The Non-BPS Black Hole Attractor Equation

Masoud Soroush; Navin Sivanandam; Renata Kallosh

REVIEW

The Non-BPS Black Hole Attractor Equation

Renata Kallosh, Navin Sivanandam, Masoud Soroush

High Energy Physics - Theory · Submitted 31 Jan 2006 · 26 references

Pith's verdict: A clean rewriting of the non-BPS extremum condition plus one genuinely new double-extremal solution, but the 'equivalent to minimizing V_BH' claim is asserted rather than proved and the Section 5 checks are consistency checks of the algebra.

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CitationMasoud Soroush, Navin Sivanandam, Renata Kallosh. "The Non-BPS Black Hole Attractor Equation." arXiv:hep-th/0602005 (2006). https://pith.science/paper/47OJCRUA

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Bottom line: this is a competent technical note that gives a usable algebraic form of the non-BPS attractor condition and one new explicit solution, but the headline claim — that solving Eq. (4.12) is equivalent to minimizing V_BH — is asserted, not shown, and the Section 5 checks are consistency checks of the algebra rather than independent evidence for the physics. [abstract:s1, abstract:s4] say as much: a derivation, then examples that reproduce Tripathy-Trivedi by another route. That is what the paper actually delivers, and it is worth having in that form. What is new and useful: Eq. (4.12), which rewrites the Ferrara-Gibbons-Kallosh extremum condition directly in the charges. The double-extremal non-BPS solution in 5.2, obtained by q0 -> -q0 in the BPS solution, is concrete and checkable. Appendix B's separation of the fully covariant D (with Christoffel) from the nabla used in Tripathy-Trivedi is the kind of notational point that actually causes confusion in this literature, and it is done correctly. The mirror quintic computation landing on n1/n2 = 0.381 and -0.319 through a different route than Tripathy-Trivedi is a real cross-check. Where it is soft: the derivation of (4.12) runs through the Hodge identity (4.8), which Appendix C itself shows is trivially satisfied at every point of moduli space. So all the content is in the extremum condition (4.10), and (4.12) is a rewriting of it. The converse — every solution of (4.12) extremizes V_BH — is what makes (4.12) a faithful attractor equation, and it is stated as 'We claim...' with no proof. In 5.1 the one-parameter ansatz z^a = i p^a t gives exactly the two roots Y=4, Y=2, so no spurious roots appear there, but that is one real equation for one real unknown with p^0 = q_a = 0. For generic charges (4.12) is 2N+2 equations for 2N moduli, and whether the solution set coincides with the critical points depends on an invertibility the paper does not establish. This is a moderate gap, not a fatal one — the direction actually used (extremum implies (4.12)) is fine. The Section 2 throat argument is heuristic and the authors say so; I would not weight it. The O'Raifeartaigh analogy in the Discussion is suggestive but speculative. Who should read it: anyone working on non-BPS attractors or double-extremal solutions in N=2 special geometry. It is a useful reference for the equation form and the worked examples. Recommendation: send it out. It is a legitimate contribution to a live program, the algebra is clean, and the double-extremal non-BPS solution deserves a referee's check. Expect a request to soften the 'equivalent to minimization' claim or prove the converse.

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Major objections (3)

[Section 3 and Section 5] Section 3 states that only minima of V_BH are true attractors (citing [7,8]), but Section 5 only finds critical points and does not verify that the non-BPS solutions (Y=2 in (5.26), n1/n2 = -0.319 in (5.58)) are minima (positive Hessian). This is load-bearing for the claim that these are attractors. The paper should either compute the Hessian or explicitly state that stability is taken from [8].
[Section 4 and Appendix C] The derivation of (4.12) uses the Hodge identity (4.8) and the extremum condition (4.10). Appendix C shows that (4.8) is an identity, so (4.12) is exactly equivalent to (4.10), not merely a necessary condition. The main text says 'We claim' rather than making this equivalence explicit, which invites a circularity concern: the Section 5 checks that solving (4.12) agrees with minimizing V_BH are then consistency checks of the algebra, not independent evidence for the attractor mechanism. The paper should state the equivalence clearly and explain that the checks are algebraic consistency checks.
[Section 5.1, Eq. (5.23)] Combining (5.8) and (5.23) with W = YDt^2, the second row of (5.25) reduces to Y=4, while the third row reduces to (Y-4)(Y-2)=0; the two rows are inconsistent as printed. The final result (5.26) is correct (it agrees with (5.28)), so the discrepancy is likely a sign error in the prefactor of (5.23): it should be +Dt^2/(2W) rather than -Dt^2/(2W) to be consistent with (5.25). Please check.

