The Geometry of D=11 Killing Spinors

Jerome P. Gauntlett; Stathis Pakis

REVIEW

The Geometry of D=11 Killing Spinors

Jerome P. Gauntlett, Stathis Pakis

High Energy Physics - Theory · Submitted 2 Dec 2002 · 33 references

Pith's verdict: The timelike branch is done properly and the converse sufficiency proof is what makes it citable; the null branch is deferred despite the abstract's 'most general' framing.

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arXiv IDhep-th/0212008 (version 3)

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CitationJerome P. Gauntlett, Stathis Pakis. "The Geometry of D=11 Killing Spinors." arXiv:hep-th/0212008 (2002). https://pith.science/paper/4C56LYO4

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Editor's note

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Short version: the timelike branch of this is done properly, and the converse proof is what makes it worth citing. The null branch is not done here, whatever the abstract's phrasing suggests at first read — the body says so plainly and defers it to future work, so treat that as acknowledged rather than as a hidden gap. What is actually new: the explicit local form (4.20)-(4.22), the condition W5 = -(1/2) d log Delta as the only restriction on the D=10 SU(5) structure, and — the part I would point a student at — the sufficiency argument at (4.28). Most papers in this genre derive necessary conditions and stop. Gauntlett and Pakis show that the four-form they write down actually reproduces a Killing spinor, with F75 dropping out of the Clifford connection. That closes the classification on the timelike branch instead of leaving it as a conjecture. The worked examples (M5, resolved M2, Godel) are concrete and checkable, and the rotating resolved-membrane family is a genuine, if small, addition. On the reader's analyticity worry: I think it is misplaced, and the stress-test note has it right. The geodesic argument already gives vanishing on an open set, and the Killing spinor equation is first-order linear, so unique continuation does the rest without analyticity. More to the point, the main theorem is local — K timelike in a neighbourhood is an assumption, not a conclusion — so global non-vanishing is not needed for (4.20)-(4.22). I would not send this back over that. The real soft spot is the sufficiency computation. 'After a lengthy computation' at (4.28) is carrying a lot of weight, and it leans on Table 1's eigenvalues of Theta. Two things make me comfortable: the 75 eigenvalue of 0 is forced by representation theory, and (4.28) matches the independently derived (C.12) coefficient for coefficient. That is a real cross-check, not a restatement. A referee should still verify Table 1 line by line. Minor: the vanishing theorems in section 4 are conditional on compactness and on restricting d-omega and H — narrow but honestly stated. The X8 correction is set aside throughout, which the authors flag. Who should read it: anyone working on supersymmetric M-theory backgrounds, and anyone doing G-structure classifications in other dimensions — the D=5 paper is the template and this is the D=11 instance. Worth a serious referee; the timelike result will be a standard reference.

Referee report

Significance

If correct, this is a foundational result: it converts the search for supersymmetric D=11 backgrounds from an ansatz-by-ansatz exercise into a structured local classification, and the local form (4.20)-(4.22) has become the template for subsequent classifications in other dimensions and for the null case. The paper is unusually careful for this genre in that it does not stop at necessary conditions: it supplies an explicit sufficiency argument, and the reduced equation (4.28) is independently cross-checked against the intrinsic-contorsion computation (C.12) in appendix C, with matching coefficients. The eigenvalue table for the map Θ (Table 1) is a genuine computation, and the observation that F75 drops out of the Clifford connection because the 75 contains no singlet is a clean structural statement. The worked examples, in particular the demonstration in §5.1 that the M5-brane requires a non-vanishing F75 to be recovered from (4.22), are useful consistency checks rather than decoration. The paper also delivers new solutions (rotating resolved membranes, generalised Gödel backgrounds) and conditional no-go theorems, which broadens its value beyond the classification programme.

