Supersymmetric AdS_5 Solutions of Type IIB Supergravity

Daniel Waldram; Dario Martelli; James Sparks; Jerome P. Gauntlett

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Supersymmetric AdS_5 Solutions of Type IIB Supergravity

Jerome P. Gauntlett, Dario Martelli, James Sparks, Daniel Waldram

High Energy Physics - Theory · Submitted 17 Oct 2005 · 46 references

Pith's verdict: A careful, essentially complete local classification of supersymmetric AdS5 x M5 type IIB backgrounds; the sufficiency step assumes an invertibility the paper only argues generically, but the local framing mostly covers it.

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arXiv IDhep-th/0510125 (version 1)

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CitationDaniel Waldram, Dario Martelli, James Sparks, Jerome P. Gauntlett. "Supersymmetric AdS_5 Solutions of Type IIB Supergravity." arXiv:hep-th/0510125 (2005). https://pith.science/paper/65S2ZGWI

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This is the type IIB counterpart to their M-theory paper, and it delivers what the abstract promises: any warped AdS5 x M5 background with SO(4,2)-compatible flux admits a local identity structure, and supersymmetry is equivalent to the torsion and algebraic conditions collected in section 3.6. That is the reference result, and it is the reason to read the paper. The new content is the identity-structure analysis with all fluxes on (dilaton, axion, three-form), the reduction to the ODE (5.6) under the constant-dilaton, vanishing-axion, SU(2) x U(1) x U(1) ansatz, and the numerical family of section 5.2. The Pilch-Warner recovery is a check against [19], not a new result, and they present it that way. The derivation is checkable: the bilinear algebra is in appendix B, and appendix D gives the integrability argument that supersymmetry implies the equations of motion. Section 5.2 is honest in a way I wish were more common: the metrics close up on S5, but the three-form flux and the spinors are not globally defined, and they say so, and appendix F labels the central charge as computed while ignoring those problems. That is the right way to report a negative result. The stress-test concern about the completeness relation (3.31) is real but bounded. Inverting m_{ab} = eta-bar_a eta_b needs {xi1, xi2, xi1^c, xi2^c} to be a basis, and appendix C only rules out xi2 in span{xi1, xi1^c}, using conditions that were themselves derived from the algebraic constraints. That is not a clean independent proof. But the paper is explicit that this is a local, generic statement, and the algebraic conditions are continuous, so a degenerate locus does not break the local classification. The abstract's 'local identity structure' already carries the qualifier. I would not call this load-bearing. The other soft spot is minor: the numerical family ships no code or tabulated data, so 'continuous family' is asserted rather than reproducible. For 2005 that is unsurprising. This is for anyone working on AdS5/CFT geometry, G-structures, or flux compactifications. Send it to referees, and ask them to tighten the generic-versus-global wording in the abstract.

Referee report

Significance

This is the type IIB counterpart of the authors' M-theory classification and is a foundational reference for the G-structure approach to AdS$_5$ holography: it supplies the general local geometry underlying any $N=1$ superconformal dual beyond the Sasaki–Einstein class, and it is the framework in which later constructions of explicit AdS$_5$ type IIB backgrounds are phrased. The paper earns credit for several concrete things: the exact closed-form recovery of the Pilch–Warner solution from the ODE, the explicit reduction of the full ten-dimensional problem to a single second-order ODE in a well-defined corner, and the integrability argument of Appendix D, which is what makes the classification practically useful by removing the need to solve the equations of motion separately. The negative global result for the numerical family in §5.2 — regular $S^5$ metric but non-globally-defined flux and spinors — is reported honestly and is itself a useful statement about what this ansatz can and cannot deliver.