Minor objections (7)

[Eq. (4.12)] Eq. (4.12) has Dbar_a Dbar_b Z (unbarred Z) in the second term, while Section 5.1 uses Dbar_a Dbar_b Zbar; the numerator should be Zbar to be consistent with (4.11) and (5.23).
[Eq. (2.11)] Eq. (2.11) asserts |DZ|^2 - |Sigma|^2 > 0 for non-BPS extremal black holes without justification; this positivity should be explained or qualified.
[Section 2] In Section 2, the derivation of (2.28) assumes that all derivatives of z^a with respect to omega vanish at the horizon, which requires z^a to be analytic in e^omega near the horizon. This assumption should be stated.
[Appendix A] Appendix A uses mu and nu for Im N and Re N, which is unusual and could be confused with spacetime indices; a clarifying sentence would help.
[Eq. (5.58)] Eq. (5.58) gives the numerical coefficients 0.063 and 0.121 without exact expressions; providing the exact coefficients would make the result easier to check.
[Section 5.3] Section 5.3 keeps only linear terms in psi; the paper should state more explicitly that the non-BPS solution is established only at linear order and discuss whether it persists at higher order.
[Section 4] Section 4 claims that (4.12) is valid for 'any non-BPS black holes in special geometry'; the derivation uses the specific Hodge decomposition (4.7) of N=2 special geometry, and the scope of the claim should be stated more precisely.

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The paper

The paper claims

The paper derives a single algebraic equation that fixes the shape moduli of a non-supersymmetric extremal black hole at its horizon, and argues that only zero-temperature black holes can do this at all.

The reading

A black hole made of charged fluxes in a Calabi–Yau compactification drags the surrounding shape moduli of the internal space to fixed values at its horizon, whatever those moduli were far away. For supersymmetric black holes this attractor behaviour is classical and well understood; the question here is whether it survives when supersymmetry is broken. The authors argue physically that what matters is not supersymmetry but extremality: an extremal black hole has an infinite throat, so the proper distance to the horizon is infinite, and a scalar rolling down the black hole potential has unlimited 'time' to settle at the critical point. A non-extremal black hole has a finite distance to its horizon, so the scalar arrives with its initial conditions still imprinted. They then derive an explicit attractor equation, $h = 2Im[ZΠ - (D̄_aD̄_bZ)G^{ac}G^{bd}D_cZ D_dΠ/(2Z)]$, whose solutions are supposed to be exactly the horizon values of the moduli, and check it against three worked examples.

What carries the argument

The load-bearing object is the Hodge decomposition of the three-form flux threading the Calabi–Yau, $H_3 = 2Im[ZΩ_3 + D̄_aZ G^{āa}D_aΩ_3]$, which is valid at any point of moduli space and therefore carries no information about the horizon by itself. The dynamical input comes from the extremum condition $2(D_aZ)Z̄ + iC_{abc}G^{bd}G^{ce}D̄_dZ̄ D̄_eZ̄ = 0$, which lets the authors replace $D_aZ$ by a quadratic expression in $D̄Z̄$ and substitute back into the decomposition. A special-geometry identity, $D_aD_bΩ = iC_{abc}G^{c d̄}D̄_{d̄}Ω$, removes the second derivatives of $Ω$ from the expansion so that only $Ω$, $DΩ$ and $D̄Ω$ appear. The resulting equation is algebraic in the charges and the moduli, so the horizon values can be read off without solving any flow.

