Supersymmetry and Attractors

Renata Kallosh; Sergio Ferrara

REVIEW

Supersymmetry and Attractors

Sergio Ferrara, Renata Kallosh

High Energy Physics - Theory · Submitted 26 Feb 1996 · 31 references

Pith's verdict: The attractor paper: S = π|Z_fix|² is derived cleanly for N=2, and the thin spots are all in the N=8 and 5d asides, not the main claim.

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arXiv IDhep-th/9602136 (version 3)

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CitationRenata Kallosh, Sergio Ferrara. "Supersymmetry and Attractors." arXiv:hep-th/9602136 (1996). https://pith.science/paper/KSZ4ZYXU

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Editor's note

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This is the paper that turned the attractor mechanism from a collection of examples into a principle, and the N=2 core is solid. What is new relative to [1] (magnetic case) and Strominger's [2]: the general dyonic statement, the symplectic-invariant formula S = π|Z_fix|² with Z_fix the central charge at the point where D_i Z = 0, the observation that hypermultiplets drop out of the area, and the N=4/N=8/5d extensions. The derivation is short and checkable: the gaugino Killing condition gives D_i Z = 0 (eq. 38), the Appendix shows that implies ∂_i|Z| = 0, and eq. (46) identifies I₁ at the fixed point with |Z_fix|². Eq. (48) then reproduces the known magnetic-case area from [1], which is the check that pins the overall coefficient. That is a real cross-check, not decoration, and it is what makes me trust the normalization of the entropy formula. On the stress-test note: half right. The BR-mass-to-|Z_fix|² identification is terse, one sentence after eq. (22), but the ingredients are all present (eq. 22 for the curvature, eq. 32 for Z as the S² flux of T⁻, eqs. 26–27 for M²_BR = A/4π), and eq. (48) closes the coefficient. I would call it a step the reader fills in, not a gap. The reader's footnote-4 worry is real but points the other way: for a BPS solution D_i Z = 0 follows from the Killing condition, so the fixed point exists by assumption. What footnote 4 actually leaves open is the converse — whether every charge vector yields a black hole, and whether the extremum is a minimum. The authors state that caveat plainly, and it should be read as acknowledged. Where the paper is genuinely thin: the N=8 reduction to eq. (56) is asserted (This finally can be reduced to) and gives the area only up to an overall constant, with ∼ rather than =. The 5d section says Details will be given elsewhere, and footnote 8 explicitly flags eq. (85) as a conjecture. The N=8 cross-check leans on [22], at the time the authors' own preprint. None of this touches the N=2 result. Who should read it: anyone working on black hole entropy, special geometry, or BPS states; the notation is still in use. Recommendation: send it out. The central claim is correct and reproducible; the peripheral sections are sketches that later work filled in.

Referee report

Significance

If correct, this is a genuinely model-independent statement: the macroscopic Bekenstein-Hawking entropy of any extremal N=2 black hole is computable from the quantized electric and magnetic charges alone, via a variational principle on the moduli space, with no need to know the asymptotic couplings. The formula is manifestly symplectic (duality) invariant, which is a real structural gain over the case-by-case area computations it replaces. The dilatonic example is a clean, exactly checkable demonstration: extremizing eq. (3) gives g^2_fix = |p/q| and S = π|pq|, reproducing the known result of [4, 5] by an entirely different route. The N=8 truncation is cross-checked against the independent E_{7(7)}-invariant formula of [22], and the charge-only character of the entropy is exactly what the microscopic D-brane counting of [29]-[32] requires. The paper is also honest about where it is incomplete, which I weigh below rather than treat as concealment.

Major objections (5)