Major objections (2)

[§3.3, Eqs. (3.31)–(3.33)] The sufficiency direction of the classification rests on inverting the 4×4 matrix $m_{αβ} = ēta_α η_β$ with $η_α ∈ {ξ_1, ξ_2, ξ_1^c, ξ_2^c}$ via the completeness relation (3.31). This requires the four spinors to be linearly independent, which is stronger than the statement proved in Appendix C, namely that $ξ_2$ cannot be proportional to $ξ_1$ or $ξ_1^c$. Section 3.1 says only that the set 'generically' forms a complete basis, which is the language of an open dense set. Since the abstract's headline claim is 'necessary and sufficient', the scope of the equivalence should be stated explicitly in the main text: the algebraic conditions (3.34)–(3.38) are equivalent to (3.5)–(3.8) on the open set where the canonical frame (3.53) is non-degenerate, and the degenerate loci are exactly where the combinations $cos^2ζ - |S|^2$ or $sin^2ζ + |S|^2$ appearing as denominators in (3.53) vanish, i.e. where $K_3$ or $K_5$ degenerates. Those are presumably the cases treated separately (Sasaki–Einstein and the SU(2) case of Appendix C). I do not think this is a gap in the result, but it is load-bearing for the precise wording of the central claim and deserves a sentence or two in §3.3 or §3.6 rather than being left implicit.
[§5.2 and Appendix E] The numerical family is presented as giving complete metrics on $S^5$ while the three-form flux and the Killing spinors fail to be globally defined, and the central charge in Appendix E is computed 'blindly ignoring these problems'. The authors are explicit about this and I do not regard it as a defect of the paper, but the logical status of the resulting $G_5^{(sec)}$ and of the ratio (E.10) should be stated once, plainly, in the main text: these quantities are computed for the local solution and are not claimed to be the central charge of any dual CFT. As written, a reader skimming §5.2 and Appendix E could take the monotonicity of Newton's constant in $p$ as a physical statement about a family of fixed points.

Minor objections (7)

[§5.2, Eq. (5.19) vs. Appendix E] There is an internal inconsistency in the value of $h(0)$. Equation (5.19) gives $h = 2p + ...$, so $h(0) = 2p$, whereas Appendix E states 'recall that $h(0) = 2/p$'. The range quoted in §5.2, $3/2 < h(0) ≤ 2$ if and only if $4/3 > p ≥ 1$, is consistent only with $h(0) = 2/p$. Please correct whichever expression is wrong and check that the $λ^{2-p}$ coefficient in (5.19) is consistent with the corrected leading term.
[§5.2, Eq. (5.20)] I was unable to reproduce the coefficients in (5.20) from (5.17) together with the expansions (5.19). The $dλ^2$ coefficient in (5.17) is $1/(g^2-1-λ^2) ≈ 1/(2βλ^p)$ near $λ=0$, not $12βλ^p$, and the $(σ'_1)^2+(σ'_2)^2$ coefficient in (5.17) is $3h^2/g$, which tends to $12p^2$ (or $12/p^2$ if the Appendix E normalization is used), not $3p$. If an implicit rescaling of $λ$ or of the $σ'$ has been made between (5.17) and (5.20), please state it; otherwise the displayed asymptotics need correction.
[Notation, §3.1 and §5] The symbol $A$ is used for the real scalar bilinear fixed to 1 in (3.13), for the function in the ansatz $ρ = Aσ_3$ of (5.1) and in $h = -A(1+λ^2)$ of (5.2), and again for the complex two-form potential in (E.7) and for the metric function in (5.17)–(5.18). Similarly $f$ denotes the five-form constant of (3.2) and appears in the combination '$-ifG$' in (3.45). Renaming at least the §5 function and the Appendix E potential would remove a real source of confusion for a reader trying to follow the reduction.
[§3.5, Eq. (3.45)] Please confirm the factor of $i$ and that $f$ in '$-ifG$' is the same real constant introduced in (3.2); as printed the term reads as a three-form multiplied by a constant, which is not the form one expects from the $G$ equation of motion in (2.4).
[§3.1] The conclusion that the only solutions with compact $M_5$ admitting a local SU(2) structure are Sasaki–Einstein uses harmonicity of $C^{(0)}$ and of the dilaton on a compact manifold. It would help to say explicitly that compactness is essential here and that the statement is not claimed for non-compact $M_5$, since the rest of the paper is careful to keep the local analysis separate from global statements.
[§3.5] The derivation of the $G$ Bianchi identity from (3.46) is stated rather than shown. A sentence indicating how the differential condition on $W$ is extracted from (3.3)–(3.8) would make the argument self-contained, since this is the step that closes the 'supersymmetry implies the equations of motion' claim.
[Typesetting] There are several typographical errors that should be fixed: 'Binachi' below (2.6), 'particulalry' in the Conclusions, and a number of broken words and stray spaces in the abstract and body text (for example 'warp f actor', 'supersymmetr ic', 'identi ty').