Extended reading notes

Core claim

The central claim is that the non-supersymmetric attractor equation (4.12) holds in $N=2$ special geometry for BPS and non-BPS extremal black holes alike, and that solving it in terms of the electric and magnetic charges is equivalent to extremizing the black hole potential $V_{BH}=|Z|^2+|DZ|^2$. For BPS solutions the second term vanishes because $D_cZ=0$ and the equation collapses to the familiar $h=2Im[ZΠ]$; for non-BPS solutions the two terms must conspire so that the moduli become functions of the charges. The authors verify this in a large-volume Calabi–Yau with only D0 and D4 charges, where both the supersymmetric branch $t^2=q_0/D$ and the non-supersymmetric branch $t^2=-q_0/D$ fall out of a single algebraic condition $(Y-4)(Y-2)=0$; in a double-extremal non-BPS solution obtained by flipping the sign of $q_0$; and in the mirror quintic near the Gepner point, where the ratio $n_1/n_2$ solves a quadratic with roots $0.381$ (supersymmetric) and $-0.319$ (non-supersymmetric).

Load-bearing premise

The equation is obtained by feeding the condition that the black hole potential has a critical point into a flux decomposition that the authors themselves note holds everywhere in moduli space, so the equation's entire content is the potential-minimization picture, and the agreement with minimizing that potential in the examples is guaranteed rather than independent.

Editorial extensions

If this is right

Horizon moduli of non-BPS extremal black holes are computable from charge data alone, by solving one algebraic equation rather than an ordinary differential equation along the radial flow.
The same charge vector can admit both a supersymmetric and a non-supersymmetric attractor, so which one is realized is a separate dynamical question rather than a consequence of the charges.
Double-extremal non-BPS black holes exist and are obtained from their BPS counterparts by continuing the electric charge $q_0 \to -q_0$, with Reissner–Nordström geometry and constant scalars everywhere.
Because the non-BPS vacuum has no supersymmetric minimum to decay into, and because the black hole is at zero temperature, such configurations are stable against both decay and evaporation.
Attractor behaviour is tied to extremality rather than to unbroken supersymmetry, which reframes the mechanism as a statement about near-horizon geometry.

Reading between the lines

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

The infinite-throat argument suggests a sharp diagnostic in other settings: any horizon with a finite proper distance should show scalars that remember their asymptotic values, which is in principle checkable numerically by shooting from several initial conditions.
Since the attractor equation is algebraic in the charges, it can be read as a map from a charge vector to a point in moduli space, and one could ask about the image of that map — which moduli values are reachable by some choice of fluxes.
The paper's analogy with the O'Raifeartaigh model suggests looking for string compactifications whose effective superpotential admits no supersymmetric critical point at all, which would make the non-supersymmetric vacuum stable by the same logic.
The authors flag in the discussion that absence of a spherical BPS state is not enough, because multi-centred BPS composites could still exist for the same charges; this is a concrete gap that a systematic search over charge vectors could close.

Original abstract

We study the attractor mechanism for extremal non-BPS black holes with an infinite throat near horizon geometry, developing, as we do so, a physical argument as to why such a mechanism does not exist in non-extremal cases. We present a detailed derivation of the non-supersymmetric attractor equation. This equation defines the stabilization of moduli near the black hole horizon: the fixed moduli take values specified by electric and magnetic charges corresponding to the fluxes in a Calabi Yau compactification of string theory. They also define the so-called double-extremal solutions. In some examples, studied previously by Tripathy and Trivedi, we solve the equation and show that the moduli are fixed at values which may also be derived from the critical points of the black hole potential.

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, 4,305 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.70 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.81 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We study the attractor mechanism for extremal non-BPS black holes with an infinite throat near horizon geometry, developing, as we do so, a physical argument as to why such a mechanism does not exist in non-extremal cases. We present a detailed derivation of the non-supersymmetric attractor equation. This equation defines the stabilization of moduli near the black hole horizon: the fixed moduli take values specified by electric and magnetic charges corresponding to the fluxes in a Calabi Yau compactification of string theory. They also define the so-called double-extremal solutions. In some examples, studied previously by Tripathy and Trivedi, we solve the equation and show that the moduli are fixed at values which may also be derived from the critical points of the black hole potential.

