Charged AdS Black Holes and Catastrophic Holography

TL;DR

The paper analyzes charged black holes in anti-de Sitter space within Einstein-Maxwell truncations arising from spinning branes, revealing rich thermodynamic phase structures that mirror van der Waals-like behavior. By examining fixed-potential and fixed-charge ensembles, it derives Euclidean actions and maps out cusp and swallowtail catastrophes in the free-energy landscape, establishing a holographic interpretation for the dual field theories with background U(1) currents. The work connects gravitational thermodynamics to confinement/deconfinement phenomena in the dual CFTs and highlights universal phase-structure features that appear across dimensions related to D3-, M2-, and M5-branes. It also discusses infinite-volume limits and potential extensions to other AdS/CFT setups, offering a framework for categorizing holographic phase transitions via catastrophe theory.

Abstract

We compute the properties of a class of charged black holes in anti-de Sitter space-time, in diverse dimensions. These black holes are solutions of consistent Einstein-Maxwell truncations of gauged supergravities, which are shown to arise from the inclusion of rotation in the transverse space. We uncover rich thermodynamic phase structures for these systems, which display classic critical phenomena, including structures isomorphic to the van der Waals-Maxwell liquid-gas system. In that case, the phases are controlled by the universal `cusp' and `swallowtail' shapes familiar from catastrophe theory. All of the thermodynamics is consistent with field theory interpretations via holography, where the dual field theories can sometimes be found on the world volumes of coincident rotating branes.

Figures (6)

• Figure 1: A summary of the phase structure of the fixed potential (left) and fixed charge (right) thermodynamic ensembles. The $T{=}0$ line gives extremal black holes, although only in the fixed charge case do they not decay into AdS. The $Q{=}0$ line is the Hawking--Page system of uncharged black holes. (Other labeling is explained in sections \ref{['sec:thermo']} and \ref{['cats']}.)
• Figure 2: The inverse temperature vs. horizon radii, $r_+$, at fixed potential for $\Phi{\ge}1/c$, $\Phi{<}1/c$, and $\Phi{=}0$ respectively. (The values $n{=}4$, $G{=}1$, $l{=}10$ and $\Phi=1,0.7,0$ have been used here.) The divergence in the first graph (here, shown with a vertical line at $r_e{=}4.08$) is at zero temperature, where the black hole is extremal. This divergence goes away for $\Phi{<}1/c$, in general, and the curve is similar to that of the uncharged situation with zero potential, shown last.
• Figure 3: The inverse temperature vs. horizon radii, for $q{>}q_{\rm crit}$, $q{<}q_{\rm crit}$, and $q{=}0$, respectively. $q_{\rm crit}$ is the value of $q$ at which the turning points of $\beta(r_+)$ appear or disappear. (The values $n{=}4$, $l{=}5$ and $q=25,5,0$ have been used here.) The divergences (here, shown by the vertical lines at $r_e{=}0.98$ and $4.05$) are at zero temperature, where the black hole is extremal. The final graph, for the uncharged case, may be thought of as a limit of the previous graphs where the divergence disappears, showing that small Schwarzschild black holes have high temperature.
• Figure 4: On the left is a graph of the free energy vs. temperature for fixed potential ensemble for large $\Phi$. (The values $n{=}4$, $G{=}1$, $l{=}10$, $\Phi{=}1$ have been used here.) The center graph depicts a family of free energy curves for different values of $\Phi$. Note the crossover from the cusp ($\Phi{<}1/c$) to the single branch ($\Phi{>}1/c$) behaviour. On the right is the free energy curve for the uncharged (or $\Phi{=}0$) ensemble, showing the physics familiar from the Schwarzschild case: visible are the two branches consisting of smaller (unstable) and large (stable) black holes. The entire unstable branch has positive free energy while the stable branch's free energy goes (rapidly, on this scale) negative for all $T{>}T_c$.
• Figure 5: The first two graphs show the free energy vs. temperature for the fixed charge ensemble. The situation for $q{<}q_{\rm crit}$ and $q{\ge}q_{\rm crit}$, respectively, are plotted. (The values $n{=}4$, $G{=}1$, $l{=}5$ and $q{=}1,25$ have been used here.) The first graph is the union of three branches. Branch 1 emanates from the origin, and merges with branch 2 at a cusp. Branch 3 forms a cusp with the other end of branch 2, and continues towards the bottom right. The graph on the right shows how the branches arise from the inverse temperature curves of eqn. (\ref{['betaform']}). (See text for discussion of critical temperature $T_c$.)