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The paper

The paper claims

The paper claims a complete local classification of supersymmetric type IIB backgrounds of the form AdS$_5$ times a five-manifold, showing that two Killing spinors fix a canonical frame whose torsion conditions are exactly equivalent to the

The reading

The paper asks what the most general supersymmetric ten-dimensional type IIB geometry with an AdS$_5$ factor looks like, because such geometries are the gravity side of four-dimensional superconformal field theories. Its answer is that the two Killing spinors define, locally, a canonical orthonormal frame on the internal five-manifold $M_5$ — an identity structure — and that supersymmetry is exactly equivalent to a short list of differential and algebraic conditions on that frame and on the fluxes. If that is right, the search for new AdS$_5$ duals stops being a guess-the-ansatz exercise and becomes a matter of solving those conditions. The authors also show the conditions are strong enough to imply the equations of motion and Bianchi identities, so any solution of the supersymmetry equations is automatically a solution of the full theory. As a demonstration they restrict to constant dilaton, vanishing axion and an $S^3$-type isometry, reduce everything to one second-order nonlinear ODE, recover the known Pilch–Warner solution exactly, and find a numerical family of local solutions beyond it.

What carries the argument

The engine is the $G$-structure method: instead of solving the Killing spinor equations directly, one forms bilinears from the two spinors — a real Killing vector $K_5$, complex vectors $K$ and $K_3$, and scalars $zeta$ and $S$ — and uses Fierz identities to convert the algebraic Killing spinor equations into an operator identity, equation (3.33), which yields the flux in terms of the frame. The differential conditions become torsion conditions on the same bilinears, written in the style of generalised calibrations (equations (3.47)–(3.49)). A key simplification is the algebraic result $Z = 0$, which removes one complex scalar and makes the frame explicit. Reducing along $K_5$ and then along the coordinate $lambda$ leaves a three-dimensional metric with a local identity structure given by $(rho, sigma, sigma^*)$ plus one scalar $eta$, and the remaining equations (4.12)–(4.14) are what the section 5 ansatz then truncates to a single ODE.

Extended reading notes

Core claim

The central claim is a necessary-and-sufficient statement: a bosonic type IIB background whose metric is a warped product $ds^2_{10} = e^{2A}(ds^2(AdS_5) + ds^2_5)$ and whose fluxes respect $SO(4,2)$ is supersymmetric if and only if $M_5$ carries an identity structure built from two spinors $xi_1, xi_2$ satisfying the conditions collected in section 3.6 — equations (3.47)–(3.52) — with the metric taking the canonical form (3.53). In that frame the five-form flux is fixed by a scalar, $f = 4m e^{4A} sin zeta$, and the three-form flux is determined algebraically by the frame and the warp factor. The authors further show that these conditions imply the dilaton and three-form equations of motion and the Bianchi identities, and that the canonical vector $K_5$ is a Killing vector generating a symmetry of the whole solution, matching the $U(1)_R$ symmetry of the dual field theory. Restricting to constant dilaton, vanishing axion and one-forms proportional to the left-invariant forms on $S^3$ with coefficients depending on a single coordinate $lambda$, the whole problem collapses to the ODE $ddot{g} lambda (g^2 - 1 - lambda^2) + dot{g}(g^2 - 1 + tfrac{1}{3}lambda^2 - 2lambda g dot{g}) = 0$. One exact solution reproduces the Pilch–Warner background; numerics reveal a one-parameter family whose metrics close up into $S^5$ but whose three-form flux and spinors fail to be globally defined, a limitation the authors state explicitly.

Load-bearing premise

The explicit-solution results depend on the section 5 ansatz: constant dilaton, vanishing axion, and one-forms proportional to the left-invariant forms on $S^3$ with coefficients depending only on one coordinate, which is what reduces the problem to a single ODE.