[Sec. 2, eqs. (22), (31), (32), (49)] The physical crux of the paper, the identification M^2_BR = A/4π = (|Z|^2)_{D_i Z = 0}, is compressed into a single sentence after eq. (22). The consistency condition (22) sets R^BR_{αβα'β'} = T_{αβ} T̄_{α'β'}, and eq. (32) defines Z as minus one half the flux of T^- over the two-sphere, so the identification is plausible; but the Einstein-equation step that converts the squared flux into the BR curvature radius with coefficient exactly one is never written out. Since the normalization of T^- in eq. (31) and the factor of 1/2 in eq. (32) both enter this coefficient, the reader cannot verify the factor without reconstructing the argument. Please insert the two or three explicit lines: the flux integral of T^- for the near-horizon ansatz, and the resulting relation between |Z_fix|^2 and the coefficient in eq. (26). This is the step that makes the entropy formula physically meaningful and it is the least explicit part of an otherwise checkable derivation.
[Footnote 4; abstract sentence s1; Sec. 1] The authors acknowledge in footnote 4 that the extremum is assumed to be a minimum and that situations with different extrema, or with no solution of D_i Z = 0 at all, cannot be excluded. That acknowledgement is appropriate, but the abstract and Sec. 1 nonetheless present the result as a 'general principle' for computing the area of N=2 extremal black holes, whereas what is established is a principle conditional on the existence and character of the critical point, verified in the examples at hand. The gap is not in the derivation for BPS solutions, where D_i Z = 0 follows from the Killing condition, but in the converse direction the abstract advertises. I would ask that the abstract and the opening of Sec. 1 state the existence/minimum condition explicitly, so that the scope of the claim matches what is proved.
[Sec. 1 (no-hair claim) versus Sec. 2, eq. (20) and abstract sentence s5] Section 1 asserts a 'stronger version of the no-hair theorem: black holes lose all their scalar hair near the horizon', but the treatment of hypermultiplets in Sec. 2 only requires the hypermultiplet scalars to be constant (eq. 20, from δζ = 0); their values are not driven to charge-determined fixed points anywhere in the paper. The statement in the abstract that hypermultiplets do not affect the area formula is correct as far as it goes, but the stronger no-hair phrasing overstates what is shown: vector-multiplet moduli are attracted, hypermultiplet moduli are merely frozen. Please qualify the Sec. 1 sentence, or state explicitly that hypermultiplet scalars remain free parameters of the solution and that this is consistent with the area being charge-only.
[Sec. 3, eq. (56)] The reduction from |Z_fix|^2 = -p_0^2 Im S (1 - (Re t^2)^2) to the O(2,2) form sqrt|(p_0^2 - p_2^2)(q_3^2 - q_1^2)| is disposed of with 'This finally can be reduced to'. Because this is precisely the cross-check against the independent E_{7(7)}-invariant result of [22], the intermediate steps matter: the choice t^1, t^3 imaginary and t^2 real, the identification of the charges p_Λ = p_0 Re X_Λ and q^Λ = p_0 Re F_Λ, and the passage to the m_0, m_2, e_1, e_3 variables should each be at least indicated. As written, the reader must take the agreement with [22] on faith, which weakens what is otherwise a valuable independent confirmation.
[Sec. 5, eqs. (84)-(87), footnote 8, abstract sentence s8] The five-dimensional statement is presented in the abstract as a result ('is also seen to be obtained by extremizing the 5d central charge'), but Sec. 5 defers the derivation ('Details will be given elsewhere') and footnote 8 explicitly says 'We conjecture in analogy with ref. [22]' for the cubic invariant. The general formula (87) is therefore not derived in this paper. Either supply the short derivation of eq. (83) to eq. (87) for the N=1, d=5 case, or mark the 5d statements in the abstract and in Sec. 5 as a sketch and a conjecture respectively, so that the abstract does not claim more than the body supports.

Minor objections (7)

[Sec. 4, eqs. (71)-(72)] The expressions '|p|2/r2', '|q|3/r3', '|p|4/r4' are ambiguous: the trailing digits are charge labels, but they read as exponents or as distinct radial coordinates. Please set them as subscripts on the charges, e.g. |p_2|/r, so that the single radial coordinate is unmistakable.
[Sec. 4, eq. (65)] The near-horizon metric in eq. (65) is written with a positive coefficient on dt^2, whereas eqs. (24) and (28) use the opposite sign convention. Please make the signature convention uniform across Secs. 2 and 4.
[Sec. 2, eqs. (43)-(45)] Equations (43)-(45) are derived in a frame that splits the symplectic vector into (L^Λ, M_Λ) and uses Im N^{-1} explicitly, so the intermediate steps are frame-dependent even though the final result (46) is not. A sentence noting this would prevent the reader from inferring that (43)-(45) are themselves symplectic covariant.
[Sec. 2, eqs. (34)-(35)] The matrix F_{ΛΣ} appearing in I_2 in eq. (35) is not defined anywhere, in the text or in the Appendix, and the assertion that both N and F are Kähler invariant is stated without the definition on which it rests. Please add the definition of F_{ΛΣ} alongside that of N_{ΛΣ}.
[Figures 1 and 2] The axis labels of both figures are illegible in the version supplied (Fig. 1 reads in part 'e/BnZr /2 /'). Since Fig. 1 is the paper's illustration of the basin of attraction and Fig. 2 of the extremum in moduli space, both should be redrawn with legible labels and units.
[References and front matter] Several typographical errors should be corrected: ref. [32] 'Beckenridge' should be Breckenridge; ref. [25] 'Townsed' should be Townsend; ref. [14] 'hep-th/94031186' appears to carry an extra digit; and the second author's address reads '9430 5-4060' for 94305-4060. In Sec. 3, 'E7SU(8)' should be E_{7(7)}/SU(8) and '0(2,2)0(2)×0(2)' should be O(2,2)/(O(2)×O(2)).
[Footnote 7] The note that the charge normalization differs from [1] is helpful but insufficient for a reader comparing eq. (47) with the result of [1]. A one-line dictionary between the two conventions would make the comparison immediate.