Major objections (2)

[§4, Eq. (4.28) and the 'necessary and sufficient' claim] The load-bearing step for the paper's central structural claim is the reduction of the D=11 Killing spinor equation to the SU(5)-preserving connection displayed in (4.28), which is asserted 'after a lengthy computation and using (D.2)'. The claim that (2.18)-(2.20) and (2.23) exhaust the content of the Killing spinor equation, and hence that W5 = -(1/2) d log Δ is the only restriction on the D=10 SU(5) structure, depends entirely on this step. The independent derivation of (C.12) from the intrinsic contorsion in appendix C is a real check and the coefficients agree, which is reassuring, but the manuscript should state explicitly which SU(5) irreducible components of the Clifford connection were verified to cancel, and in particular where the F75 contribution drops out (the 75 contains no singlet under the isotropy group, so it cannot appear in a connection acting on the preserved spinor). A short paragraph or a small table in §4, or an appendix, listing the irreducible pieces checked would make the sufficiency argument verifiable by a reader rather than merely plausible.
[§4, Eqs. (4.32)-(4.37), vanishing theorems] The vanishing results are stated as 'general vanishing theorems for compactifications with flux', but the derivation proceeds by imposing by hand the restrictions dω = dω^(2,0) + dω^(0,2) and H_{5+5bar} = H_75 = 0, neither of which follows from compactness of B. The text does say 'if we now restrict to', so the logic is honest, but the word 'general' overstates the scope. The authors should state plainly that these are conditional no-go results, spell out the hypotheses as a theorem, and indicate which known compactifications with flux satisfy or violate them. It would also help to say explicitly whether the X8 term is retained in (4.32) and (4.34) and under what assumption on i_K X8, since footnote 4 flags that the analysis is only consistent when i_K X8 = 0.

Minor objections (9)

[§2, non-vanishing of the Killing spinor] The argument that the Killing spinor never vanishes invokes analyticity ('Assuming analyticity, we conclude that ε_i vanishes everywhere'). This is stronger than needed: the geodesic argument already gives vanishing on an open neighbourhood of a hypothetical zero, and the Killing spinor equation is a first-order linear PDE, so unique continuation suffices. Since the main classification is local, in a neighbourhood where K is timelike by assumption, a sentence noting this would remove an unnecessary hypothesis.
[§4, text following Eq. (4.30)] The text says the four-form equation 'gives rise to two equations', lists two lines in (4.30), and then refers to 'the third equation' being implied by the first. The counting is inconsistent; presumably the intended reference is to the second line of (4.30). Please clarify.
[§4, Eq. (4.22)] F75 is introduced in (4.22) with a subscript referring to the 75 of SU(5), but the decomposition of H into irreducible pieces only appears later in (4.16). A forward reference at the point of first use would help the reader.
[§4, Eqs. (4.2), (4.5), (4.6)] The factors of Δ inserted in the definition of χ are stated to ensure the compatibility condition (4.5) and the normalisation (4.6). A sentence showing how these two conditions fix the powers of Δ would make the construction easier to follow.
[§5.1, M5-branes] The statement that a non-vanishing F75 given by (5.6) is needed to recover (5.1) from (4.22) is an interesting check. It would strengthen the example to indicate which component of (4.22) fails when F75 is set to zero, rather than only reporting the value of F75 that works.
[Abstract and §1] The abstract's 'most general bosonic geometries' is established only at the level of the isotropy group; the detailed classification is carried out for the timelike case, with the null case explicitly deferred. Making this distinction explicit in the abstract would avoid over-reading.
[§3, p. 11] Typographical: 'there may be a another mathematical language' should read 'there may be another mathematical language'.
[§2, p. 7] Typographical: 'inteteresting' should read 'interesting'.
[References] Several reference entries need correction: ref. [19] reads 'varXiv:hep-th/0012011'; ref. [28] reads 'Pitma n Reseacrh'; ref. [7] gives 'arXiv:math.DG/011220', which appears truncated; ref. [29] has a stray space in 'math.DG/010 7146'.