Editorial extensions

If this is right

Any claimed supersymmetric AdS$_5$ solution of type IIB with $SO(4,2)$-invariant fluxes can be checked against equations (3.47)–(3.52) rather than by re-solving the Killing spinor equations from scratch.
Because supersymmetry implies the equations of motion and Bianchi identities in this class, one only needs to solve the first-order torsion conditions and the algebraic flux relations to obtain a genuine background.
The canonical Killing vector $K_5$ gives a geometric origin for the $U(1)_R$ symmetry of the dual superconformal field theory, and it acts trivially on the dilaton, axion and warp factor while rotating the complex scalar $S$.
The Pilch–Warner solution sits inside the ODE as the special point $p = 1$, so its known dual Leigh–Strassler fixed point is recovered as one member of a broader family of local solutions.
The numerical family shows that the ODE admits regular $S^5$ metrics beyond Pilch–Warner, but their flux and spinors are not globally defined, so they do not by themselves give new AdS/CFT duals.

Reading between the lines

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

If the classification is complete, then any future AdS$_5$ type IIB solution with $SO(4,2)$ symmetry — including the beta-deformed and interpolating solutions the authors mention as contemporaneous work — must be expressible in the canonical frame (3.53), which gives a practical test for whether a proposed background is genuinely new or a coordinate rewrite of a known one.
The obstruction found in the numerical family is a global one, not a local one: the metric closes up on $S^5$ while the complex flux picks up a phase under the shift of the fibre coordinate. That pattern suggests the interesting question is which discrete periods make the phase trivial, i.e. a quantisation condition on the parameter $p$, rather than whether the ODE has more solutions.
A natural next test of the framework is to impose $N = 2$ supersymmetry as an extra condition on the same identity structure, mirroring what was done for the M-theory case; if that over-determines the system enough, it could yield further closed-form solutions.
The claim that supersymmetry implies the equations of motion rests on integrability of the Killing spinor equations plus the Bianchi identities; the argument in appendix D isolates a single Einstein-equation component, which is where an independent check would most usefully be aimed.

Original abstract

We analyse the most general bosonic supersymmetric solutions of type IIB supergravity whose metrics are warped products of five-dimensional anti-de Sitter space AdS_5 with a five-dimensional Riemannian manifold M_5. All fluxes are allowed to be non-vanishing consistent with SO(4,2) symmetry. We show that the necessary and sufficient conditions can be phrased in terms of a local identity structure on M_5. For a special class, with constant dilaton and vanishing axion, we reduce the problem to solving a second order non-linear ODE. We find an exact solution of the ODE which reproduces a solution first found by Pilch and Warner. A numerical analysis of the ODE reveals an additional class of local solutions.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.82 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We analyse the most general bosonic supersymmetric solutions of type IIB supergravity whose metrics are warped products of five-dimensional anti-de Sitter space AdS_5 with a five-dimensional Riemannian manifold M_5. All fluxes are allowed to be non-vanishing consistent with SO(4,2) symmetry. We show that the necessary and sufficient conditions can be phrased in terms of a local identity structure on M_5. For a special class, with constant dilaton and vanishing axion, we reduce the problem to solving a second order non-linear ODE. We find an exact solution of the ODE which reproduces a solution first found by Pilch and Warner. A numerical analysis of the ODE reveals an additional class of local solutions.

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 analyse the most general bosonic supersymmetric solutions of type IIB supergravity whose metrics are warped products of five-dimensional anti-de Sitter space AdS_5 with a five-dimensional Riemannian manifold M_5.	contribution 0.95	0.94	0.09	asserted 0.91
s1	All fluxes are allowed to be non-vanishing consistent with SO(4,2) symmetry.	method 0.47	0.48	0.04	asserted 0.75
s2	We show that the necessary and sufficient conditions can be phrased in terms of a local identity structure on M_5.	result 0.85	0.94	0.61	asserted 1.00
s3	For a special class, with constant dilaton and vanishing axion, we reduce the problem to solving a second order non-linear ODE.	method 0.87	0.92	0.06	asserted 1.00
s4	We find an exact solution of the ODE which reproduces a solution first found by Pilch and Warner.	result 0.97	0.92	0.13	asserted 1.00
s5	A numerical analysis of the ODE reveals an additional class of local solutions.	result 0.98	0.95	0.15	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

not examined

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levels run

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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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Forward citations

Cited by 30 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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How later papers used it

60 later papers cite it

Evidence: substantial. 39 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 30 of the 220 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

2 later papers by other authors are listed below: 1 used its method or tool and 1 cited it as support. 3 more papers by the same authors are listed: 3 used its method or tool.