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 study the attractor mechanism for extremal non-BPS black holes with an infinite throat near horizon geometry, developing, as we do so, a physical argument as to why such a mechanism does not exist in non-extremal cases.	contribution 0.99	0.93	0.37	asserted 0.99
s1	We present a detailed derivation of the non-supersymmetric attractor equation.	contribution 0.95	0.96	0.19	asserted 0.99
s2	This equation defines the stabilization of moduli near the black hole horizon: the fixed moduli take values specified by electric and magnetic charges corresponding to the fluxes in a Calabi Yau compactification of string theory.	interpretation 0.57	0.58	0.27	asserted 1.00
s3	They also define the so-called double-extremal solutions.	interpretation 0.69	0.37	0.09	asserted 1.00
s4	In some examples, studied previously by Tripathy and Trivedi, we solve the equation and show that the moduli are fixed at values which may also be derived from the critical points of the black hole potential.	result 0.73	0.82	0.12	hedged possibility 0.56

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 v2

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.** On the left there is an Euclidean section of the near horizon “cigar” geometry of the nonextremal black holes with non-vanishing temperature. On the right there is an Euclidean section of an infinite throat near horizon geometry of the extremal black holes with vanishing temperature. We start by deriving the equations of motion for the scalar field from the Lagrangian: L

**Figure 2.** Evolution of the dilaton field e −2z for various initial conditions at ρ = 0 (r = ∞) to a common fixed point at ρ = −∞ (r = 0) where ρ = − 1 r . The special case of the horizontal line represents a double-extremal black hole. We call these black holes double-extremal since they pick up the values of the moduli at infinity that extremize the black hole mass; values that are equal to those at the horizon. There is no energy stored in the scalars, so ∂z ∂τ = 0 everywhere. While for the BPS case DZ = 0 everywhere, the most general conditions are that double-extremal black holes (BPS

Figures reproduced from arXiv:hep-th/0602005 by the authors.

References

Works cited (26)

[8]

P. K. Tripathy and S. P. Trivedi, “Non-Supersymmetric Attract ors in String Theory,” [arXiv:hep-th/0511117]

View on Pith Pith review arXiv
[1]

S. Ferrara, R. Kallosh and A. Strominger, “N=2 extremal black h oles,” Phys. Rev. D 52, 5412 (1995) [arXiv:hep-th/9508072]

View on Pith Pith review arXiv 1995
[2]

A. Strominger, “Macroscopic Entropy of N = 2 Extremal Black Holes,” Phys. Lett. B 383, 39 (1996) [arXiv:hep-th/9602111]

View on Pith Pith review arXiv 1996
[3]

S. Ferrara and R. Kallosh, “Supersymmetry and Attractors,” P hys. Rev. D 54, 1514 (1996) [arXiv:hep-th/9602136]; S. Ferrara and R. Kallosh, “Universality of Supersymmetric Attractors,” Phys. Rev. D 54, 1525 (1996) [arXiv:hep-th/9603090]

View on Pith Pith review arXiv 1996
[4]

G. W. Moore, “Les Houches lectures on strings and arithmetic,” a rXiv:hep-th/0401049; G. W. Moore, “Arithmetic and attractors,” [arXiv:hep-th/9807087 ]

View on Pith Pith review arXiv
[5]

F. Denef, B. R. Greene and M. Raugas, “Split attractor ﬂows an d the spectrum of BPS D-branes on the quintic,” JHEP 0105, 012 (2001) [arXiv:hep-th/0101135]

View on Pith Pith review arXiv 2001
[6]

S. Ferrara, G. W. Gibbons and R. Kallosh, “Black holes and critical points in moduli space,” Nucl. Phys. B 500, 75 (1997) [arXiv:hep-th/9702103]. G. W. Gibbons, “Supergravity vacua and solitons,” Prepared for A Newton Institute Euroconference on Duality and Supersymmetric Theories, Ca mbridge, England, 7-18 Apr 1997, PROCEEDINGS. Edited by D.I. Olive, P.C...

