2D Gravity on $AdS_2$ with Chern-Simons Corrections

We study 2D Maxwell-dilaton gravity with higher order corrections given by the Chern-Simons term. The model admits three distinctive $AdS_2$ vacuum solutions. By making use of the entropy function formalism we find the entropy of the solutions which is corrected due to the presence of the Chern-Simons term. We observe that the form of the correction depends not only on the coefficient of the Chern-Simons term, but also on the sign of the electric charge; pointing toward the chiral nature of the dual CFT. Using the asymptotic symmetry of the theory as well as requiring a consistent picture we can find the central charge and the level of U(1) current. Upon uplifting the solutions to three dimensions we get purely geometric solutions which will be either $AdS_3$ or warped $AdS_3$ with an identification.