Holographic Description of AdS_2 Black Holes

TL;DR

This work provides a systematic holographic renormalization for AdS$_2$ gravity coupled to a Maxwell field, deriving boundary counterterms and a gauge-invariant boundary mass term. It computes the renormalized boundary stress tensor and central charge, obtaining $c = -24 \alpha e^{-2\phi} = \frac{3}{G_2}$ and validating it against dimensional reduction from AdS$_3$ and Brown–Henneaux results. The paper establishes a precise 3D/2D dictionary via KK reduction, showing that 2D currents and central charges reproduce their 3D progenitors on-shell, and uses these tools to analyze black hole thermodynamics, obtaining the Bekenstein–Hawking entropy $S = \frac{1}{4G_2}$ and connecting it to Cardy-like ground-state counting. It also extends the analysis to backgrounds with non-constant dilaton, demonstrating robustness of the variational principle and counterterms, and discusses implications for the dual chiral CFT and potential universality of the central charge.

Abstract

We develop the holographic renormalization of AdS_2 gravity systematically. We find that a bulk Maxwell term necessitates a boundary mass term for the gauge field and verify that this unusual term is invariant under gauge transformations that preserve the boundary conditions. We determine the energy-momentum tensor and the central charge, recovering recent results by Hartman and Strominger. We show that our expressions are consistent with dimensional reduction of the AdS_3 energy-momentum tensor and the Brown--Henneaux central charge. As an application of our results we interpret the entropy of AdS_2 black holes as the ground state entropy of a dual CFT.