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The paper

The paper claims

Any eleven-dimensional supergravity background that preserves at least one supersymmetry must carry one of two specific geometric structures, and in the timelike case that structure fixes almost the entire metric and four-form up to a small

The reading

The paper asks whether every supersymmetric bosonic configuration of eleven-dimensional supergravity — every metric-plus-four-form that admits at least one Killing spinor — can be written down in one uniform way. The authors answer yes, with a split by causal character: the bilinears built from a Killing spinor always produce a Killing vector $K$, and $K$ is either timelike or null, never spacelike. In the timelike case they show the whole geometry is fixed by a ten-dimensional $SU(5)$ structure (a metric, a Kähler form $Omega$ and a $(5,0)$-form $chi$) up to one component of the four-form that the Killing spinor equation simply cannot see. They then prove the converse, so the local form is necessary as well as sufficient, and add the extra equations that turn a supersymmetric geometry into a genuine solution. If correct, this turns 'find all supersymmetric M-theory backgrounds' from an open-ended search into a structured problem with a known answer template.

What carries the argument

The machinery is a chain of three steps. First, the bilinears of a Killing spinor — a one-form $K$, a two-form $Omega$ and a five-form $Sigma$ — are shown by Fierz rearrangement to satisfy a closed set of algebraic identities and, via the Killing spinor equation, a closed set of exterior-derivative relations such as $dOmega = i_K F$ and $dSigma = i_K {}^*F - Omega wedge F$. Second, these forms are read as the invariant tensors of a $G$-structure, whose intrinsic torsion is organised into the $SU(5)$ modules $W_1$ through $W_5$ defined by decomposing $dOmega$ and $dchi$. Third, a linear map $Theta$ on four-forms, $alpha mapsto alpha - {}^*(Omega wedge alpha)$, is diagonalised on the irreducible pieces of the $mathbf{210}$ of $SO(10)$; its eigenvalue table lets the authors invert the relation between the field strength and the structure, which is how almost all of $F$ gets solved for. The vanishing modules then translate into the single constraint $W_5 = -tfrac{1}{2} dlogDelta$.

Extended reading notes

Core claim

The central claim is a normal-form theorem. Any bosonic geometry of $D=11$ supergravity admitting a Killing spinor has, in a neighbourhood where the associated Killing vector is timelike, the metric $ds^2_{11} = -Delta^2(dt+omega)^2 + Delta^{-1} g_{mn}dx^m dx^n$, where the ten-dimensional base carries an $SU(5)$ structure $(g,Omega,chi)$ whose only restriction is $W_5 = -tfrac{1}{2} dlogDelta$. The four-form is then fixed by the paper's equation (4.22) up to an arbitrary piece $F_{75}$ sitting in the $mathbf{75}$ of $SU(5)$. The authors prove both directions: the spinor forces this form, and this form forces a spinor, with $F_{75}$ dropping out of the Clifford connection entirely. Imposing the Bianchi identity $domega wedge dOmega + dH = 0$ and the four-form equation of motion then gives a full supergravity solution, because the Einstein equations follow from the integrability of the Killing spinor equation. In the null case the structure is instead $(mathrm{Spin}(7) ltimes mathbb{R}^8) times mathbb{R}$, which the paper identifies but explicitly leaves for future work.

Load-bearing premise

The claim that the Killing spinor and its Killing vector are nowhere zero, and therefore that the $SU(5)$ structure is defined everywhere, rests on an analyticity assumption the authors state only in passing: 'Assuming analyticity, we conclude that the spinor vanishes everywhere.'

Editorial extensions

If this is right

Every supersymmetric $D=11$ background with a timelike Killing spinor can be generated by choosing a ten-dimensional $SU(5)$ structure with exact Lee form and a closed two-form $omega$, rather than by guessing an ansatz.
Because $F_{75}$ is invisible to the Killing spinor equation, the supersymmetry conditions alone can never fix the full four-form; the Bianchi identity and the four-form equation of motion are what pin it down.
When the ten-dimensional base is compact and the flux vanishes, the argument forces $dOmega = 0$ and hence $F = 0$, ruling out that corner of the landscape outright.
The same bilinear identities reproduce, and extend to stationary and null Killing vectors, the generalised calibration conditions previously derived for static membranes and fivebranes.
Rotating generalisations of the resolved membrane solutions and further Gödel-type backgrounds follow as special cases of the normal form rather than as separate constructions.

