REVIEW
Supersymmetry of Black Strings in D=5 Gauged Supergravities
High Energy Physics - Theory · Submitted 20 Jan 2000 · 32 references
Pith's verdict: A solid, incremental BPS black-string paper with one minor coordinate typo and a standard presentation gap; worth a referee.
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Referee report
The paper
The paper claims
Klemm and Sabra show that a five-dimensional magnetic black string with a hyperbolic transverse space keeps a quarter of the theory's supersymmetry, and that its near-horizon geometry, $AdS_3$ times $H^2$, keeps half.
The reading
What carries the argument
The load-bearing device is the pair of Killing spinor equations obtained by setting the gravitino variation (3) and gaugino variation (4) to zero in a purely bosonic background, then imposing constant moduli chosen to minimise the magnetic central charge $Z = q_I X^I$ subject to $V_I X^I = 1$. The minimisation conditions $X^I = q^I/Z$ and $Z = (tfrac{1}{6}C_{IJK}q^Iq^Jq^K)^{1/3}$ make the gaugino variation vanish identically, so the whole analysis reduces to the gravitino equation. Two constant projectors, $Gamma^3Gamma^4epsilon = iepsilon$ and $Gamma^2epsilon = -epsilon$, reduce that equation to a set of algebraic and first-order conditions on the metric functions, whose solvability yields the horizon relation $e^{-U} = -Z/r + gr$ and the generalised Dirac quantisation condition $Z = 1/(3g)$ (in pure supergravity, $q^0 = 1/(3g)$). The same machinery, run in a $3+2$ split with $Gamma^{hatalpha} = gamma^{hatalpha}otimes sigma_3$, produces the topological selection rule: only negative internal curvature survives, with $R = -6g^2$.
Extended reading notes
Core claim
The central claim is that gauged $D=5$, $N=2$ supergravity coupled to abelian vector multiplets contains a BPS magnetic black string whose transverse space is hyperbolic, and that this object exhibits supersymmetry enhancement at its horizon. Concretely, the metric in Eq. (30) has $e^{-U} = -1/(3gr) + gr$, an event horizon at $r_+ = 1/(gsqrt{3})$, constant scalar moduli $X^I = 3gq^I$, and magnetic gauge fields $A^I_varphi = -q^I cosh theta$; the Killing spinor is $epsilon(r) = e^{V/2}epsilon_0$ with two constant projectors $Gamma^3Gamma^4epsilon_0 = iepsilon_0$ and $Gamma^2epsilon_0 = -epsilon_0$, leaving one quarter of the supersymmetries. In the near-horizon limit the geometry becomes $AdS_3 times H^2$, which the authors show is half-supersymmetric, so the fraction of preserved supersymmetry doubles at the horizon. The companion result is a general $3+2$ decomposition argument: for any product $M_3 times M_2$ with constant moduli and magnetic flux on $M_2$, supersymmetry forces the magnetic central charge to equal the gauge coupling, $Z = g$, and the internal scalar curvature to be $R = -6g^2$, so $M_2$ must be hyperbolic. Applied to $M_3 = $ BTZ, this gives a quarter-supersymmetric extremal rotating BTZ black hole times $H^2$.
Load-bearing premise
The paper identifies its black string by solving the Killing spinor equations with constant moduli and does not display a check of the full Einstein, Maxwell and scalar equations of motion for the metric (30), so the claim rests on the standard but unstated implication that a bosonic background annihilating every fermion variation automatically solves those equations.
Editorial extensions
If this is right
- BPS black strings with regular horizons exist in gauged five-dimensional supergravity, in contrast with the spherical magnetic string of [16], which is a naked singularity; the difference is purely the sign of the transverse curvature.
- The near-horizon geometry $AdS_3 times H^2$ offers a two-dimensional conformal symmetry to count entropy against, but the authors note that the near-horizon BTZ has $M = J = 0$, so the naive Cardy formula gives a wrong answer and a different counting, closer to that used for extremal three-dimensional black strings, is required.
- Supersymmetric product compactifications of this theory are topologically restricted: the internal two-manifold must be hyperbolic, and its curvature is fixed by the gauge coupling through $R = -6g^2$ and $Z = gR^2$.
- The extremal rotating BTZ black hole tensored with $H^2$ preserves one quarter of supersymmetry, giving a rotating BPS background in the same class.
- Because the $U(1)^3$ truncation embeds into type IIB supergravity, these strings lift to ten dimensions with a distorted five-sphere, suggesting a dual $N=4$ super Yang–Mills on $R^2 times H^2$ in the presence of background $U(1)^3$ currents.
Reading between the lines
- The result hints at a general pattern in gauged supergravities: BPS objects with regular horizons tend to require hyperbolic horizon topology, so searches for holographically useful BPS black objects should concentrate on negative-curvature foliations rather than spheres.
- Since the full string is quarter-supersymmetric and its horizon is half-supersymmetric, the interpolating solution behaves like a flow between two supersymmetric vacua; a natural next calculation is a supergravity $c$-function along that flow, compared against the central charges of the two endpoints.
- The acknowledged failure of the naive Cardy count suggests the entropy is not captured by the asymptotic $AdS_3$ symmetry algebra alone; a direct check would be whether the horizon area of Eq. (30) can be reproduced by a state count in the dual theory with background currents turned on.