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The paper

The paper claims

The horizon area of an extremal supersymmetric black hole equals the square of its central charge evaluated where that charge is extremized over the scalar fields, so the entropy depends only on electric and magnetic charges.

The reading

Extremal black holes in string theory carry scalar fields whose values far from the hole are continuous parameters, so naively the horizon area could depend on them. This paper argues that it does not: as one falls toward the horizon, the scalars are driven to a fixed point determined entirely by the electric and magnetic charges, and the area is read off from the central charge $Z$ (the charge-dependent quantity that equals the ADM mass for a supersymmetric state) evaluated at that fixed point. The rule is $S = A/4 = |Z|^2$, where $Z$ is extremized over the moduli space of the theory, equivalently at points where the Kähler-covariant derivative $D_i Z$ vanishes. If the rule holds, the macroscopic Bekenstein-Hawking entropy of any extremal $N=2$ black hole becomes a model-independent, duality-invariant function of charges alone, and the same variational principle reproduces known results in $N=4$, $N=8$ and five dimensions.

What carries the argument

The load-bearing object is the central charge $Z = e^{K/2}(X^Lambda q_Lambda - F_Lambda p^Lambda)$, built from the holomorphic symplectic sections of special geometry and the electric and magnetic charge vector. Its modulus equals the ADM mass of a half-supersymmetric state, and its covariant derivative $D_i Z$ measures how far the scalar configuration is from the attractor. Setting $D_i Z = 0$ produces an algebraic relation expressing the charges in terms of the sections, $p^Lambda = i(bar Z L^Lambda - Z bar L^Lambda)$ and similarly for $q_Lambda$, which shows the fixed point is determined by charge ratios up to a Kähler gauge choice. The symplectic invariant $I_1 = |Z|^2 + |D_i Z|^2$, written as $-tfrac12 P^t M(N) P$ with the vector kinetic matrix at the fixed point, then delivers the area in a manifestly duality-invariant form.

Extended reading notes

Core claim

The paper's central claim is that the horizon area of an extremal $N=2$ black hole in four dimensions is fixed by a variational principle in the moduli space: extremize the modulus of the central charge, $|Z(z,bar z,p,q)|$, with respect to the scalar fields, and the value at the extremum gives the entropy through $S = A/4 = |Z_{fix}|^2$. The condition for the extremum is the duality-covariant equation $D_i Z = (partial_i + K_i/2) Z = 0$, which is exactly the condition that the vector-multiplet field strengths vanish near the horizon and that the scalars sit at a fixed point of attraction. Because the fixed-point equations are homogeneous in the charges, the critical moduli depend only on charge ratios, so the area forgets the asymptotic values of the coupling constants. The authors further show that the near-horizon geometry is of Bertotti-Robinson type and is conformally flat, which is what restores the second supersymmetry there, doubling the unbroken supersymmetry from one half to all of $N=2$. Hypermultiplets drop out of the area formula entirely.

Load-bearing premise

The derivation assumes that the equation $D_i Z = 0$ actually has a solution and that the stationary point so found is a minimum; the paper's own footnote concedes that neither point is established in general.

Editorial extensions

If this is right

Every extremal $N=2$ black hole entropy becomes computable from its electric and magnetic charges alone, with no need to know the asymptotic values of the scalar fields.
The result gives a model-independent macroscopic formula that any microscopic state count must reproduce, and it is manifestly invariant under electric-magnetic duality rotations.
The attractor mechanism extends to $N=4$ and $N=8$ supergravity, where the area reduces to expressions such as $sqrt{|p_0 p_1|}$ or $sqrt{|m_0 m_2 e_1 e_3|}$ built from products of charges.
In five dimensions the same extremization of the central charge yields $S sim Z^{3/2}$, reproducing the charge-product formulas discussed for five-dimensional black holes.
Hypermultiplet scalars do not enter the area formula, so the entropy is insensitive to that whole sector of the theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the extremum is always a minimum, the moduli space near the horizon acts like a basin of attraction whose shape is set by the charges, which suggests the entropy is a kind of distance function on the charge lattice rather than a dynamical quantity.
The same variational structure looks like it should survive beyond static extremal holes, for instance stationary rotating ones, where the relevant function would be extremized over moduli with the angular momentum held fixed.
Because the area is invariant under duality but the underlying charges are discrete, the formula is a natural bridge to microscopic counting: any candidate microstate degeneracy must be a duality-invariant function of the same charge invariants.
The conformal flatness of the near-horizon geometry is doing more work than the paper exploits: it hints that the near-horizon region is a maximally symmetric vacuum, so the attractor fixed point may be a statement about which vacua a given charge configuration can interpolate between.

