REVIEW
General Rotating Five Dimensional Black Holes of Toroidally Compactified Heterotic String
High Energy Physics - Theory · Submitted 15 Mar 1996 · 21 references
Pith's verdict: A solid, checkable solution-generating paper: the explicit 27-charge rotating metric is real and the BPS limits reproduce known results, but 'most general' rides on the no-hair conjecture the authors themselves flag.
T0 · AI review
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Referee report
The paper
The paper claims
The paper claims to write down, in explicit closed form, the most general rotating black hole of five-dimensional toroidally compactified heterotic string theory: 27 electric charges, two independent angular momenta, and an ADM mass obeying
The reading
What carries the argument
The engine is the $O(8,24)$ symmetry of the three-dimensional effective action obtained by reducing the stationary five-dimensional theory on the time direction and one rotation angle, with the remaining gauge fields dualized into scalars (Eq. (11)). Because that action is a nonlinear sigma model on a coset, any symmetry transformation of the three-dimensional scalar matrix $M$ maps one solution to another. The authors first fix $M_infty = I_{4,28}$ by a constant transformation, then apply three boosts $delta_e, delta_{e_1}, delta_{e_2}$ whose parameters become the three charges, and finally the internal coset rotations whose dimension count, $4+20$, supplies the remaining charges. The load-bearing object is therefore the orbit of the neutral Kerr solution under $O(8,24)$, together with the charge-to-boost dictionary $Q=2m coshdelta sinhdelta$ that makes the mass formula and the horizon area computable in closed form.
Extended reading notes
Core claim
The central claim is that the full stationary black-hole spectrum of the five-dimensional $N=4$ toroidally compactified heterotic vacuum is reachable from a single seed by duality. Concretely: the neutral Myers-Perry rotating solution, boosted three times inside $O(8,24)$, gives a six-parameter generating solution with mass $m$, two rotation parameters, and three charges $Q^{(1)}_1$, $Q^{(2)}_1$, $Q$; acting further with $[SO(5)times SO(21)]/[SO(4)times SO(20)]subset O(5,21)$, which does not touch the five-dimensional space-time, adds $4+20$ charges for a total of 27. The resulting configuration has ADM mass $M=sqrt{m^2+(Q^{(1)}_1)^2}+sqrt{m^2+(Q^{(2)}_1)^2}+sqrt{m^2+Q^2}$ and horizons at $r^2_pm = m - tfrac{1}{2}l_1^2 - tfrac{1}{2}l_2^2 pm tfrac{1}{2}sqrt{(l_1^2-l_2^2)^2+4m(m-l_1^2-l_2^2)}$, existing whenever $m ge (|l_1|+|l_2|)^2$. Taking $m to 0$ with the charges and $J$ held fixed produces a BPS-saturated rotating black hole whose area is $A_{text{BPS}} = 4pi^2 sqrt{Q^{(1)}_1 Q^{(2)}_1 Q - tfrac{1}{4}J^2}$, a quantity independent of the asymptotic moduli and dilaton.
Load-bearing premise
The word 'most general' rests on the no-hair conjecture, which the authors themselves label a conjecture, plus a counting step: that three boosts and the 24 internal rotations exhaust the 27 charges of any stationary black hole with these asymptotics, with nothing in the paper proving that the symmetry orbit actually reaches every such solution.
Editorial extensions
If this is right
- Every rotating BPS black hole in this vacuum has horizon area $A=4pi^2sqrt{Q^{(1)}_1 Q^{(2)}_1 Q - J^2/4}$, so its entropy depends only on quantized charges and one rotational parameter, not on the 115 moduli or the dilaton.
- The general solution's mass $M=sqrt{m^2+(Q^{(1)}_1)^2}+sqrt{m^2+(Q^{(2)}_1)^2}+sqrt{m^2+Q^2}$ meets the Bogomol'nyi bound exactly as $m to 0$, so the non-extremality parameter $m$ is the natural dial for departures from extremality.
- Near the BPS limit with all three charges nonzero, the mass shifts at order $m^2$ while the area shifts already at order $m$; when one charge is taken small the mass shifts at order $m$ and the area scales as $m^{1/2}$, so the two regimes are distinguishable by how the horizon grows.
- The two angular momenta separate off extremality: $J_varphi$ and $J_psi$ are equal and opposite only at the BPS point, differing by an $O(m)$ term proportional to $(L_1+L_2)$ that vanishes when the two rotation parameters are tuned together.
- Because the area formula is written in terms of quantized charges dressed by asymptotic moduli (Eqs. (30)-(32)), the same expression covers arbitrary points of the $N=4$ moduli and coupling space without recomputation.
Reading between the lines
- If the area is genuinely moduli independent, the obvious next test is a microscopic D-brane state count for a configuration with two angular momenta; the $J^2/4$ subtraction in the entropy suggests the count should be organized as an index with one quarter of the angular momentum squared removed.