- Because the $g to 0$ limit does not exist, these strings are solitonic with no ungauged analogue; one could test whether their mass saturates a BPS bound that interpolates between the $AdS_5$ and $AdS_3$ central charges.
Original abstract
Supersymmetry of five dimensional string solutions is examined in the context of gauged D=5, N=2 supergravity coupled to abelian vector multiplets. We find magnetic black strings preserving one quarter of supersymmetry and approaching the half-supersymmetric product space AdS_3\times H^2 near the event horizon. The solutions thus exhibit the phenomenon of supersymmetry enhancement near the horizon, like in the cases of ungauged supergravity theories, where the near horizon limit is fully supersymmetric. Finally, product space compactifications are studied in detail, and it is shown that only for negative curvature (hyperbolic) internal spaces, some amount of supersymmetry can be preserved. Among other solutions, we find that the extremal rotating BTZ black hole tensored by H^2 preserves one quarter of supersymmetry.
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
Supersymmetry of five dimensional string solutions is examined in the context of gauged D=5, N=2 supergravity coupled to abelian vector multiplets. We find magnetic black strings preserving one quarter of supersymmetry and approaching the half-supersymmetric product space AdS_3\times H^2 near the event horizon. The solutions thus exhibit the phenomenon of supersymmetry enhancement near the horizon, like in the cases of ungauged supergravity theories, where the near horizon limit is fully supersymmetric. Finally, product space compactifications are studied in detail, and it is shown that only for negative curvature (hyperbolic) internal spaces, some amount of supersymmetry can be preserved. Among other solutions, we find that the extremal rotating BTZ black hole tensored by H^2 preserves one quarter of supersymmetry.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Supersymmetry of five dimensional string solutions is examined in the context of gauged D=5, N=2 supergravity coupled to abelian vector multiplets. | background 0.58 | 0.64 | 0.05 | asserted 0.81 |
| s1 | We find magnetic black strings preserving one quarter of supersymmetry and approaching the half-supersymmetric product space AdS_3\times H^2 near the event horizon. | result 1.00 | 0.95 | 0.45 | asserted 1.00 |
| s2 | The solutions thus exhibit the phenomenon of supersymmetry enhancement near the horizon, like in the cases of ungauged supergravity theories, where the near horizon limit is fully supersymmetric. | interpretation 0.97 | 0.49 | 0.27 | asserted 1.00 |
| s3 | Finally, product space compactifications are studied in detail, and it is shown that only for negative curvature (hyperbolic) internal spaces, some amount of supersymmetry can be preserved. | result 0.94 | 0.91 | 0.23 | asserted 0.95 |
| s4 | Among other solutions, we find that the extremal rotating BTZ black hole tensored by H^2 preserves one quarter of supersymmetry. | result 1.00 | 0.92 | 0.11 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How the literature uses it
34 arXiv papers cite this in 53 sentences
Papers citing it · 40 sentences without a confident label
Builds on it 3 papers
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We now would like to construct supersymmetric generalizations of the magnetic string solutions of [ref] , that carry momentum along the string.
Black strings in AdS_5 · Construction of the solution · 1.00
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In a similar fashion we can also take the known $AdS_3×Σ_g$ solution of minimal $D=5$ gauged supergravity [ref] , which is a topological twist construction, and then uplift to $D=7$ to obtain $AdS_3×Σ_g⋉_2$ solutions which are associated with M5-branes wrapping the four-dimensional orbifold $M_4=Σ_g⋉_2$.
M5-branes wrapped on four-dimensional orbifolds · Introduction · 0.96
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The solutions with $w(r)=0$, $H(r)=const.$ represent Abelian black strings, generalizing the exact BPS solutions in [ref] .
Magnetic solutions in AdS$_5$ and trace anomalies · Introduction · 0.98
Compares with it 2 papers
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Unlike any other known supersymmetric solutions of five-dimensional gauged supergravity [ref] , the solutions we find have a hyper-Kähler base space!
Black Holes and Fourfolds · Introduction · 0.96
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Using 10ant we find that the ten dimensional type IIB metric takes the form ds^2 = 1/r^2(1-m/r^4/3) ds^2(^1,1 ) + 1/4(1-m/r^4/3) ds^2(H_2/Γ) + dr^2/r^2(1-m/r^4/3)^2 + 1/4 [ (dz+P)^2 + ds^2(CP^2). ] If we take $m>0$, $0≤r≤m^3/4$ we essentially have the solution of [ref] .
Geometries with Killing Spinors and Supersymmetric AdS Solutions · An ansatz for the geometry in $2n+2$ dimensions · 0.84
Cites it as background 6 papers
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Since the pioneering work [ref] , a number of similar solutions have been found [ref] .
New supersymmetric $AdS_5$ black strings from 5D $N=4$ gauged supergravity · Introduction · 0.83
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[ref] for a select overview.
Charged Nariai black holes on the dark bubble · The 5d black string · 1.00
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This will yield predictions for black strings in AdS$_5$, for which at present only partial results are known [ref] .
Dyonic AdS black holes from magnetohydrodynamics · Black strings in AdS$_5$ · 0.81
Discussion
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