Original abstract

We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for N=2 black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit model independent expression for the macroscopic Bekenstein-Hawking entropy of N=2 black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in N=4,8 supersymmetries and the relation to the N=2 case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,902 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.64 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for N=2 black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit model independent expression for the macroscopic Bekenstein-Hawking entropy of N=2 black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in N=4,8 supersymmetries and the relation to the N=2 case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge.	contribution 0.51	0.95	0.80	asserted 1.00
s1	One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction).	method 0.96	0.48	0.12	asserted 0.95
s2	The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges.	result 1.00	0.83	0.40	asserted 1.00
s3	The doubling of unbroken supersymmetry at the fixed point of attraction for N=2 black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry.	result 0.91	0.84	0.07	asserted 1.00
s4	These results provide an explicit model independent expression for the macroscopic Bekenstein-Hawking entropy of N=2 black holes which is manifestly duality invariant.	result 0.61	0.89	0.48	asserted 1.00
s5	The presence of hypermultiplets in the solution does not affect the area formula.	result 0.99	0.77	0.05	asserted 1.00
s6	Various examples of the general formula are displayed.	contribution 0.89	0.85	0.02	asserted 0.97
s7	We outline the attractor mechanism in N=4,8 supersymmetries and the relation to the N=2 case.	contribution 0.92	0.74	0.05	asserted 0.89
s8	The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge.	result 0.80	0.67	0.06	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Evolution of the dilaton from various initial cond

**Figure 2.** Extremum of the central charge in the moduli space.

Figures reproduced from arXiv:hep-th/9602136 by the authors.

References

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The topologically twisted index of a general 3d N=2 quiver gauge theory scales as N^{3/2} or N^{5/3} at large N and can be extracted from the Bethe potential via a universal index theorem.
Models of AdS_2 Backreaction and Holography
hep-th 2014-02 accept novelty 7.0 of 10

In a solvable 1+1 dimensional dilaton gravity model, matter backreaction is a relevant effect at low energies that explicitly breaks the conformal symmetry of AdS2 holography for compact transverse spaces.
Near-horizon symmetries of extremal black holes
hep-th 2007-05 accept novelty 7.0 of 10

Extremal black holes with D-3 rotational symmetries necessarily develop SO(2,1) near-horizon symmetry, proving an assumption of the rotating attractor mechanism.
Higher Derivative Corrections to Non-supersymmetric Extremal Black Holes in N=2 Supergravity
hep-th 2006-03 accept novelty 7.0 of 10

For non-supersymmetric extremal black holes in N=2 supergravity, higher-derivative corrected entropies computed in four dimensions disagree with the five-dimensional Kraus-Larsen prediction, indicating missing terms i...
Exact Counting of Black Hole Microstates
hep-th 2004-09 conditional novelty 7.0 of 10

Including all higher-curvature F-term corrections in Wald's entropy formula reproduces, to all orders in a large-charge expansion, the microscopic degeneracy of two-charge BPS black holes in N=4 string theory.
U(1) Charges and Moduli in the D1-D5 System
hep-th 1999-05 accept novelty 7.0 of 10

The D1-D5 conformal field theory's moduli space has one weakly coupled cusp for each factorization of N into D1 and D5 charges, with the singular loci forming arcs that connect pairs of cusps.
Universality of Sypersymmetric Attractors
hep-th 1996-03 conditional novelty 7.0 of 10

The horizon area of supersymmetric black holes in N=2,4,8 supergravity is universally given by the square of the largest central charge extremized in moduli space.
How to get an attractive model
hep-th 2026-10 conditional novelty 6.0 of 10

At phi = 1/8, a Calabi-Yau operator's zeta data factors into pieces matching two modular forms, giving evidence for a rank-two attractor, though the operator was already in a thesis.
Scalarized Einstein-Euler-Heisenberg black holes at the approach to extremality
gr-qc 2026-09 accept novelty 6.0 of 10

The authors derive a universal logarithmic accumulation law for scalarized Einstein-Euler-Heisenberg black holes approaching extremality and construct extremal solutions with constant and nonconstant scalar hair.
IR Black Hole Instabilities Trigger Species-Scale Particle Production
hep-th 2026-08 conditional novelty 6.0 of 10