View on Pith Pith review arXiv 1997
[7]

K. Goldstein, N. Iizuka, R. P. Jena and S. P. Trivedi, “Non-super symmetric attractors,” [arXiv:hep-th/0507096]; A. Sen, “Black hole entropy function and t he attractor mechanism in higher derivative gravity,” JHEP 0509, 038 (2005) [arXiv:hep-th/0506177]; M. Alishahiha and H. Ebrahim, “Non-supersymmetric attractors an d entropy function,” [arXiv:hep-th/0601016]

View on Pith Pith review arXiv 2005
[9]

R. Kallosh, M. Shmakova and W. K. Wong, “Freezing of moduli by N = 2 dyons,” Phys. Rev. D 54, 6284 (1996) [arXiv:hep-th/9607077]

View on Pith Pith review arXiv 1996
[10]

K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W. K. Won g, “STU black holes and string triality,” Phys. Rev. D 54, 6293 (1996) [arXiv:hep-th/9608059]

View on Pith Pith review arXiv 1996
[11]

K. Behrndt, G. Lopes Cardoso, B. de Wit, R. Kallosh, D. Lust an d T. Mohaupt, Nucl. Phys. B 488, 236 (1997) [arXiv:hep-th/9610105]

View on Pith Pith review arXiv 1997
[12]

R. Kallosh, “New attractors,” JHEP 0512, 022 (2005) [arXiv:hep-th/0510024]

View on Pith Pith review arXiv 2005
[13]

A. Giryavets, “New attractors and area codes,” [arXiv:hep-th /0511215]

View on Pith
[14]

S. Ferrara, M. Bodner and A. C. Cadavid, “Calabi-Yau Supermo duli Space, Field Strength Duality And Mirror Manifolds,” Phys. Lett. B 247, 25 (1990)

View on Pith 1990
[15]

F. Denef and M. R. Douglas, “Distributions of ﬂux vacua,” JHEP 0405, 072 (2004) [arXiv:hep-th/0404116]

View on Pith Pith review arXiv 2004
[16]

G. W. Gibbons and R. E. Kallosh, “Topology, entropy and Witten in dex of dilaton black holes,” Phys. Rev. D 51, 2839 (1995) [arXiv:hep-th/9407118]

View on Pith Pith review arXiv 1995
[17]

R. Kallosh, A. D. Linde, T. Ortin, A. W. Peet and A. Van Proeyen, “Supersymmetry as a cosmic censor,” Phys. Rev. D 46, 5278 (1992) [arXiv:hep-th/9205027]. – 30 –

View on Pith Pith review arXiv 1992
[18]

B. de Wit, P. G. Lauwers and A. Van Proeyen, “Lagrangians Of N =2 Supergravity - Matter Systems,” Nucl. Phys. B 255, 569 (1985)

View on Pith 1985
[19]

A. Ceresole, R. D’Auria, S. Ferrara and A. Van Proeyen, “Dualit y transformations in supersymmetric Yang-Mills theories coupled to supergravity,” Nucl. Phys. B 444, 92 (1995) [arXiv:hep-th/9502072]

View on Pith Pith review arXiv 1995
[20]

B. de Wit and A. Van Proeyen, “Potentials And Symmetries Of Gen eral Gauged N=2 Supergravity - Yang-Mills Models,” Nucl. Phys. B 245, 89 (1984); A. Ceresole, R. D’Auria and S. Ferrara, “The Symplectic Structure of N=2 Supergravity a nd its Central Extension,” Nucl. Phys. Proc. Suppl. 46, 67 (1996) [arXiv:hep-th/9509160]

View on Pith Pith review arXiv 1984
[21]

R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, Class. Qu ant. Grav. 17, 4269 (2000) [Erratum-ibid. 21, 5017 (2004)] [arXiv:hep-th/0006179]

View on Pith Pith review arXiv 2000
[22]

