Central Charge for 2D Gravity on AdS(2) and AdS(2)/CFT(1) Correspondence

TL;DR

This work studies 2D Maxwell–dilaton gravity on $AdS_2$ and demonstrates that consistent boundary conditions require a twisted energy-momentum tensor with a nonzero central charge, enabling entropy to be computed via the Cardy formula. It analyzes two models: Type I, which cannot be lifted to $AdS_3$, and Type II, which can, deriving central charges $c=12kG^2Q^2l^4$ (Type I) and $c=\frac{3}{2G}$ (Type II) and fixing current levels to reproduce the gravity entropy, thereby supporting an $AdS_2/CFT_1$ correspondence. In Type I, a near-horizon CFT with the same central charge as the asymptotic CFT emerges, suggesting a horizon–boundary CFT correspondence. In Type II, the AdS$_2$ result connects to AdS$_3$ via a lift, linking the twisted boundary dynamics to Brown–Henneaux central charges. Together, these results imply that the holographic dual of gravity on $AdS_2$ is a chiral half of a 2D CFT and motivate further exploration of 2D chiral gravity and potential Chern–Simons deformations.

Abstract

We study 2D Maxwell-dilaton gravity on AdS(2). We distinguish two distinctive cases depending on whether the AdS(2) solution can be lifted to an AdS(3) 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 $AdS(2)/CFT(1) correspondence.