- The dimension-counting step is where a classification theorem would belong: proving that the stationary Killing sector's $O(8,24)$ orbit is transitive on the 27-dimensional charge space would convert 'most general' from a conjecture-shaped statement into a theorem, and that is a well-posed problem independent of the explicit metric.
- The same orbit technique should produce general rotating solutions for $T^6$ and $K3times S^1$ compactifications, where the three-dimensional symmetry groups are larger; comparing the resulting horizon areas would test whether the $Q^{(1)}Q^{(2)}Q-J^2/4$ structure is a feature of five dimensions or of the duality group.
- The linear-in-$m$ correction to the area suggests a concrete perturbative check: adding a single low-lying mode to the BPS configuration should reproduce the $O(m)$ term in Eq. (34) from a microscopic calculation.
Original abstract
We present the most general rotating black hole solution of five-dimensional N=4 superstring vacua that conforms to the ``no hair theorem''. It is chosen to be parameterized in terms of massless fields of the toroidally compactified heterotic string. The solutions are obtained by performing a subset of O(8,24) transformations, i.e., symmetry transformations of the effective three-dimensional action for stationary solutions, on the five-dimensional (neutral) rotating solution parameterized by the mass m and two rotational parameters $l_1$ and $l_2$. The explicit form of the generating solution is determined by three $SO(1,1)\subset O(8,24)$ boosts, which specify two electric charges $Q_1^{(1)}, Q_{2}^{(2)}$ of the Kaluza-Klein and two-form U(1) gauge fields associated with the same compactified direction, and the charge Q (electric charge of the vector field, whose field strength is dual to the field strength of the five-dimensional two-form field). The general solution, parameterized by 27 charges, two rotational parameters and the ADM mass compatible with the Bogomol'nyi bound, is obtained by imposing $[SO(5)\times SO(21)]/[SO(4)\times SO(20)]\subset O(5,21)$ transformations, which do not affect the five-dimensional space-time. We also analyze the deviations from the BPS-saturated limit.
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.
The abstract, sentence by sentence
We present the most general rotating black hole solution of five-dimensional N=4 superstring vacua that conforms to the ``no hair theorem''. It is chosen to be parameterized in terms of massless fields of the toroidally compactified heterotic string. The solutions are obtained by performing a subset of O(8,24) transformations, i.e., symmetry transformations of the effective three-dimensional action for stationary solutions, on the five-dimensional (neutral) rotating solution parameterized by the mass m and two rotational parameters $l_1$ and $l_2$. The explicit form of the generating solution is determined by three $SO(1,1)\subset O(8,24)$ boosts, which specify two electric charges $Q_1^{(1)}, Q_{2}^{(2)}$ of the Kaluza-Klein and two-form U(1) gauge fields associated with the same compactified direction, and the charge Q (electric charge of the vector field, whose field strength is dual to the field strength of the five-dimensional two-form field). The general solution, parameterized by 27 charges, two rotational parameters and the ADM mass compatible with the Bogomol'nyi bound, is obtained by imposing $[SO(5)\times SO(21)]/[SO(4)\times SO(20)]\subset O(5,21)$ transformations, which do not affect the five-dimensional space-time. We also analyze the deviations from the BPS-saturated limit.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present the most general rotating black hole solution of five-dimensional N=4 superstring vacua that conforms to the ``no hair theorem''. | contribution 0.93 | 0.94 | 0.76 | asserted 1.00 |
| s1 | It is chosen to be parameterized in terms of massless fields of the toroidally compactified heterotic string. | method 0.57 | 0.79 | 0.06 | asserted 1.00 |
| s2 | The solutions are obtained by performing a subset of O(8,24) transformations, i.e., symmetry transformations of the effective three-dimensional action for stationary solutions, on the five-dimensional (neutral) rotating solution parameterized by the mass m and two rotational parameters $l_1$ and $l_2$. | method 0.99 | 0.90 | 0.14 | asserted 1.00 |
| s3 | The explicit form of the generating solution is determined by three $SO(1,1)\subset O(8,24)$ boosts, which specify two electric charges $Q_1^{(1)}, Q_{2}^{(2)}$ of the Kaluza-Klein and two-form U(1) gauge fields associated with the same compactified direction, and the charge Q (electric charge of the vector field, whose field strength is dual to the field strength of the five-dimensional two-form field). | method 0.49 | 0.88 | 0.11 | asserted 1.00 |
| s4 | The general solution, parameterized by 27 charges, two rotational parameters and the ADM mass compatible with the Bogomol'nyi bound, is obtained by imposing $[SO(5)\times SO(21)]/[SO(4)\times SO(20)]\subset O(5,21)$ transformations, which do not affect the five-dimensional space-time. | method 0.51 | 0.91 | 0.26 | asserted 1.00 |
| s5 | We also analyze the deviations from the BPS-saturated limit. | contribution 0.98 | 0.92 | 0.06 | asserted 0.79 |
What was examined, and what was not
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References
How later papers used it
88 later papers cite it
1 later paper by other authors is listed below: 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.