A. Giryavets, S. Kachru, P. K. Tripathy and S. P. Trivedi, “Flux compactiﬁcations on Calabi-Yau threefolds,” JHEP 0404, 003 (2004) [arXiv:hep-th/0312104]

View on Pith Pith review arXiv 2004
[23]

P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, “A Pa ir Of Calabi-Yau Manifolds As An Exactly Soluble Superconformal Theory,” Nucl. Phys . B 359, 21 (1991)

View on Pith 1991
[24]

L. O’Raifeartaigh, “Spontaneous Symmetry Breaking For Chira l Scalar Superﬁelds,” Nucl. Phys. B 96, 331 (1975)

View on Pith 1975
[25]

K. Intriligator and N. Seiberg, “The runaway quiver,” [arXiv:hep- th/0512347]

View on Pith
[26]

B. Bates and F. Denef, “Exact solutions for supersymmetric s tationary black hole composites,” arXiv:hep-th/0304094. – 31 –

View on Pith Pith review arXiv

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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.
Black Holes, Qubits and Octonions
hep-th 2008-09 accept novelty 4.0 of 10

Black hole entropy and qubit entanglement measures turn out to be the same invariant polynomials, and the review organizes this correspondence while adding explicit dictionaries between three descriptions of E7.
Black Hole Entropy Function, Attractors and Precision Counting of Microstates
hep-th 2007-08 accept novelty 4.0 of 10

The statistical entropy of a class of quarter-BPS dyons in N=4 string theories matches the extremal black hole entropy, including the first inverse-charge correction from a Gauss-Bonnet term.
Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
hep-th 2006-07 conditional novelty 4.0 of 10

Radial quantization of BPS black hole attractor flows yields an exact wave function in terms of a Bessel function, and suggests a one-parameter generalization of the topological string amplitude tied to minimal repres...
${\cal N}=2$ Supergravity in $D=4,5,6$ Dimensions
hep-th 2020-04 accept novelty 2.0 of 10

A book-length, self-contained account of how N=2 supergravity in D=4,5,6 is constructed by superconformal methods, and of the special and quaternionic-Kahler geometries that the scalar sectors realise.
Extremal Black Holes in Supergravity
hep-th 2006-11 accept novelty 1.5 of 10

Extremal black holes in four-dimensional extended supergravity exhibit an attractor mechanism: scalar fields flow to charge-determined fixed points at the horizon, making the Bekenstein-Hawking entropy a U-duality invariant.

How the literature uses it

45 arXiv papers cite this in 74 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 · 41 sentences without a confident label

Uses its method or tool 3 papers

Putting this ansatz into ([ref]) we obtain a one dimensional effective Lagrangian for the radial evolution of the quantities $U(τ)$, $z^a(τ)$, as well as the electric $ξ^I(τ)$, and magnetic $χ_I(τ)$ potentials defined as [ref]

A three-qubit interpretation of BPS and non-BPS STU black holes · STU black holes · 0.85
A convenient metric is e.g. [ref] [equation] Details on this parametrization are given in an appendix of [ref] .

Effective action for the field equations of charged black holes · The black hole potential · 0.87
For this example we will follow [ref] .

Optimal paths across potentials on scalar field space · One-modulus attractor flows · 0.86

Uses its data or values 1 paper

From ([ref]), the superpotential for this system reads [equation] The attractor values of the moduli in the non-SUSY branch have been previously found in [ref] by minimizing the black hole potential $V_BH$ and they are given by [equation] As in the previous example, the signs of the solutions are chosen such that the Kähler volume is positive.

New Perspectives on Attractor Flows and Trees from CFT · D0-D4 system · 0.81

Compares with it 1 paper

Because of the factorized structure of the $stu$ model, the attractor equations [ref] [equation] can be rewritten as a set of equations [equation] which coincide with the $N=8$ algebraic attractor equations [equation] for the four complex eigenvalues $z_i$ of $Z_AB$ [ref] .

Universality of the superpotential for d = 4 extremal black holes · Connection with $N=8$ · 0.84

Cites it as support 2 papers

This confirms that there is no attractor mechanism for general non-extremal STU black holes [ref] .