A mechanism is proposed in which black hole instability at the tower scale converts a fraction of the mass into particles at the species scale, with Hawking evaporation subdominant.
Intersection Bounds for BPS Strings in Six-Dimensional Supergravity
hep-th 2026-07 conditional novelty 6.0 of 10

Using a lattice Zariski decomposition of the gravitational charge and E8 current-algebra embeddings, the paper bounds previously uncontrolled BPS-string intersections: C_-3·e ≤ 3, C_-2·e ≤ 7, and isolated gauged C_-2·...
The Fate of Black Hole-Induced Moduli Excursions in the Presence of Scalar Potentials
hep-th 2026-07 conditional novelty 6.0 of 10

Black-hole moduli throats survive or fail according to how a potential’s force, sign, oscillations, or barrier distance behave along the GHS scalar trajectory, not by the mere presence of a mass.
Moduli-dependent one-loop entropy of hyperbolic BPS black hole in AdS$_4$
hep-th 2026-02 conditional novelty 6.0 of 10

The one-loop logarithmic entropy correction for a hyperbolic BPS black hole in AdS4 depends on the unfixed horizon scalar modulus and acts as a quantum potential that lifts the classical flat direction.
Only Flat Spacetime is Full BPS in Four Dimensional N=3 and N=4 Supergravity
hep-th 2025-05 unverdicted novelty 6.0 of 10

Flat spacetime is the only fully supersymmetric solution in four-dimensional N=3 and N=4 higher derivative Poincaré supergravity, unlike N=2 where Bertotti-Robinson geometry also qualifies.
Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes
hep-th 2020-04 conditional novelty 6.0 of 10

BPS states in infinite-distance limits of N=2 Calabi-Yau compactifications lie on a two-radius ellipsoid in charge-to-mass space, saturate the black hole extremality bound, and force the Swampland Distance Conjecture ...
Gluing gravitational blocks for AdS black holes
hep-th 2019-09 conditional novelty 6.0 of 10

A universal entropy functional built from glued gravitational blocks reproduces the entropy and horizon scalar values of rotating BPS black holes and strings in AdS4 and AdS5.
Black holes in Sol minore
hep-th 2019-08 conditional novelty 6.0 of 10

A magnetically charged five-dimensional gauged-supergravity black hole with a Sol horizon is constructed, with no-go results against its supersymmetric limit and an attractor entropy formula for the extremal case.
Einstein-Maxwell-scalar black holes: classes of solutions, dyons and extremality
hep-th 2019-05 accept novelty 6.0 of 10

Dyonic (electrically and magnetically charged) scalarised black holes admit an extremal limit with nonzero horizon area, and the attractor/entropy-function formalism shows that both charges are necessary for such a li...
Topologically twisted indices in five dimensions and holography
hep-th 2018-08 conditional novelty 6.0 of 10

A contour-integral gluing formula for the 5D topologically twisted index on M4 x S1, with large-N limits reproducing AdS7 holography and predicting AdS6 black hole entropy.
Asymptotically-flat supergravity solutions deep inside the black-hole regime
hep-th 2017-11 conditional novelty 6.0 of 10

Smooth asymptotically-flat supergravity solutions with the same charges as D1-D5-P black holes are constructed, with arbitrarily small angular momentum and identified dual CFT states.
An extremization principle for the entropy of rotating BPS black holes in AdS$_5$
hep-th 2017-05 conditional novelty 6.0 of 10

The Bekenstein-Hawking entropy of 1/16 BPS rotating black holes in AdS5 x S5 equals the Legendre transform of a Casimir-energy-like functional of four chemical potentials, subject to a linear constraint.
The Cardy limit of the topologically twisted index and black strings in AdS$_5$
hep-th 2016-11 conditional novelty 6.0 of 10

The high-temperature topologically twisted index of a 4d N=1 quiver gauge theory is proportional to the left-moving central charge of the 2d (0,2) SCFT obtained by compactification on S squared.
Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates
hep-th 2011-08 accept novelty 6.0 of 10

Half-BPS black holes in N=2 supergravity carry a logarithmic entropy correction of (1/12)(23 + n_H - n_V) ln(A_H/G_N), fixed entirely by the massless spectrum and computable from the near-horizon geometry.
Static supersymmetric black holes in AdS_4 with spherical symmetry
hep-th 2010-12 conditional novelty 6.0 of 10