Uses its method or tool 4 of 8 papers with this label · 3 by the same authors

Now we move on to determine the dependence of the Killing spinors $ϵ_r̂$ on the internal space parametrised by $ψ_i = 1,2,⋯, 6$ and $θ$ by numerically solving the algebraic KSE [ref], which can be written more explicitly as:[In practice, we use the Eigenvectors[] function of Mathematica and take the eigenvectors whose eigenvalue is zero. ] [equation] Following similar ansatz in [ref] , our spinor may have the following form: [equation] where $ϵ_(4)$ and $ϵ_(6)$ are Killing spinors on AdS$_4$ and $S^6$ respectively.

New $G_2$-invariant supersymmetric AdS$_4$ vacua in 11d supergravity and holography (2025) · The Killing spinors for the dWNW saddle · 0.95

By the same authors

Considered the spinor bi-linears that characterise the identity structure on M_5 from this paper and found that all of the above conditions are satisfied for a chosen equation.

Consistent Kaluza-Klein Reductions for General Supersymmetric AdS Solutions (2007) (same authors) · 0.87 · paraphrased: the licence does not allow quoting
Used the supersymmetry equations in d=5 directly from this paper by setting G=0 to obtain its equations.

F-theory and AdS_3/CFT_2 (2,0) (2017) (same authors) · 0.82 · paraphrased: the licence does not allow quoting
Concluded from the result in appendix D of this paper that any solution to its differential equations will also solve the type IIB Einstein equations.

Flowing from $AdS_5$ to $AdS_3$ with $T^{1,1}$ (2014) (same authors) · 0.85 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 4 papers with this label

Used this paper's general result for IIB as evidence that the equations of motion for the metric and dilaton are satisfied.

AdS_6 solutions of type II supergravity (2014) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

59 arXiv papers cite this in 219 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 · 153 sentences without a confident label

Contests it 3 papers

In this expression the $ρ^α$ generate Cliff(4,1) and satisfy $ρ_01234=-i$ (this corrects a sign in [ref] ).

Consistent Kaluza-Klein Reductions for General Supersymmetric AdS Solutions · IIB reduction: fermions · contest 0.84
Attempts to set $F_5=0$ in the final equations of [ref] run into inconsistencies as it involves dividing by zero.

Supersymmetric AdS$_{5}$ solutions of type IIB supergravity without D3 branes · Introduction · contest 0.72
We use exactly[Although it will not be relevant in this paper we point out that there is a typo in [ref] : the $ρ_a$ matrices generating $(4,1)$ actually satisfy $ρ_01234=-$.] the same conventions as in [ref] , up to some simple relabelling.

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry · Conventions and 6D to 5D map · contest 0.82

Builds on it 4 papers

We therefore restrict to the case of them being independent and admitting an identity structure.]; we can therefore import the bilinear equations from [ref] where we set $sinζ≡0$ and $f≡0$.

Supersymmetric AdS$_{5}$ solutions of type IIB supergravity without D3 branes · Bilinear equations · 0.84
Since the $N=2$ case we are considering here is a special case of the $N=1$ geometries studied in [ref] , and since $X=Y=0$ implies that $K^GMSW_5=0$, the argument presented in this section will also imply that no solutions exist when $X$ and $Y$ are set to zero.

A search for AdS5 X S2 IIB supergravity solutions dual to N = 2 SCFTs · New R-symmetry directions? · 0.87
Here we will extend this result by showing that for the most general $AdS_5×_w M_5$ supersymmetric solution of type IIB supergravity with all of the fluxes active, that were analysed in [ref] , the KK reduction is also consistent.

