Universal low temperature theory of charged black holes with AdS$_2$ horizons

TL;DR

The paper shows that charged black holes in asymptotically AdS spacetimes possess a universal low-temperature description governed by a 0+1D action combining a Schwarzian for time reparametrizations with a phase mode conjugate to charge. This effective theory is derived from the full Einstein-Maxwell action through a two-step reduction to AdS$_2$ and then to a Schwarzian boundary theory, with couplings ($\gamma$, $K$, $\mathcal{E}$) fixed by the thermodynamics of the higher-dimensional black hole. The authors demonstrate that the Schwarzian coefficient matches the thermodynamic prediction and that the phase mode arises from a Wilson line, yielding a consistent, gauge-invariant effective action that remains valid for $T \ll 1/R_h$ and regardless of the AdS$_{D}$ radius $L$. They further show how finite-$T$ effects and diffeomorphism modes couple to this phase sector, producing a complete low-energy description that reproduces known thermodynamic and spectral properties, including a controlled density-of-states envelope akin to complex SYK-like behavior. The work provides a concrete bridge between bulk Einstein-Maxwell dynamics and a universal 0+1D boundary theory, with potential extensions to supersymmetric and string-theoretic setups.

Abstract

We consider the low temperature quantum theory of a charged black hole of zero temperature horizon radius $R_h$, in a spacetime which is asymptotically AdS$_{D}$ ($D > 3$) far from the horizon. At temperatures $T \ll 1/R_h$, the near-horizon geometry is AdS$_2$, and the black hole is described by a universal 0+1 dimensional effective quantum theory of time diffeomorphisms with a Schwarzian action, and a phase mode conjugate to the U(1) charge. We obtain this universal 0+1 dimensional effective theory starting from the full $D$-dimensional Einstein-Maxwell theory, while keeping quantitative track of the couplings. The couplings of the effective theory are found to be in agreement with those expected from the thermodynamics of the $D$-dimensional black hole.

Figures (2)

• Figure 1: Charged black holes in asymptotically AdS$_{d+2}$ space with a near-horizon AdS$_2 \times S_d$ geometry. We choose the compact space $M_d$ to be $S_d$, the $d$-sphere. The intermediate geometry is described by the effective action in Eq. (\ref{['Seff']}).
• Figure 2: Low temperature crossovers outside a black hole of charge $\mathcal{Q}$. The value of $R_h$ is determined from $\mathcal{Q}$ via Eq. (\ref{['Q1']}), and we describe $T \ll 1/R_h$ at fixed $\mathcal{Q}$, $\mu$ specified by Eq. (\ref{['muET']}), and $R_2 \sim R_h$. We denote contributions to the free energy $F = \Omega + \mu \mathcal{Q}$ in each region ($E_0 = \Omega_0 + \mu_0 \mathcal{Q}$ is the ground state energy).