Static quarter-BPS asymptotically AdS4 black holes with spherical horizons exist in N=2 Fayet-Iliopoulos gauged supergravity, provided the scalar fields run with radius and are given by harmonic functions.
Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function
hep-th 2010-05 accept novelty 6.0 of 10

The one-loop contribution of all matter multiplet fields to the logarithmic correction of quarter BPS black hole entropy cancels to zero, making the correction independent of the number of matter multiplets.
Supersymmetric AdS_4 black holes and attractors
hep-th 2009-11 accept novelty 6.0 of 10

Constructs the first explicit supersymmetric AdS4 black holes with nonconstant scalar fields and derives the attractor equations for gauged supergravity.
Almost BPS black holes
hep-th 2008-12 conditional novelty 6.0 of 10

A new class of non-BPS black hole solutions in 4D and 5D supergravity is constructed by patching locally supersymmetric solutions, yielding first-order flow equations and multi-centre configurations.
Flow Equations for Non-BPS Extremal Black Holes
hep-th 2007-02 accept novelty 6.0 of 10

A constant symplectic matrix commuting with the black hole potential's symplectic matrix yields a fake superpotential whose critical points are stable non-BPS extremal black holes obeying first-order flow equations.
Rotating Attractors
hep-th 2006-06 accept novelty 6.0 of 10

The entropy function formalism extends to rotating extremal black holes: extremizing a function of charges and angular momentum yields the entropy and near-horizon geometry, independent of asymptotic moduli when the e...
On N=8 attractors
hep-th 2006-03 conditional novelty 6.0 of 10

The N=8 black hole attractor conditions reduce to a simple algebraic system with exactly two regular solutions, BPS and non-BPS, yielding the known entropy formulas from an explicit derivation.
Entropy Maximization in the Presence of Higher-Curvature Interactions
hep-th 2006-03 conditional novelty 6.0 of 10

Higher-curvature corrections preserve and enhance entropy maxima at conifold-type singularities for real topological string coupling, but destroy them for complex coupling.
Real symplectic formulation of local special geometry
hep-th 2006-03 conditional novelty 6.0 of 10

For local special geometry, the real symplectic metric g_IJ is explicitly derived as -1/(2S) H_IJ + 1/(4S^2)(S_I S_J + (HΩS)_I(HΩS)_J).
The supersymmetric configurations of N=2, d=4 supergravity coupled to vector supermultiplets
hep-th 2006-03 accept novelty 6.0 of 10

All supersymmetric configurations of ungauged N=2, d=4 supergravity with n vector multiplets are either timelike Behrndt-Lust-Sabra configurations or null Brinkmann pp-waves, with equations of motion reducing to harmo...
Partition function of non-supersymmetric black holes in the supergravity limit
hep-th 2006-02 conditional novelty 6.0 of 10

The paper proposes Z_ns = exp(pi^2 D(p)/(2*phi0)) as the mixed partition function of non-supersymmetric attractors, engineered so its Legendre transform equals the Bekenstein-Hawking entropy.
Observations on the Darboux coordinates for rigid special geometry
hep-th 2006-02 conditional novelty 6.0 of 10

In rigid special geometry in Darboux coordinates, the matrix M(Re F, Im F) equals minus the Hessian of the Hamiltonian S(p,q), so the Kähler metric and the c-map hyperkähler metric both follow from S.
Holographic Gravitational Anomalies
hep-th 2005-08 accept novelty 6.0 of 10

A gravitational Chern-Simons term in AdS3 shifts the left and right central charges oppositely, gives empty AdS3 a nonzero angular momentum of 4 pi beta, and fixes black hole entropy corrections exactly.
General Concentric Black Rings
hep-th 2004-08 accept novelty 6.0 of 10

Explicit supersymmetric black ring solutions of five-dimensional supergravity coupled to an arbitrary number of vector multiplets are constructed, including concentric multi-ring configurations with regular S1 x S2 ho...
Permeable conformal walls and holography
hep-th 2001-11 accept novelty 6.0 of 10

The Casimir energy between two permeable conformal interfaces in two dimensions is an exact dilogarithm of the reflection probability, and matches a classical stretched-string computation in AdS3 in the supergravity limit.
Six-Dimensional Supergravity on S^3 X AdS_3 and 2d Conformal Field Theory
hep-th 1998-06 conditional novelty 6.0 of 10

The Kaluza-Klein spectrum of 6d supergravity on S^3 x AdS_3 agrees with the K3^N/S_N orbifold CFT, and the non-chiral primaries Vafa counted via the elliptic genus arise as products of descendants of chiral primaries.
Machine learning automorphic forms for black holes
hep-th 2025-05 conditional novelty 5.0 of 10

