Supersymmetry of topological Kerr-Newmann-Taub-NUT-aDS spacetimes

N. Alonso-Alberca; P.Meessen; T. Ortin

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Supersymmetry of topological Kerr-Newmann-Taub-NUT-aDS spacetimes

N. Alonso-Alberca, P.Meessen, T. Ortin

High Energy Physics - Theory · Submitted 9 Mar 2000 · 31 references

Pith's verdict: Solid technical extension with a clean PD factorization and a genuinely interesting SO(2) duality; the exact supersymmetry fractions for the rotating and PD cases are asserted from a necessary condition rather than constructed.

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arXiv IDhep-th/0003071 (version 2)

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CitationN. Alonso-Alberca, P.Meessen, T. Ortin. "Supersymmetry of topological Kerr-Newmann-Taub-NUT-aDS spacetimes." arXiv:hep-th/0003071 (2000). https://pith.science/paper/62GT4YUS

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Two things here are worth your time: an explicit NUT-charged extension of the topological KN-aDS family, and a clean factorization of the supersymmetry conditions for the general Plebanski-Demianski solution that exposes an SO(2) rotation mixing (M,N) with (Q,P). The rest is competent bookkeeping around those two results. What's new: the NUT-charged topological KN-aDS metric and field strength (2.7), the topological RB-aDS solutions with the constraint (2.9), Table 1's classification for the non-rotating case, and the PD analysis in §3.4. The PD result is the nicest piece: B± = W² − (E ± 2gα^{1/2})Z² is simple, duality-invariant, and reduces to the standard ungauged bound. The duality observation is stated with appropriate modesty — the authors call its meaning obscure and note the rotated charges aren't the physical ones. That's the right tone, and it's the part I'd actually cite. Soft spots, in order of how much they matter. The main one is the one your reader flagged: the fractions in Table 1 and §3.3–3.4 come from the rank of the integrability condition, which the paper correctly calls necessary. For the non-rotating cases the spinors are in Romans and Caldarelli-Klemm, so Table 1 is fine. For the rotating and PD cases no explicit spinors are constructed, and the 'enhancement from 1/4 to 1/2 at the RB horizon' is asserted rather than shown. The abstract's 'at most 1/4' is the safe version, so the headline survives; the exact fractions are the part asserted beyond what's proved. That's a moderate gap, not a fatal one, and the authors are honest elsewhere that they haven't found a 1/2-BPS rotating case. On the stress-test's secondary check: I don't think the (3.12) vs (3.20) mismatch lands. The two are related through the (A.2) redefinitions, not by setting a=0 in (3.20), and the square root is controlled by the paper's own α ≥ 0 bound (α^{1/2} ≤ ±E/2g). That said, the printed (3.20) does look like it's missing a square on the NUT term — worth a query to the authors, but a typo-level issue. Verdict: solid technical paper in a specialized corner. Send it out. The PD bound and the duality observation are the citable bits; the rotating-case fractions should be tightened in revision.

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Major objections (2)

[§3, Table 1, §3.3–3.4] The paper correctly states in §3 that the integrability condition [D,D]ε=0 is only necessary for the existence of Killing spinors, but the supersymmetry fractions are then read off from the rank of that condition. The abstract's 'at most 1/4' is consistent with a necessary condition, but the Table 1 caption ('for which there are Killing spinors') and the statements in §3.3 that the determinant condition 'break[s] half of the available supersymmetry' and that there is 'supersymmetry enhancement at the horizon from 1/4 to 1/2' are existence and exactness claims. For the non-rotating RN-TN-AdS cases the existence of the spinors is supported by the cited Romans and Caldarelli-Klemm results, but for the rotating KN-TN-AdS and general PD cases no explicit Killing spinors are constructed. The authors should either construct the spinors (or show that the integrability kernel is saturated) or explicitly state that the quoted fractions are upper bounds.
[§3.2, Eq. (3.20)] The a→0 limit of the KN-TN-AdS bound (3.20) does not reproduce the RN-TN-AdS bound (3.12). In particular, the argument of the square root in (3.20) appears to omit the P² term that is present in α in Eq. (A.2), and as printed the argument becomes negative (hence the bound complex) for ℵ=+1, N≠0, a=0. A concrete check is to set a=0 in (3.20) and compare with (3.12); the two should agree because the KN-TN-AdS metric reduces to the RN-TN-AdS metric in that limit. The authors should verify the reduction and correct the expression if the omission is real.