Reading between the lines

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

If the normal form is exhaustive, the search for new supersymmetric $M$-theory backgrounds becomes a problem in ten-dimensional complex geometry: pick a five-fold with a prescribed exact Lee form, and the eleven-dimensional lift is automatic.
The undetermined $mathbf{75}$ component is the natural place to look for a counting of flux degrees of freedom: it is the only part of $F$ that supersymmetry does not constrain, so it plausibly controls how many distinct solutions share a given $SU(5)$ structure.
A useful test of the framework would be to take a known solution with more than one Killing spinor and check that the extra spinors correspond to further reductions of the structure group inside $SU(5)$, which the paper sketches but does not carry out.
The null case, left open here, is where the classification is most likely to be tested against horizons and plane-wave limits, since those are exactly the regions where the Killing vector is expected to degenerate.

Original abstract

We propose a way to classify all supersymmetric configurations of D=11 supergravity using the G-structures defined by the Killing spinors. We show that the most general bosonic geometries admitting a Killing spinor have at least a local SU(5) or an (Spin(7)\ltimes R^8)x R structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the former case we determine what kind of local SU(5) structure is present and show that almost all of the form of the geometry is determined by the structure. We also deduce what further conditions must be imposed in order that the equations of motion are satisfied. We illustrate the formalism with some known solutions and also present some new solutions including a rotating generalisation of the resolved membrane solutions and generalisations of the recently constructed D=11 Godel solution.

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, 4,378 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.25 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We propose a way to classify all supersymmetric configurations of D=11 supergravity using the G-structures defined by the Killing spinors. We show that the most general bosonic geometries admitting a Killing spinor have at least a local SU(5) or an (Spin(7)\ltimes R^8)x R structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the former case we determine what kind of local SU(5) structure is present and show that almost all of the form of the geometry is determined by the structure. We also deduce what further conditions must be imposed in order that the equations of motion are satisfied. We illustrate the formalism with some known solutions and also present some new solutions including a rotating generalisation of the resolved membrane solutions and generalisations of the recently constructed D=11 Godel solution.

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 propose a way to classify all supersymmetric configurations of D=11 supergravity using the G-structures defined by the Killing spinors.	contribution 0.88	0.88	0.21	hedged possibility 0.46
s1	We show that the most general bosonic geometries admitting a Killing spinor have at least a local SU(5) or an (Spin(7)\ltimes R^8)x R structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively.	result 0.99	0.93	0.73	asserted 1.00
s2	In the former case we determine what kind of local SU(5) structure is present and show that almost all of the form of the geometry is determined by the structure.	result 0.90	0.95	0.14	asserted 0.99
s3	We also deduce what further conditions must be imposed in order that the equations of motion are satisfied.	result 0.59	0.95	0.06	asserted 1.00
s4	We illustrate the formalism with some known solutions and also present some new solutions including a rotating generalisation of the resolved membrane solutions and generalisations of the recently constructed D=11 Godel solution.	contribution 0.80	0.94	0.06	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.74 · 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

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S sections
T body sentences
C citations
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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

References

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Smooth toric Kähler surfaces with a torus-invariant self-dual twistor 2-form fall into exactly six explicit local families: product-toric, Calabi-toric, orthotoric, elliptic, parabolic, and hyperbolic.
Accelerating Black Holes and Spinning Spindles
hep-th 2020-12 accept novelty 6.0 of 10