Feed-forward neural networks trained on Fourier coefficients can predict modular weights for negative-weight powers of eta and E2, and for simple Jacobi theta products, within the training range.
Interpolating between multi-center microstate geometries
hep-th 2021-05 unverdicted novelty 5.0 of 10

Constructs a 2-center helical-profile solution that interpolates between two circular-profile Lunin-Mathur microstate geometries while exhibiting charge delocalization and transfer between centers.
M-theory, Black Holes and Cosmology
hep-th 2020-09 conditional novelty 5.0 of 10

New algebraic identities connect STU black hole entropy to Bhargava cube triality, and fermion mass eigenvalues in octonion cosmology to the exceptional Jordan eigenvalue problem.
Extremal and Near-Extremal Black Holes and Near-$CFT_1$
hep-th 2018-08 conditional novelty 5.0 of 10

Charged extremal black holes respond at low energies through a time-reparametrisation mode and a gauge phase mode, both suppressed by the horizon scale, with higher partial waves contributing at somewhat higher energies.
A double copy for ${\cal N}=2$ supergravity: a linearised tale told on-shell
hep-th 2016-09 conditional novelty 5.0 of 10

An on-shell double copy dictionary is constructed for linearised N=2 supergravity with one vector multiplet, and shown to reproduce the weak-field dyonic BPS black hole from an electric gauge-theory configuration.
Integrable Scalar Cosmologies I. Foundations and links with String Theory
hep-th 2013-07 conditional novelty 5.0 of 10

Nine families of exactly solvable one-scalar cosmologies with exponential potentials are derived and solved in closed form, and a string-theoretic origin for such potentials is proposed with a discrete formula for the...
State Operator Correspondence and Entanglement in AdS_2/CFT_1
hep-th 2011-01 conditional novelty 5.0 of 10

In the AdS2/CFT1 correspondence the black hole entropy equals both the log degeneracy of CFT1 ground states and the entanglement entropy between the two AdS2 boundaries, and the dual bulk states are U(N)-twisted Hartl...
Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
hep-th 2008-09 conditional novelty 5.0 of 10

The degeneracy of an extremal black hole's microstates equals the finite part of a string-theory path integral on its near-horizon AdS2 geometry, with a Wilson-line insertion, reducing to Wald entropy in the classical limit.
Born-Infeld Corrections to the Entropy Function of Heterotic Black Holes
hep-th 2006-04 conditional novelty 5.0 of 10

Born-Infeld terms change the near-horizon geometry and entropy of two-charge heterotic black holes, but only yield a consistent attractor when Gauss-Bonnet terms are present.
Species Entropy and Thermodynamics
hep-th 2023-05 conditional novelty 4.0 of 10

Particle species in quantum gravity are assigned an entropy equal to the Bekenstein-Hawking entropy of a minimal black hole and a temperature equal to the species scale, and these quantities are argued to obey laws an...
Multi-centered ${\mathcal N}=2$ BPS black holes: a double copy description
hep-th 2016-11 conditional novelty 4.0 of 10

An on-shell double copy dictionary for linearised N=2 supergravity with any number of vector multiplets is built and used to re-express multi-centered BPS black holes as convolutions of gauge-theory data.
Classification of near-horizon geometries of extremal black holes
hep-th 2013-06 accept novelty 4.0 of 10

A unified survey of near-horizon geometries of extremal black holes, with new complete classifications in three-dimensional vacuum and Einstein-Maxwell-Chern-Simons theory and for homogeneous five-dimensional horizons.
Using the D1D5 CFT to Understand Black Holes
hep-th 2010-12 conditional novelty 3.0 of 10

The D1D5 orbifold CFT reproduces, mode by mode, the emission spectrum of the JMaRT horizonless geometries, supporting the reading of their ergoregion instability as Hawking radiation from black hole microstates.
Black Holes, Black Rings and their Microstates
hep-th 2007-01 conditional novelty 3.0 of 10

Smooth horizonless three-charge geometries with black-hole and black-ring charges exist in large numbers, and heuristic mass-gap matching suggests many are dual to CFT states in the same sector as typical black-hole m...
Lectures on AdS Black Holes, Holography and Localization
hep-th 2019-02 accept novelty 2.0 of 10

Localization computations of a topologically twisted index in the dual conformal field theory reproduce, at large N, the Bekenstein-Hawking entropy of dyonic AdS4 black holes, and the lectures survey the analogous pro...
The Kerr/CFT correspondence and its extensions: a comprehensive review
hep-th 2012-03 conditional novelty 2.0 of 10

A chiral half of Cardy's formula reproduces the Bekenstein-Hawking entropy of essentially every known extremal black hole, but the review argues no standard dual CFT exists.