Minor objections (7)

[§3.3] The sentence 'Obviously, for g=0 one finds Robinson-Bertotti which will not break any supersymmetry' is misleading: the Robinson-Bertotti solution in ungauged N=2, d=4 supergravity is 1/2-BPS, not maximally supersymmetric. What is unbroken at g=0 is the integrability condition, not the full supersymmetry; the wording should be clarified.
[§2.1] The sentence 'These solutions are valid in the g=0 case. In that limit (with N=0) so we can speak of black holes...' is grammatically incomplete and should be rewritten.
[Eq. (3.11)] Equation (3.11) is blank in the manuscript; presumably it is a continuation of the preceding condition and should be displayed or removed.
[§2.3] The sentence 'The ℵ=−1, K²=2L² solution, which has special supersymmetry properties has been also given in [30]' needs a comma after 'properties'.
[Abstract] The abstract spells 'Kerr-Newmann' with a double n; the standard spelling is 'Kerr-Newman'.
[§2.2, Eq. (2.7)] The definition 'S(u) = S(u) + (a²g²u²+4ag²Nu)(u²−ℵ²)' uses the same symbol on both sides, which is confusing; a different symbol for the deformed function would improve readability.
[Table 1] The table caption should be qualified in light of major comment 1; 'for which there are Killing spinors' asserts existence, whereas the analysis establishes only that the integrability condition admits solutions.

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The paper

The paper claims

Adding NUT charge and spin to anti-de Sitter black holes leaves at most a quarter of the supersymmetry, and the surviving sector has a hidden rotation symmetry.

The reading

The paper asks which members of a broad family of four-dimensional anti-de Sitter solutions — charged, rotating black holes with NUT charge, their non-rotating limits, and the near-horizon Robinson-Bertotti-type geometries — can still be supersymmetric in gauged $N=2$, $d=4$ supergravity. The authors first write down the missing solutions: the topological Kerr-Newman-anti-de Sitter metrics extended with NUT charge, plus cosmological Robinson-Bertotti geometries built from $AdS_2$ times a sphere, torus, or higher-genus Riemann surface, and they show these all arise as limits of the general Plebanski-Demianski solution. They then translate the Killing spinor equation into algebraic conditions on the parameters by computing the integrability condition, and find that the constraints collapse to a pair of factorized equations, $g[MP+NQ]=0$ and $B_+B_-=0$. The concrete payoff is a table of parameter choices that admit Killing spinors, with the result that generic solutions keep only one quarter of the supersymmetry, while half-supersymmetric cases are non-rotating. A second payoff is that the supersymmetric Plebanski-Demianski sector is invariant under a simultaneous rotation of mass with NUT charge and electric with magnetic charge, even though the gauged theory as a whole has no electric-magnetic duality.

What carries the argument

The engine is the integrability condition $[D_+ , D_-]$-style commutator of the supercovariant derivative, which must vanish for a Killing spinor to exist and which reduces to an algebraic expression in the Weyl tensor, the covariant derivative of the field strength, and terms proportional to the gauge coupling $g$. Evaluating it on a suitable vierbein turns a functional differential problem into a matrix condition on the parameters, and the determinant of that matrix factorizes into $B_+B_-$, which is what makes an exhaustive parameter classification possible. The factorization is the load-bearing structure: it is what lets the authors read off the Bogomol'nyi bound, spot the mass-NUT and electric-magnetic rotation symmetry, and check that the rotating, non-rotating and Plebanski-Demianski results agree under the coordinate and parameter redefinitions collected in the appendix.

Extended reading notes

Core claim

The central claim is that supersymmetry of these solutions is governed by two algebraic conditions. The first, $g[MP+NQ]=0$, is a constraint linking mass $M$, NUT charge $N$, electric charge $Q$ and magnetic charge $P$; the second, $B_+B_-=0$ with $B_+B_-$ built from $M^2+N^2$ and $Q^2+P^2$, plays the role of a saturated Bogomol'nyi bound. Solving them for the non-rotating topological case yields four classes: two that are locally $AdS_4$ and preserve all supersymmetry, one half-supersymmetric class generalizing $M=|Q|$ with an induced magnetic charge proportional to the NUT charge, and one quarter-supersymmetric class that exists only for nonzero gauge coupling and generalizes Romans' global monopole. For the rotating case the same factorization holds with $M$ and $N$ shifted by angular momentum terms, and the authors report that in every case they examined, switching on rotation takes a half-supersymmetric solution down to one quarter. For the general Plebanski-Demianski family the conditions become $M^2=(E_+)(Q^2)$ and $N^2=(E_+)(P^2)$-type relations that rotate into each other under $M' = M - N$ mixing and $Q' = Q + P$ mixing with a common angle, so the supersymmetric locus itself, not the full theory, carries a continuous $SO(2)$ duality.