Entropy of non-extremal STU black holes: the F-invariant unveiled · Introduction · 0.99
Once the attractor mechanism is at work, any given point outside the horizon has actually an infinite physical distance away from the horizon (see, for example, [ref] ) and so the condition $ρ_B > r_+ = r_-$ with fixed $ρ_B$, the radial location of the wall of the cavity, will be enough.

Phase structure of black branes in grand canonical ensemble · Black $p$-brane solution and the action · 0.98

Cites it as background 24 papers

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
This was first proposed in [ref] , where also the concept of a black hole potential was introduced, and studied in detail later in [ref] .

Effective action for the field equations of charged black holes · Introduction · 0.87
Such non-supersymmetric attractors have been studied predominantly in $𝒩=2$ supergravity [ref] and type II string theory and $M$-theory compactified on Calabi-Yau threefold [ref] .

Supersymmetry and Attractors in N = 4 Supergravity: The Superconformal Approach · Introduction · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [8]

P. K. Tripathy and S. P. Trivedi, “Non-Supersymmetric Attract ors in String Theory,” [arXiv:hep-th/0511117]

View on Pith Pith review arXiv

[2] [1]

S. Ferrara, R. Kallosh and A. Strominger, “N=2 extremal black h oles,” Phys. Rev. D 52, 5412 (1995) [arXiv:hep-th/9508072]

View on Pith Pith review arXiv 1995

[3] [2]

A. Strominger, “Macroscopic Entropy of N = 2 Extremal Black Holes,” Phys. Lett. B 383, 39 (1996) [arXiv:hep-th/9602111]

View on Pith Pith review arXiv 1996

[4] [3]

S. Ferrara and R. Kallosh, “Supersymmetry and Attractors,” P hys. Rev. D 54, 1514 (1996) [arXiv:hep-th/9602136]; S. Ferrara and R. Kallosh, “Universality of Supersymmetric Attractors,” Phys. Rev. D 54, 1525 (1996) [arXiv:hep-th/9603090]

View on Pith Pith review arXiv 1996

[5] [4]

G. W. Moore, “Les Houches lectures on strings and arithmetic,” a rXiv:hep-th/0401049; G. W. Moore, “Arithmetic and attractors,” [arXiv:hep-th/9807087 ]

View on Pith Pith review arXiv

[6] [5]

F. Denef, B. R. Greene and M. Raugas, “Split attractor ﬂows an d the spectrum of BPS D-branes on the quintic,” JHEP 0105, 012 (2001) [arXiv:hep-th/0101135]

View on Pith Pith review arXiv 2001

[7] [6]

S. Ferrara, G. W. Gibbons and R. Kallosh, “Black holes and critical points in moduli space,” Nucl. Phys. B 500, 75 (1997) [arXiv:hep-th/9702103]. G. W. Gibbons, “Supergravity vacua and solitons,” Prepared for A Newton Institute Euroconference on Duality and Supersymmetric Theories, Ca mbridge, England, 7-18 Apr 1997, PROCEEDINGS. Edited by D.I. Olive, P.C...

View on Pith Pith review arXiv 1997

[8] [7]

K. Goldstein, N. Iizuka, R. P. Jena and S. P. Trivedi, “Non-super symmetric attractors,” [arXiv:hep-th/0507096]; A. Sen, “Black hole entropy function and t he attractor mechanism in higher derivative gravity,” JHEP 0509, 038 (2005) [arXiv:hep-th/0506177]; M. Alishahiha and H. Ebrahim, “Non-supersymmetric attractors an d entropy function,” [arXiv:hep-th/0601016]

View on Pith Pith review arXiv 2005

[9] [9]

R. Kallosh, M. Shmakova and W. K. Wong, “Freezing of moduli by N = 2 dyons,” Phys. Rev. D 54, 6284 (1996) [arXiv:hep-th/9607077]

View on Pith Pith review arXiv 1996

[10] [10]

K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W. K. Won g, “STU black holes and string triality,” Phys. Rev. D 54, 6293 (1996) [arXiv:hep-th/9608059]