Uplifting accelerating, rotating, dyonic AdS4 black holes to eleven-dimensional supergravity on a regular Sasaki-Einstein seven-manifold gives smooth backgrounds whose horizons are weighted projective spaces, plus a n...
Aspects of AdS$_2$ classification in M-theory: Solutions with mesonic and baryonic charges
hep-th 2019-08 conditional novelty 6.0 of 10

The paper gives necessary and sufficient geometric conditions for N=(1,0) AdS2 solutions in M-theory with SU(4)-structure, and presents numerical evidence for a new Q^{1,1,1} solution with both mesonic and baryonic charges.
F-theory and AdS_3/CFT_2 (2,0)
hep-th 2017-12 conditional novelty 6.0 of 10

Varying the Type IIB axio-dilaton over the internal space yields new AdS3 solutions dual to 2d (0,2) SCFTs, with holographic and field-theory central charges agreeing to subleading order in N.
6D microstate geometries from 10D structures
hep-th 2013-06 conditional novelty 6.0 of 10

A general null-Killing-vector formalism yields the BPS equations for D1-D5-P microstates and an explicit nonlinear, circle-dependent solution claimed to be regular and asymptotically flat.
Supergravity as Generalised Geometry II: $E_{d(d)} \times \mathbb{R}^+$ and M theory
hep-th 2012-12 accept novelty 6.0 of 10

Eleven-dimensional supergravity restricted to a d-dimensional internal space is shown to be exactly the generalised Ricci-flat geometry of an exceptional generalised connection, with the fermion equations, action and ...
Adding momentum to supersymmetric geometries
hep-th 2012-08 conditional novelty 6.0 of 10

A single harmonic function deforms any GMR supersymmetric background into a new solution carrying a travelling wave, and for D1-D5 microstates the deformation is matched to a U(1) current in the dual CFT.
All homogeneous N=2 M-theory truncations with supersymmetric AdS4 vacua
hep-th 2012-08 accept novelty 6.0 of 10

Eleven-dimensional supergravity on any seven-dimensional SU(3)-structure manifold reduces to four-dimensional N=2 gauged supergravity with vector-multiplet, Heisenberg-algebra, and hyperscalar isometries, and the fram...
Four-Dimensional SCFTs from M5-Branes
hep-th 2012-03 accept novelty 6.0 of 10

An infinite family of 4D N=1 superconformal field theories arises from M5-branes on complex curves with decomposable normal bundles, with central charges computed consistently by anomaly, holography, and quiver methods.
Lifshitz Solutions of D=10 and D=11 supergravity
hep-th 2010-08 accept novelty 6.0 of 10

Infinite families of exact z = 2 Lifshitz solutions of type IIB, type IIA and D = 11 supergravity are constructed from five- and seven-dimensional Einstein spaces, and are supersymmetric when those spaces are Sasaki-Einstein.
Quantum Criticality and Holographic Superconductors in M-theory
hep-th 2009-12 accept novelty 6.0 of 10

A new consistent Kaluza-Klein truncation of M-theory yields fully back-reacted holographic superconductors whose zero-temperature limit is a charged domain wall ending at a universal conformal fixed point.
Supersymmetric sources, integrability and generalized-structure compactifications
hep-th 2007-06 conditional novelty 6.0 of 10

Orientifolds are compatible with dynamic SU(3)xSU(3)-structure, and source-corrected Einstein and dilaton equations follow from supersymmetry plus form Bianchi identities.
Matrix String Description of Cosmic Singularities in a Class of Time-dependent Solutions
hep-th 2006-03 conditional novelty 6.0 of 10

A class of 1/2 BPS time-dependent supergravity solutions, including backgrounds with four-form flux, can be carried into weak-coupling matrix string theory, with three example families satisfying the decoupling conditions.
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...
Supersymmetric AdS(4) compactifications of IIA supergravity
hep-th 2004-12 conditional novelty 6.0 of 10

N=1 supersymmetric AdS4 compactifications of massive IIA supergravity require a half-flat internal six-manifold with torsion in two specific classes, and all form fluxes are fixed by the internal geometry.
Superstrings with Intrinsic Torsion
hep-th 2003-02 accept novelty 6.0 of 10