Consistent Kaluza-Klein Reductions for General Supersymmetric AdS Solutions · Introduction · 0.96

Uses its method or tool 8 papers

After considering the spinor bi-linears that characterise the identity structure on $M_5$ [ref] , we find that all of the above conditions are satisfied if we choose [equation]

Consistent Kaluza-Klein Reductions for General Supersymmetric AdS Solutions · IIB reduction: bosonic sector · 0.87
We use the well known method of analysing the G-structure determined by the Killing spinors as was employed, for example, in [ref] (and references therein) from which some of this work is derived.

Supersymmetric AdS$_{5}$ solutions of type IIB supergravity without D3 branes · Introduction · 0.99
We will use the spinor conventions of [ref] (see also [ref] ) under the understanding that we are in addition decomposing the internal $M_5$ space into a direct product of $S^2$ and $M_3$.

A search for AdS5 X S2 IIB supergravity solutions dual to N = 2 SCFTs · Spinor and $γ$-matrix conventions · 0.83

Uses its data or values 2 papers

We have $Y=S^5$ with non-trivial metric g_Y = 1/9[6ϑ^2 + 6cos^2ϑ/3-cos2ϑ(σ_1^2+σ_2^2) + 6sin^22ϑ/(3-cos2ϑ)^2σ_3^2 +4(φ+2cos^2ϑ/3-cos2ϑσ_3)^2] , where $0≤ϑ≤π2$, $0≤φ≤2π$, and $σ_i$, $i=1,2,3$, are left-invariant one-forms on $SU(2)$ (denoted with hats in [ref] ).

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry · The Pilch-Warner solution · 0.94
The latter, in turn, was computed in appendix E of [ref] , and is given by the supergravity formula G_5 = κ^2_10/8πV_5 , V_5 ≡∫_Y ^8Δ_Y .

The central charge of supersymmetric AdS_5 solutions of type IIB supergravity · Central charge · 0.95

Compares with it 3 papers

Contrast this with the $F_5≠0$ case [ref] where one finds the infinitely many Sasaki-Einstein solutions and the Pilch-Warner solution (which has $P=0$), whilst in type IIA [ref] one finds infinitely many massive IIA solutions and recovers previously known massless solutions such as the Maldacena-Núñez solution.

Supersymmetric AdS$_{5}$ solutions of type IIB supergravity without D3 branes · Conclusions · 1.00
The conditions for AdS$_4$ solutions are more restrictive than the ones for AdS$_5$; for example, the Bianchi identities do not follow from supersymmetry as for AdS$_5$ solutions in IIB [ref] .

Perturbing gauge/gravity duals by a Romans mass · Introduction · 0.80
Indeed, there is a similarity between the equations SHeqs we have studied here and the general study of $AdS_5$ solutions of IIB supergravity in [ref] .

On Supersymmetric Flux Solutions of M-theory · Discussion · 0.91

Cites it as support 3 papers

In fact, for $AdS_5$ solutions it was shown in [ref] that supersymmetry implies all of the equations of motion and Bianchi identities.

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry · Reformulation as generalized complex geometries · 0.97
It is important to stress that all the supergravity equations of motion are satifsfied once the above supersymmetry conditions and the RR Bianchi identity ([ref]) are imposed [ref] .

On moduli and effective theory of N=1 warped flux compactifications · Pure spinors, $N=1$ conditions and generalized complex structure · 0.96
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

Cites it as background 18 papers

In [ref] AdS$_5$ solutions of IIB with non-vanishing $F_5$ Ramond-Ramomd (R-R) flux were classified.

Supersymmetric AdS$_{5}$ solutions of type IIB supergravity without D3 branes · Introduction · 0.83
We begin by summarising the results of [ref] .

Consistent Kaluza-Klein Reductions for General Supersymmetric AdS Solutions · Internal geometry and fluxes · 0.80
Our starting point is the most general class of supersymmetric $AdS_5$ solutions of type IIB supergravity, as studied in [ref] .

AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry · Supersymmetric $AdS_5$ backgrounds · 0.86

Discussion

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