See all 62 Pith citations

How later papers used it

186 later papers cite it

Evidence: some. 179 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 8 of the 259 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 built on it and 1 used its method or tool.

Builds on it 1 of 4 papers with this label

Used this to uncover new general properties of near horizon regions of small black holes, regarded as an extension of the attractor mechanism for large extremal black holes.

Fake supersymmetry versus Hamilton-Jacobi (2012) · 0.99 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 8 papers with this label

Used this paper's attractor equations to fix the moduli at the black hole horizon in terms of the charges and compute the entropy.

A note on modular invariant species scale and potentials (2023) · 0.90 · paraphrased: the licence does not allow quoting

How the literature uses it

175 arXiv papers cite this in 247 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 136 sentences without a confident label

Builds on it 4 papers

We are interested in finding a generalization of the attractor mechanism of [ref] for cases which include not only constant values of the scalars $ϕ^i$ and tangent space field strengths $F_ab= e_a^μ e_b^ν F_μν$ (fluxes), but also constant values of the tangent space vectors $A_a= e^μ_a A_μ$.

Generalized Attractor Points in Gauged Supergravity · Generalized Attractors in D=4 Supergravity · 0.95
We use this in section [ref] to uncover new general properties of the near horizon regions of small black holes, which can be regarded as an extension of the attractor mechanism for large extremal black holes [ref] .

Fake supersymmetry versus Hamilton-Jacobi · Introduction · 0.99
In [ref] the authors present a generalization of the well-known black hole attractor mechanism in $N=2$ ungauged supergravity introduced in [ref] to extremal static black holes in four-dimensional $N=2$ $FI$ supergravity coupled to abelian vector multiplets.

Supersymmetric objects in gauged supergravities · Attractors and Hypermultiplets · 0.89

Uses its method or tool 9 papers

It turns out that in these cases known techniques (the “attractor mechanism" [ref] ) available from study of macroscopic black holes and black strings for large charges (where we rescale the cycle class by a large integer) can give reliable predictions about properties of non-BPS objects and these, in turn, lead to predictions about existence, stability, and asymptotic count for for minimal volume 2- and 4-cycles in the Calabi-Yau.

Non-Holomorphic Cycles and Non-BPS Black Branes · Introduction · 0.95
To find the area correction $δA_h$ we use the attractor mechanism [ref] .

Derivative Corrections to Extremal Black Holes with Moduli · The area correction via the attractor mechanism · 0.99
The key ingredient is the lattice Zariski decomposition $b_0=P+N$ of the gravitational charge $b_0$, whose physical content is the attractor flow of the gravitational BPS black string [ref] .

Intersection Bounds for BPS Strings in Six-Dimensional Supergravity · Discussion · 0.80

Compares with it 2 papers

Furthermore, it is easy to see that in the limit $g →0$ where the gauging vanishes one recovers the well-known first order attractor flow equations of black holes in ungauged $N=2$ supergravity [ref] .

Static supersymmetric black holes in AdS_4 with spherical symmetry · BPS conditions and attractor flow · 0.80
With this ansatz, the equations give = -2 = + i = i/R e^iψ_0 These equations are in fact identical to the attractor equations for solutions of the form $AdS_2×S^2$ in ungauged $=2$ supergravity [ref] with the obvious replacement of the gauging parameters $$ with charges $$.

Supersymmetric Black Holes in AdS4 from Very Special Geometry · General $AdS_4$ solutions · 0.87

Cites it as background 80 papers

The entropy function method was developed by Sen [ref] , and was first applied for the computation of the entropy and the study of the attractor mechanism [ref] for black holes with spherical symmetry.

Conductivities from attractors · Entropy function · 0.88
The first one is the “background”: our starting point is an AdS$_2$ solution, characterized by a set of electric and magnetic charges, and the six scalar fields are controlled by the attractor mechanism [ref] .

Near-AdS$_2$ Spectroscopy: classifying the spectrum of operators and interactions in $\mathcal{N} = 2$ 4D supergravity · Introduction · 0.87
In fact, by defining the Freudenthal transformation acting on the e.m. black hole charges as [equation] it holds that [ref] [equation] In this framework, the Freudenthal map ([ref]) is anti-involutive since it can be recast in the following form: [equation] where $ℳ_H(𝒬)$ denotes the matrix $ℳ$ of special Kähler geometry (see e.g. [ref] and Refs. therein), evaluated at the (unique) event horizon of the extremal black hole (at which the attractor mechanism [ref] takes place; see also ).

Near-Extremal Freudenthal Duality · $T=0$ : invariance of extremal entropy · 0.86

Discussion

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