---
title: Supersymmetric Black Holes in AdS5
url: https://www.emergentmind.com/topics/supersymmetric-black-holes-in-ads-_5
type: topic
---

# Supersymmetric Black Holes in AdS5

Supersymmetric black holes in AdS$_5$ are a central subject in gauge/gravity duality, combining intricate gravitational solutions with nontrivial BPS state-counting in $\mathcal{N}=4$ super Yang-Mills. They form a unique bridge between five-dimensional supergravity, the geometry of higher-dimensional horizons, and exact field-theoretic indices. The corpus of recent research has addressed the explicit construction and classification of such black holes—with various horizon geometries, symmetries, and matter content—their uniqueness, microstate counting via AdS/CFT, the structure of their near-horizon geometries in the EVH (extremal vanishing horizon) limit, and the role of higher-derivative and quantum corrections.

## 1. Definition, BPS Structure, and Basic Solutions

Supersymmetric AdS$_5$ black holes are stationary, asymptotically locally AdS$_5$ solutions of (gauged) five-dimensional supergravity admitting at least one preserved supersymmetry (Killing spinor). In minimal $\mathcal{N}=2$ gauged SUGRA (or its U(1)$^3$ extension), the general family is encapsulated in the four-parameter Chong–Cvetič–Lü–Pope (CCLP) solution, involving two angular momenta $(a, b)$ and three electric charges $Q_I$ [2208.00896, 2311.06124, 2408.11529]. Supersymmetric solutions lie on a codimension-two subspace of the parameter space, saturating the BPS bound
\[
E = J_1 + J_2 + Q_1 + Q_2 + Q_3
\]
with all charges and parameters further constrained by a highly nontrivial algebraic relation required for regularity and absence of closed timelike curves. Not all values of $(Q_i, J_{a})$ can be realized; only those lying on a specific algebraic hypersurface are allowed (see Section 3).

This class of solutions can be embedded in type IIB on $S^5$ or in 11d SUGRA via wrapped M5-brane reductions [2212.10360]. Their near-horizon regions and broader generalizations cover cohomogeneity-one solutions (Gutowski–Reall type [2105.08542]) and those admitting toric Kähler symmetry [2208.00896, 2311.06124].

## 2. Geometric and Symmetric Classification

Timelike supersymmetric AdS$_5$ black holes universally admit a fibration structure,
\[
ds_5^2 = -f^2(dt+\omega)^2 + f^{-1}h_4,
\]
where $h_4$ is a four-dimensional Kähler metric and $f$, $\omega$ determined by BPS conditions [2208.00896]. This geometry accommodates uplifts with toric symmetry in the base, described by Kähler metrics with toric Hamiltonian Killing vectors—a setup reducible to a “symplectic potential” problem. The CCLP black hole is the maximal, regular, toric solution with spherical horizon, described by a simple explicit symplectic potential $g(x_1, x_2)$ [2208.00896]. Separable toric Kähler geometry provides a classification framework: all regular solutions with toric symmetry and locally spherical horizon are Calabi-toric, and no further black hole families exist beyond the known CCLP solution and its near-horizon limits [2311.06124].

The uniqueness of solutions under either $SU(2)$ or toric $U(1)^2$ symmetry has been rigorously established: under these symmetry and analyticity conditions, any supersymmetric, timelike solution outside a horizon is locally isometric to the CCLP (or Gutowski–Reall, for cohomogeneity-one) black hole or its near-horizon geometry [2208.00896, 2105.08542].

## 3. BPS Charge Constraint and Regularity Conditions

The BPS constraint manifests as a highly nontrivial nonlinear algebraic relation among the three R-charges and two angular momenta—the supersymmetric AdS$_5$ black hole exists only if
\[
Q_1 Q_2 Q_3 + \frac{N^2}{2} J_1 J_2 = (Q_1 + Q_2 + Q_3 + \tfrac{N^2}{2}) \left[ Q_1 Q_2 + Q_2 Q_3 + Q_3 Q_1 - \frac{N^2}{2}(J_1 + J_2) \right]
\]
where $N^2$ is set by the AdS$_5$ radius through the holographic map to the gauge theory [2405.17648, 2409.03862]. Equivalently, within the free $\mathcal{N}=4$ SYM, this constraint emerges from statistical ensembles in which all states related by the preserved supercharge are weighted equally. This shows that the constraint is not solely a property of classical gravity, but is microscopically accounted for by the combinatorics of BPS “letters” in the field theory, with interaction-induced rescalings of $N^2$ [2405.17648, 2409.03862].

The classic black hole mass is $E = J_1 + J_2 + Q_1 + Q_2 + Q_3$ (BPS bound), and further regularity excludes regions of parameter space that violate this nonlinear constraint, as CTCs or pathologies would otherwise arise [2405.17648].

## 4. Near-Horizon, EVH, and Attractor Geometries

The near-horizon structure of supersymmetric AdS$_5$ black holes generically features an AdS$_2$ fibered by two $U(1)$ isometries, reflecting the $J_1, J_2$ quantum numbers [0708.3695, 2408.11529]. In the so-called EVH (Extremal Vanishing Horizon) limit, where one angular momentum vanishes (e.g., $b\to0$), the near-horizon geometry degenerates into a locally AdS$_3$ region with a pinching angular direction. In the near-EVH regime, a pinching extremal BTZ factor emerges, with the third U(1) quantum number fractionated and the entropy scaling as $S_{\text{EVH}}\sim T$ [1910.14293].

The spectrum admits an attractor mechanism, fixing all moduli at the horizon algebraically in terms of charges [2007.06582, 2408.11529]: the entropy thus depends only on the conserved charges, and both gravity and CFT extremization computations yield matching attractor values.

