Phases of R-charged Black Holes, Spinning Branes and Strongly Coupled Gauge Theories

TL;DR

This paper analyzes the thermodynamic stability of charged AdS black holes in D=5,4,7 gauged supergravity, deriving Hawking–Page transition criteria and local stability conditions in both grand-canonical (fixed potentials) and canonical (fixed charges) ensembles. By performing explicit calculations of the Gibbs and Helmholtz Euclidean actions, it maps large R-charged black holes to spinning D3-, M2-, and M5-branes, revealing precise equivalences between their masses, charges, and entropies and those of the corresponding branes. The authors obtain analytic stability boundaries, including critical lines and Hawking–Page boundaries, and demonstrate that in the large-charge limit, the D=4 and D=7 stability constraints mirror known spinning brane results, with intriguing dualities and self-dualities in D=5. They also discuss the physical implications of ensemble choice for quantum gravity: fixed charge configurations tend to favor a transition to AdS gas, while fixed potentials can leave large, unstable black holes as saddle points, hinting at possible multi-black-hole or fragmented configurations in AdS. Overall, the work strengthens the connection between black hole thermodynamics in AdS, brane dynamics, and the gauge/gravity correspondence, highlighting how stability analyses encode features of the dual gauge theories.

Abstract

We study the thermodynamic stability of charged black holes in gauged supergravity theories in D=5, D=4 and D=7. We find explicitly the location of the Hawking-Page phase transition between charged black holes and the pure anti-de Sitter space-time, both in the grand-canonical ensemble, where electric potentials are held fixed, and in the canonical ensemble, where total charges are held fixed. We also find the explicit local thermodynamic stability constraints for black holes with one non-zero charge. In the grand-canonical ensemble, there is in general a region of phase space where neither the anti-de Sitter space-time is dynamically preferred, nor are the charged black holes thermodynamically stable. But in the canonical ensemble, anti-de Sitter space-time is always dynamically preferred in the domain where black holes are unstable. We demonstrate the equivalence of large R-charged black holes in D=5, D=4 and D=7 with spinning near-extreme D3-, M2- and M5-branes, respectively. The mass, the charges and the entropy of such black holes can be mapped into the energy above extremality, the angular momenta and the entropy of the corresponding branes. We also note a peculiar numerological sense in which the grand-canonical stability constraints for large charge black holes in D=4 and D=7 are dual, and in which the D=5 constraints are self-dual.

Figures (6)

• Figure 1: In Figure 1a stability domains are plotted as ${q}$ vs. ${{r_+}}$ for the grand-canonical ensemble of $D=5$ R-charged black holes with one charge turned on. In Figure 1b the same stability plots are exhibited, but now as ${\tilde{q}}$ (the physical charge) vs. $M$ (the ADM mass). Only the region satisfying the BPS bound, $M\ge {\tilde{q}}$, corresponds to physical black hole solutions. The vertically shaded areas correspond to the regions where the $AdS_5$ is the preferred solution and the horizontally shaded area correspond to the regions where the black hole solutions are local minimum of the Gibbs action. The checkered area is the domain of common overlap: there black holes still correspond to the local maxima of the entropy (local minima of the Euclidean action); however, anti-de Sitter space is preferred globally. In the unshaded area, neither the black hole solution nor pure anti-de Sitter space are global minima of the action.
• Figure 2: In Figure 2a we plot the stability domains as $q$ vs. $r_+$ for the canonical ensemble of $D=5$ R-charged black holes with one non-zero charge. In Figure 2b the same stability plots are exhibited, but now plotting ${{\tilde{q}}}$ vs. ${M}$. The vertically shaded areas are the regions where anti-de Sitter space is the preferred solution. The horizontally shaded areas are the regions where the black hole solution is a local minimum of the Helmholtz action. The checkered area is the domain of common overlap: meta-stable black holes in the intepretation of section \ref{['Gibbs']}.
• Figure 3: The stability domains as ${q}$ vs. ${{r_+}}$ for the grand-canonical ensemble of $D=4$ R-charged black holes with one non-zero charge are given. The vertically shaded areas correspond to the regions where the $AdS_4$ is the preferred solution and the horizontally shaded area correspond to the regions where the black hole solutions are local minimum of the Gibbs action. The checkered area is the domain of common overlap. The unshaded area corresponds to black holes that are unstable but nevertheless favored over pure anti-de Sitter space. Explicit powers of $L$ can be restored by replacing $r_+$ with ${r_+\over L}$ and $q$ with ${{q}\over L}$.
• Figure 4: The stability domains are plotted for ${q}$ vs. ${{r_+}}$ for the canonical ensemble of $D=4$ R-charged black holes with one non-zero charge. The vertically shaded areas indicates the regions where $AdS_4$ is the preferred solution, and the horizontally shaded area shows the regions where the black hole solutions are local minima of the Helmholtz action. The checkered area is the domain of common overlap. Explicit powers of $L$ can be restored by replacing $r_+$ with ${r_+\over L}$, and $q$ with ${{q}\over L}$.
• Figure 5: The stability domains as ${q}$ vs. ${{r_+}}$ for the grand-canonical ensemble of $D=7$ R-charged black holes with one non-zero charge. The vertically shaded areas correspond to the regions where the $AdS_7$ is the preferred solution and the horizontally shaded area correspond to the regions where the black hole solutions are local minimum of the Gibbs action. The checkered area is the domain of common overlap. The unshaded area is where neither black holes nor anti-de Sitter space are stable. Powers of $L$ can be restored by replacing $r_+$ with ${r_+\over L}$ and $q$ with ${{q}\over L^4}$.