On the √ 2 puzzle in AdS 2 / CFT 1

In this letter we analyze the Hamiltonian formulation of the Jackiw-Teitelboim model of 2D gravity and calculate the central charge associated with the asymptotic symmetries, taking care of boundary terms. For black hole solutions, we show that there is no √ 2 discrepancy between the thermodynamical entropy and the statistical one obtained via Cardy’s formula. PACS numbers: 04.20.-q, 04.70.-s, 04.70.Dy Two-dimensional dilatonic models of gravity received recently much attention. Their importance relays on their role in the conjectured AdS/CFT correspondence [1, 2] and in describing some properties of higher-dimensional black-holes (see e.g. [3]). Two distinct realizations of the AdS2/CFT correspondence are known [4]. The simplest one is known as AdS2/CFT1 and can be understood in terms of boundary fields [3], but is believed to be plagued by a value of the central charge that leads to a mismatch between statistical and thermodynamical entropy of black hole solutions of the Jackiw-Teitelboim model. In a previous paper [5] we have shown that if the U(1) timelike symmetry of a stationary solutions has fixed points, then the hamiltonian for gravity has a boundary term coming from the fixed point set, and moreover that this term can be interpreted in a CFT language. This gave a statistical entropy matching correctly to the usual thermodynamics of black holes. It was clear from [5] that this horizon boundary term is a general feature of gravity in any dimension. Take for example the BTZ[6] black hole, with metric