Uses its method or tool 1 of 3 papers with this label · 1 by the same authors
By the same authors
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Used the coordinates t, r, θ, ϕ̃ and ψ of this paper, with the redefinition u=cosθ.
Gaussian Null Coordinates for Rotating Charged Black Holes and Conserved Charges (2020) (same authors) · 0.88 · paraphrased: the licence does not allow quoting
Compares with it 1 of 9 papers with this label
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Said that the solution it reached in the scaling limit is essentially identical to the original extremal black string of this paper.
Gravitational index of the D1-D5-P black string (2026) · 0.98 · paraphrased: the licence does not allow quoting
How the literature uses it
80 arXiv papers cite this in 130 sentences
Papers citing it · 84 sentences without a confident label
Builds on it 4 papers
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In this Section, we extend the previous analysis of CT transformations to eSTURBHs (extremal rotating BHs in STU supergravity) [ref] .
Turning rotating D-branes and BHs inside out their photon-halo · Extremal Rotating BHs in STU supergravity · 0.90
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At this point it is straightforward to guess the expression for the supersymmetric on-shell action in the more general case where two independent rotation parameters are turned on in the asymptotically flat solution of [ref] , also allowing for a generic $C_IJK$.
Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography · Supersymmetric on-shell action for asymptotically flat black holes · 0.98
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Finally, we would like to conclude this study of solutions of the $6D$ Einstein-ModMax system with a preliminary result towards the finding of the first rotating ModMax black-string solutions that would generalize those found in $D=6$ supergravities [ref] .
Waves and strings in an interacting conformal chiral 2-form theory in six dimensions · Towards rotating strings in $6D$ Einstein-ModMax theory · 0.90
Uses its method or tool 2 papers
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These solution generating techniques can be used to both generate all the charged black holes of the previous subsection from the non-extremal Kerr solution[ They were employed in [ref] to generate four charge rotating black holes in four dimension with two magnetic and two electric charges.
General black holes, untwisted · Solution-generating transformations · 0.86
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The method presented here, which adds charges to a generic uncharged metric, is essentially the same as that presented in [ref] and more pertinently applied to the singly spinning ring in [ref] .
A Charged Doubly Spinning Black Ring · Introduction · 0.92
Uses its data or values 1 paper
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The corresponding chemical potentials were also calculated in [ref] ,
Black hole thermodynamics in natural variables: Quadrophenia · 5d Einstein-Maxwell · 0.81
Compares with it 9 papers
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In the extremal solution [ref] , both $$ and $$ are taken to zero with $/r_0$ and $/r_0$ held fixed in the limit.
Gravitational index of the D1-D5-P black string · Index condition · 0.99
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Now, we compare the capped black hole with the Cvetič-Youm solution [ref] having the same conserved charges, the mass, angular momenta and electric charge.
Solution Generation of a Capped Black Hole · Non-uniqueness of spherical black holes · 1.00
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The black string obtained from [ref] with $r_S^2 ≥r_B^2$ is identical to the Cvetic-Youm non-rotating D1-D5 black string with equal charges [ref] .
Topological Stars, Black holes and Generalized Charged Weyl Solutions · The embedding of the six-dimensional solutions · 0.99
Cites it as background 24 papers
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According to the uniqueness theorem for charged rotating black holes in the bosonic sector of five-dimensional minimal supergravity [ref] , under the assumptions of the existence of two commuting axial isometries and spherical topology of horizon cross-sections, an asymptotically flat, stationary charged rotating black hole with a non-extremal horizon is uniquely characterized by the mass, charge, and two independent angular momenta and therefore is described by the five-dimensional Cvetič-Youm solution [ref] .
A Capped Black Hole in Five Dimensions · Introduction · 0.86
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Restricting our attention to the supersymmetric case, the first-order $α'$ corrections to the heterotic version of the “Strominger–Vafa” and “Maldacena–Strominger–Witten” black holes were obtained in [ref] , more than two decades after the corresponding two-derivative solutions had been constructed [ref] and their entropy successfully reproduced microscopically [ref] .[The $α'$ corrections to the finite-temperature black holes with three and four charges were later studied in [ref] .] Subsequently, the two-charge case —namely, the so-called small black holes— was investigated in [ref] .
BMPV black hole at first order in $\alpha'$ · Introduction · 0.89
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The supergravity D1-D5-P black hole solution was found in [ref] and a non-extremal family of fuzzballs was constructed in [ref] .
Modave Lectures on Fuzzballs and Emission from the D1-D5 System · Gravity Solutions: Black Holes and Smooth Solutions · 0.91
Discussion
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