the fact that the boundary of AdS2 is one-dimensional. In a previous work [2] , we have been able, working in the context of 2D dilaton gravity, to show that the asymptotic symmetry group of AdS2 is generated by a Virasoro algebra. Using a canonical realization of the symmetry we have also computed the central charge of the algebra. Unfortunately, using our value of the central charge for the computation of the statistical entropy of 2D black holes, we found a discrepancy of a factor of √ 2 with respect to the thermodynamical entropy. Later Navarro-Salas and Navarro [3], using a boundary field realization of the symmetry, found a value of the central charge (half of our result) that produces a statistical entropy in agreement with the thermodynamical one. In this letter we try to clarify this controversial point. We show that in models where AdS2 is endowed with a non-constant dilaton, the origin of the central charge of the Virasoro algebra can be traced back to the breakdown of the SL(2, R) isometry group of AdS2 . The authors of Ref. [3] use instead a non-scalar dilaton, which enables them to keep the SL(2, R) symmetry of AdS2 unbroken. In this case the origin of the central charge is the same as in the 3D case and the 2D model has a sensible interpretation as compactification of AdS3, explaining the value of the central charge found in Ref. [3]. But keeping the dilaton to transform as a scalar, and hence considering AdS2 as intrinsically 2D, our previous result of Ref. [2] for the central charge can be recovered also adopting a boundary field realization of the asymptotic symmetry of AdS2 . Let us consider a generic two-dimensional (2D) dilaton gravity model, S = 1 2 ∫ dx √ −g [ ηR + λV (η) ] , (1) where η is a scalar field related to the usual definition of the dilaton φ by η = exp(−2φ). In the discussion of the symmetries of the model a crucial role is played by the scalar η. In general, the scalar character of η implies that the symmetries (isometries) of the 2D spacetime are broken by a non-costant dilaton. In fact under the isometric transformations generated by a Killing vector χ, δη = Lχη = χ∂μη. (2)