Load-bearing premise

The paper reads the number of supersymmetries off the kernel of a condition that is only necessary for a Killing spinor to exist, and does not display explicit spinors for the rotating or Plebanski-Demianski cases, so the stated fractions assume that every solution of that condition really lifts to a spinor.

Editorial extensions

If this is right

Any member of the topological Kerr-Newman-Taub-NUT-anti-de Sitter family that satisfies $g[MP+NQ]=0$ and $B_+B_-=0$ admits Killing spinors, giving a finite checklist rather than a search over continuous parameters.
Half-supersymmetric solutions in this family are necessarily non-rotating, so angular momentum costs exactly one further half of the supersymmetry.
The near-horizon Robinson-Bertotti-type geometry of the quarter-supersymmetric magnetic monopole solution has enhanced supersymmetry, preserving one half instead of one quarter at the horizon.
The supersymmetric Plebanski-Demianski locus is closed under a simultaneous rotation of $(M,N)$ and $(Q,P)$, so statements about a purely electric supersymmetric solution can be transported to a purely magnetic or dyonic one within that locus.

Reading between the lines

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

If the mass-NUT and electric-magnetic rotation is a genuine symmetry of the supersymmetric sector rather than an artifact of the parameterization, it suggests the physical charge combinations that enter the superalgebra are not $(M,N,Q,P)$ separately but rotation-invariant pairings of them; the authors themselves note the transformation looks very nonlinear in terms of physical charges.
The quarter-supersymmetric solutions have no known interpretation as intersections of half-supersymmetric objects, which is the standard bookkeeping in ungauged supergravity; the gauged magnetic-charge solutions therefore mark a structural difference between the gauged and ungauged cases rather than a technical one.
A natural check would be to see whether the same factorized bound appears for topological horizons in higher-dimensional gauged supergravity, where the analogue of the genus parameter and the NUT charge both exist.
The rotation symmetry, if it survives quantization, would relate spectra of states carrying different NUT charge, which is testable in principle against any microscopic counting for these backgrounds.

Original abstract

We extend the topological Kerr-Newman-aDS solutions by including NUT charge and find generalizations of the Robinson-Bertotti solution to the negative cosmological constant case with different topologies. We show how all these solutions can be obtained as limits of the general Plebanski-Demianski solution. We study the supersymmetry properties of all these solutions in the context of gauged N=2,d=4 supergravity. Generically they preserve at most 1/4 of the total supersymmetry. In the Plebanski-Demianski case, although gauged N=2,d=4 supergravity does not have electric-magnetic duality, we find that the family of supersymmetric solutions still enjoys a sort of electric-magnetic duality in which electric and magnetic charges and mass and Taub-NUT charge are rotated simultaneously.

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We extend the topological Kerr-Newman-aDS solutions by including NUT charge and find generalizations of the Robinson-Bertotti solution to the negative cosmological constant case with different topologies. We show how all these solutions can be obtained as limits of the general Plebanski-Demianski solution. We study the supersymmetry properties of all these solutions in the context of gauged N=2,d=4 supergravity. Generically they preserve at most 1/4 of the total supersymmetry. In the Plebanski-Demianski case, although gauged N=2,d=4 supergravity does not have electric-magnetic duality, we find that the family of supersymmetric solutions still enjoys a sort of electric-magnetic duality in which electric and magnetic charges and mass and Taub-NUT charge are rotated simultaneously.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We extend the topological Kerr-Newman-aDS solutions by including NUT charge and find generalizations of the Robinson-Bertotti solution to the negative cosmological constant case with different topologies.	contribution 0.97	0.95	0.21	asserted 1.00
s1	We show how all these solutions can be obtained as limits of the general Plebanski-Demianski solution.	contribution 0.49	0.92	0.08	asserted 1.00
s2	We study the supersymmetry properties of all these solutions in the context of gauged N=2,d=4 supergravity.	contribution 0.99	0.94	0.11	asserted 0.88
s3	Generically they preserve at most 1/4 of the total supersymmetry.	result 1.00	0.87	0.26	asserted 0.85
s4	In the Plebanski-Demianski case, although gauged N=2,d=4 supergravity does not have electric-magnetic duality, we find that the family of supersymmetric solutions still enjoys a sort of electric-magnetic duality in which electric and magnetic charges and mass and Taub-NUT charge are rotated simultaneously.	result 0.97	0.95	0.30	asserted 0.99