A case-by-case classification of static supersymmetric NS-NS string backgrounds in terms of G-structures and intrinsic torsion, adding new d=8, d=9 and d=6 geometries to the known list.
Consistent Kaluza-Klein Reductions for General Supersymmetric AdS Solutions
hep-th 2007-07 accept novelty 5.0 of 10

For the most general supersymmetric AdS5 backgrounds of type IIB string theory, the authors construct a consistent Kaluza-Klein reduction to minimal five-dimensional gauged supergravity, with analogous reductions for ...
AdS(3) Solutions of IIB Supergravity from D3-branes
hep-th 2005-11 conditional novelty 5.0 of 10

Supersymmetric pure D3-brane IIB backgrounds with an AdS3 factor are locally a warped U(1) fibration over a six-dimensional Kahler manifold whose curvature obeys one fourth-order equation.
Flux compactifications in string theory: a comprehensive review
hep-th 2005-09 accept novelty 3.0 of 10

A structured survey of type II flux compactifications: which fluxes preserve N=1, why compact spaces need orientifold planes, how fluxes generate superpotentials, and how far they go toward fixing moduli.

How later papers used it

89 later papers cite it

Evidence: substantial. 73 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 31 of the 168 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

3 later papers by other authors are listed below: 3 used its method or tool. 4 more papers by the same authors are listed: 4 used its method or tool.

Uses its method or tool 7 of 12 papers with this label · 4 by the same authors

Used this paper to reduce the 11d supersymmetry conditions into differential conditions on a 10d base space.

The Near-Horizon Geometry of Supersymmetric Rotating AdS$_4$ Black Holes in M-theory (2020) · 0.98 · paraphrased: the licence does not allow quoting
Solved the Killing spinor equations, the 4-form Bianchi identity, and the 4-form equations of motion following this paper's conventions.

Aspects of AdS$_2$ classification in M-theory: Solutions with mesonic and baryonic charges (2019) · 0.99 · paraphrased: the licence does not allow quoting
Used a standard analysis, citing this paper, to write the integrability equation.

The Supersymmetry of Higher-Derivative Supergravity in AdS$_4$ Holography (2021) · 0.92 · paraphrased: the licence does not allow quoting

By the same authors

Constructed bosonic solutions of D=11 supergravity using this paper's conventions.

Supersymmetric quantum criticality supported by baryonic charges (2012) (same authors) · 0.95 · paraphrased: the licence does not allow quoting
Demanded that its solution be a supersymmetric solution of D=11 supergravity with a time-like Killing spinor by requiring an SU(5) structure satisfying the conditions given in this paper.

Lifshitz Solutions of D=10 and D=11 supergravity (2010) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Followed an argument of this paper.

Geometries with Killing Spinors and Supersymmetric AdS Solutions (2007) (same authors) · 0.99 · paraphrased: the licence does not allow quoting
Turned to the D=11 Killing spinor equation as given in this paper and decomposed the D=11 Clifford algebra.

Accelerating Black Holes and Spinning Spindles (2020) (same authors) · 0.82 · paraphrased: the licence does not allow quoting

How the literature uses it

87 arXiv papers cite this in 160 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 · 98 sentences without a confident label

Contests it 1 paper

Equation [ref] corrects equation (2.14) of [ref] and equation (B.6) of [ref] .

On the algebraic structure of Killing superalgebras · The algebraic and differential relations · contest 0.83

Builds on it 2 papers

We derive these by imposing an AdS$_3$ factor on the general geometric constraints that any $𝒩=1$ solution in $d=11$ should obey presented in [ref] .

All $\mathcal{N}=(8,0)$ AdS$_3$ solutions in 10 and 11 dimensions · Geometric supersymmetry conditions for AdS$_3×$M$_8$ · 0.86
From this data, building on earlier ideas of Calabi and Eckmann [ref] , Goldstein and Prokushkin [ref] constructed a toric fibration $π:X→Y$, equipped with a $(1,0)$-form $θ$ on $X$ satisfying $∂θ=0$, $∂̅θ=π^*(ω_1+iω_2)$.