View on Pith Pith review arXiv 1996

[11] [11]

K. Behrndt, G. Lopes Cardoso, B. de Wit, R. Kallosh, D. Lust an d T. Mohaupt, Nucl. Phys. B 488, 236 (1997) [arXiv:hep-th/9610105]

View on Pith Pith review arXiv 1997

[12] [12]

R. Kallosh, “New attractors,” JHEP 0512, 022 (2005) [arXiv:hep-th/0510024]

View on Pith Pith review arXiv 2005

[13] [13]

A. Giryavets, “New attractors and area codes,” [arXiv:hep-th /0511215]

View on Pith

[14] [14]

S. Ferrara, M. Bodner and A. C. Cadavid, “Calabi-Yau Supermo duli Space, Field Strength Duality And Mirror Manifolds,” Phys. Lett. B 247, 25 (1990)

View on Pith 1990

[15] [15]

F. Denef and M. R. Douglas, “Distributions of ﬂux vacua,” JHEP 0405, 072 (2004) [arXiv:hep-th/0404116]

View on Pith Pith review arXiv 2004

[16] [16]

G. W. Gibbons and R. E. Kallosh, “Topology, entropy and Witten in dex of dilaton black holes,” Phys. Rev. D 51, 2839 (1995) [arXiv:hep-th/9407118]

View on Pith Pith review arXiv 1995

[17] [17]

R. Kallosh, A. D. Linde, T. Ortin, A. W. Peet and A. Van Proeyen, “Supersymmetry as a cosmic censor,” Phys. Rev. D 46, 5278 (1992) [arXiv:hep-th/9205027]. – 30 –

View on Pith Pith review arXiv 1992

[18] [18]

B. de Wit, P. G. Lauwers and A. Van Proeyen, “Lagrangians Of N =2 Supergravity - Matter Systems,” Nucl. Phys. B 255, 569 (1985)

View on Pith 1985

[19] [19]

A. Ceresole, R. D’Auria, S. Ferrara and A. Van Proeyen, “Dualit y transformations in supersymmetric Yang-Mills theories coupled to supergravity,” Nucl. Phys. B 444, 92 (1995) [arXiv:hep-th/9502072]

View on Pith Pith review arXiv 1995

[20] [20]

B. de Wit and A. Van Proeyen, “Potentials And Symmetries Of Gen eral Gauged N=2 Supergravity - Yang-Mills Models,” Nucl. Phys. B 245, 89 (1984); A. Ceresole, R. D’Auria and S. Ferrara, “The Symplectic Structure of N=2 Supergravity a nd its Central Extension,” Nucl. Phys. Proc. Suppl. 46, 67 (1996) [arXiv:hep-th/9509160]

View on Pith Pith review arXiv 1984

[21] [21]

R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, Class. Qu ant. Grav. 17, 4269 (2000) [Erratum-ibid. 21, 5017 (2004)] [arXiv:hep-th/0006179]

View on Pith Pith review arXiv 2000

[22] [22]

A. Giryavets, S. Kachru, P. K. Tripathy and S. P. Trivedi, “Flux compactiﬁcations on Calabi-Yau threefolds,” JHEP 0404, 003 (2004) [arXiv:hep-th/0312104]

View on Pith Pith review arXiv 2004

[23] [23]

P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, “A Pa ir Of Calabi-Yau Manifolds As An Exactly Soluble Superconformal Theory,” Nucl. Phys . B 359, 21 (1991)

View on Pith 1991

[24] [24]

L. O’Raifeartaigh, “Spontaneous Symmetry Breaking For Chira l Scalar Superﬁelds,” Nucl. Phys. B 96, 331 (1975)

View on Pith 1975

[25] [25]

K. Intriligator and N. Seiberg, “The runaway quiver,” [arXiv:hep- th/0512347]

View on Pith

[26] [26]

B. Bates and F. Denef, “Exact solutions for supersymmetric s tationary black hole composites,” arXiv:hep-th/0304094. – 31 –

View on Pith Pith review arXiv