## 5. Entropy, Field-Theoretic Index, and Microstate Counting

The Bekenstein–Hawking entropy of these black holes,
\[
S_{\text{BH}} = 2\pi \sqrt{Q_1 Q_2 + Q_2 Q_3 + Q_3 Q_1 - \frac{N^2}{2}(J_1 + J_2)},
\]
is universally reproduced by a Legendre transform of the large-$N$ superconformal index of $\mathcal{N}=4$ SYM, subject to the constraint $\sum_I \Delta_I - \sum_a \omega_a = 2\pi i$ [2409.03862, 1806.01849, 2502.10372]. In the Cardy-like limit, the index admits an explicit saddle (with log $Z \sim (N^2/2) \Delta_1\Delta_2\Delta_3/(\omega_1\omega_2)$), and the extremization yields black hole entropy in full agreement with gravity [1910.14293].

The computation is robust: subleading $\alpha'$ string corrections to the gravitational background vanish identically at the supersymmetric (BPS) locus, ensuring precise agreement [2007.06582]. Quantum corrections, specifically logarithmic corrections to the entropy, have also been computed both from the field theory and via the Kerr/CFT correspondence, and shown to match exactly ($-2\log N$ correction) [2106.09730].

Table: Key Formulations for Black Hole Entropy

| Approach                  | Entropy Expression                                | Constraint     |
|---------------------------|---------------------------------------------------|----------------|
| Bekenstein–Hawking (GR)   | $S_{\rm BH}=2\pi \sqrt{Q_1Q_2+Q_2Q_3+Q_3Q_1-\tfrac{N^2}{2}(J_1+J_2)}$ | BPS algebraic  |
| Superconformal Index (CFT)| Legendre transform of $\log Z$                    | $\sum \Delta_I-\sum \omega_a=2\pi i$ |
| Cardy Formula (EVH limit) | $S_{\rm Cardy}=2\pi\sqrt{c(L_0-c/24)/6}$         | $c_{\text{irr}}\to0$ or $T\to0$      |

The “macroscopic-microscopic” agreement extends to cases with squashed boundaries or including matter multiplets, provided appropriate Page charges (corrected for the Chern–Simons term) are employed instead of naively defined holographic/ADM charges [1903.00021, 1809.02149, 2502.10372].

## 6. Extensions: Matter Multiplets, Hair, and Exotic Topologies

Further generalizations allow for nontrivial scalar hair (as in supersymmetric “hairy” black holes [1806.01849]), the addition of vector and hypermultiplets [2502.10372], or construction in higher $\mathcal{N}=4$ supergravity [1812.10122]. Minimal supersymmetric AdS$_5$ black holes admit only spherical (or lens-space) horizons and cannot realize black rings or toroidal horizons as global solutions; partial near-horizon constructions with $S^1\times S^2$ and $T^3$ exist in the U(1)$^3$ model but do not globally extend to asymptotically AdS$_5$ [0708.3695].

Wrapped M5-brane constructions produce supersymmetric AdS$_5$ black holes with $N^3$-scaling entropy in eleven-dimensional SUGRA, and the entropy is again exactly reproduced by the large-$N$ index of the dual class-$\mathcal{S}$ 4d SCFTs [2212.10360].

## 7. EVH/CFT Correspondence and Dimensional Reductions

The EVH limit provides a natural setting to uncover a 2d CFT in the IR of the 4d superconformal theory: the Legendre transform of the (reduced) index becomes equivalent to the Cardy formula for the dual CFT$_2$. In the strict EVH setup, the entropy and potentials agree explicitly between supergravity and the emergent CFT$_2$ [1910.14293]. Near-EVH perturbations yield a pinching BTZ geometry and the corresponding entropy scaling, with the central charge and temperatures determined holographically.

In other dimensions, analogous EVH and near-EVH limits yield near-horizon AdS$_{k+2}$ throats and associated CFT$_{k+1}$ descriptions, signaling the universality of the AdS/CFT correspondence in extremal scaling limits [1910.14293].

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References:
- [2208.00896]: On the uniqueness of supersymmetric AdS$_5$ black holes with toric symmetry
- [2311.06124]: All separable supersymmetric AdS$_5$ black holes
- [2408.11529]: Supergravity Spectrum of AdS$_5$ Black Holes
- [2105.08542]: Uniqueness of supersymmetric AdS$_5$ black holes with $SU(2)$ symmetry
- [2405.17648]: Supersymmetric Charge Constraints on AdS Black Holes from Free Fields
- [2409.03862]: Quantum Anatomy of Supersymmetric Black Holes in AdS Spacetimes
- [1910.14293]: Probing the EVH limit of supersymmetric AdS black holes
- [0708.3695]: Near-horizon geometries of supersymmetric AdS(5) black holes
- [1809.02149, 1903.00021]: Squashed boundary and Page charge dependence
- [2502.10372]: Microstates of AdS$_5$ black holes with hypermultiplets
- [2212.10360]: Wrapped M5-branes and AdS$_5$ Black Holes
- [1806.01849]: Evidence for the existence of a novel class of supersymmetric black holes with AdS$_5\times$S$^5$ asymptotics
- [2106.09730]: Logarithmic Corrections to the Entropy of Rotating Black Holes and Black Strings in AdS$_5$
- [2007.06582]: Stringy corrections to the entropy of electrically charged supersymmetric black holes with $\mathrm{AdS}_5\times S^5$ asymptotics

Source: https://www.emergentmind.com/topics/supersymmetric-black-holes-in-ads-_5