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References

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How the literature uses it

34 arXiv papers cite this in 68 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 · 46 sentences without a confident label

Contests it 1 paper

It is perhaps surprising that the Carter-Plebański family leads to spindles, since its Lorentzian version contains only the Kerr-Newman-NUT-AdS, Reissner-Nordström-NUT-AdS and the AdS-C-metric [ref] [Note that in [ref] it is incorrectly stated that the Carter-Plebański solution is the most general Plebański-Demianski solution.], whereas the conical singularities of the spindle are associated to a non-zero acceleration, which is present only in the more general Plebański-Demianski family [ref] [See also [ref] .].

NUTs, Bolts, and Spindles · Introduction · contest 0.86

Builds on it 1 paper

In this paper we want to pick up the threads on these developments and show that the recently proposed thermodynamics can readily be extended to the charged Lorentzian Taub–NUT solution of the Einstein–Maxwell theory [ref] (see also [ref] for the higher-dimensional generalizations).

Thermodynamics and Phase Transitions of NUTty Dyons · Introduction · 0.86

Uses its method or tool 2 papers

By examining the integrability conditions for [ref], as in [ref] we find that supersymmetry implies that the five parameters are constrained by the following two conditions [equation] With $α0$, and $m>0$ we must have $g>0$.

Accelerating Black Holes and Spinning Spindles · Supersymmetric and extremal limits · 0.87
First we compute the supercurrent from the Lagrangian of minimal $D=4$ $N=2$ gauged supergravity following the conventions of [ref] (which is written in 1.5-formalism): [equation] where $ψ= i ψ^† γ_0, e=√(detg_μν),$ [equation] and [equation] The spin connection satisfies [equation] for a given vielbein $e^a = e_μ^a dx^μ$.

On BPS bounds in D=4 N=2 gauged supergravity · Supercharges of minimal gauged supergravity · 0.99

Compares with it 1 paper

These are precisely the conditions on the electric and magnetic charge found in [ref] , for which the Reissner-Nordström-Taub-NUT-AdS solution is supersymmetric[Actually, the conditions given in [ref] are $Q=∓Ml/(2N)$ and $P=±(2N^2/l-l/2)$, corresponding to vanishing $B_∓$ in (3.10), (3.12) of [ref] .

Nut-charged black holes in matter-coupled N=2, D=4 gauged supergravity · Constant scalars · 0.89

Cites it as support 1 paper

That there must be supersymmetric solutions of this type is clear from the analysis for minimal gauged supergravity in [ref] .

Rotating BPS black holes in matter-coupled AdS(4) supergravity · Constant scalars · 0.97

Cites it as background 12 papers

The Euclidean supersymmetric gravity solution of interest is studied in some detail in [ref] , see also [ref] , and can be written as

Higher-Derivative Supergravity, AdS$_4$ Holography, and Black Holes · $U(1) ×U(1)$ squashed sphere · 0.81
Hence we refer the readers to [ref] as well as earlier works such as [ref] for a comprehensive analysis of the background concerning its global structure, the distinctions between NUTs and bolts, supersymmetry, moduli spaces, and constraints for uplift to 11d supergravity.

Perturbatively exact supersymmetric partition functions of ABJM theory on Seifert manifolds and holography · Supergravity dual · 0.99
Then, we solve it for an already known solution, namely the black hole carrying not only mass but also NUT charge (also called “magnetic mass"), and both electric and magnetic Maxwell charges as well (see e.g. [ref] ).

Supersymmetry and Gravitational Duality · The BPS Taub-NUT charged solution in $N=2$ supergravity · 0.80

Discussion

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