Geometric Partial Differential Equations from Unified String Theories · The Anomaly flow and the Fu-Yau solution of the Hull-Strominger system · 0.82

Uses its method or tool 12 papers

We will determine these conditions by using the results in [ref] which classified all 11d supergravity backgrounds preserving supersymmetry and admitting a timelike Killing vector.

The Near-Horizon Geometry of Supersymmetric Rotating AdS$_4$ Black Holes in M-theory · Setup · 0.87
However, a more efficient approach is build on an existing geometric classification, namely that of the geometry following from a single arbitrary Killing spinor in 11 dimensions [ref] .

Aspects of AdS$_2$ classification in M-theory: Solutions with mesonic and baryonic charges · Supersymmetric AdS$_2$ · 0.93
The relevant conditions that need[In the language of [ref] note that these imply, in particular, that $W_4=3dlogΔ$ and $W_5=-12dlogΔ$.] to be imposed are [ref] [equation] The first equation implies that $W=W_1+i W_2$ is primitive and the second one implies that its type $(0,2)$ component is missing.

Lifshitz Solutions of D=10 and D=11 supergravity · Checking supersymmetry for Lif$_4$($z=2$) solutions in $D=11$ · 0.95

Compares with it 4 papers

The 10d base admits a canonical SU$(5)$ structure which we denote by $(j,ω)$[In comparison to [ref] one should identify $(a,^2ϕj,^5ϕω,^5ϕ[ω])_here ↔(ω, Ω, θ, χ)_there$.

The Near-Horizon Geometry of Supersymmetric Rotating AdS$_4$ Black Holes in M-theory · Timelike structures in 11d supergravity · 0.97
Our four-form flux, $F_(4)$, is normalized such that $F_(4)= - 12 G_(4)$, where $G_(4)$ is the flux in [ref] .

Supergravity Instabilities of Non-Supersymmetric Quantum Critical Points · A brief outline · 0.81
We note that some of our identities differ in sign from those in [ref] and [ref] ; this is due to our conventions on the metric, which is “mostly minus”, and Clifford algebras.

On the algebraic structure of Killing superalgebras · The algebraic and differential relations · 0.96

Cites it as support 3 papers

Indeed, as shown in [ref] , the field equations are precisely the vanishing of the gamma-trace of the curvature of $D$.

Spencer cohomology and eleven-dimensional supergravity · Infinitesimal deformations of g · 1.00
Its equation of motion can be seen to follow from the supersymmetry equations and the Bianchi identities [ref] – as do the equations of motion for the metric and dilaton [ref] .

Domain walls from ten dimensions · $G_2×G_2$ structure and flow equations: the general case · 0.97
This has been proven for many supergravity theories coupled to matter, and, in particular, for $M$-theory (see, for example, [ref] ).

Lectures on Microstate Geometries · Some general observations about the supersymmetric solution · 0.96

Cites it as background 32 papers

In [ref] the geometry that follows from a single Majorana Killing spinor $ϵ$ in M-theory is classified.

Aspects of AdS$_2$ classification in M-theory: Solutions with mesonic and baryonic charges · Review of the Geometry of 11 dimensional Killing spinors · 0.94
We begin this section by reviewing the conditions for a supersymmetric geometry in 11d supergravity to admit a timelike Killing vector following [ref] .

The Near-Horizon Geometry of Supersymmetric Rotating AdS$_4$ Black Holes in M-theory · Setup · 0.84
At present there exist a classification for $N=32$ [ref] , non-existence results for $N=31$ [ref] and $N=30$ [ref] , a structure result for $N=1$ [ref] , and a huge zoo of solutions for other values of $N$, but no claim of classification.

Spencer cohomology and eleven-dimensional supergravity · Introduction · 0.85

Discussion

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