Central charge for 2D gravity on AdS2 and AdS2/CFT1 correspondence

We study 2D Maxwell-dilaton gravity on AdS2. We distinguish two distinctive cases depending on whether the AdS2 solution can be lifted to an AdS3 geometry. In both cases, in order to get a consistent boundary condition we need to work with a twisted energy momentum tensor which has non-zero central charge. With this central charge and the explicit form of the twisted Virasoro generators we compute the entropy of the system using the Cardy formula. The entropy is found to be the same as that obtained from gravity calculations for a specific value of the level of the U(1) current. The agreement is an indication of AdS2/CFT1 correspondence.