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Gauged Six-dimensional Supergravity from Massive Type IIA

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M. Cvetic, H. Lu, C.N. Pope

High Energy Physics - Theory · Submitted 28 Jun 1999 · 16 references

Pith's verdict: The paper that closed the loop on Romans F(4): an explicit warped S^4 ansatz from massive IIA, with the Einstein-equation check reported rather than shown.

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arXiv IDhep-th/9906221 (version 1)
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CitationC.N. Pope, H. Lu, M. Cvetic. "Gauged Six-dimensional Supergravity from Massive Type IIA." arXiv:hep-th/9906221 (1999). https://pith.science/paper/KDNC7LDQ
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The paper

The paper claims

The paper claims a complete, non-linear reduction ansatz that turns the bosonic equations of massive type IIA supergravity in ten dimensions into the equations of Romans' F(4) gauged supergravity in six dimensions, via a warped four-sphere.

The reading

Six-dimensional gauged supergravity has long lacked a higher-dimensional origin, unlike its four-, five- and seven-dimensional cousins. This paper proposes one: the bosonic sector of massive type IIA supergravity, reduced on a warped four-sphere, is claimed to truncate consistently to Romans' F(4) gauged supergravity with an SU(2) gauge group. The authors write down a complete non-linear ansatz for the metric, dilaton and all form fields, then substitute it into the ten-dimensional equations of motion and Bianchi identities and report that the ten-dimensional equations hold exactly when the six-dimensional ones do. The mass parameter of the ten-dimensional theory and the gauge coupling of the six-dimensional theory are tied together by $m = (2/3)^{1/2} g$, so the two theories are not merely similar but matched. If the claim holds, every solution of the six-dimensional gauged theory has a ten-dimensional life, including the warped $AdS_6$ geometry that appears as the near-horizon limit of a D4-D8 brane configuration.

What carries the argument

The load-bearing object is the reduction ansatz itself, equation (6): a metric with warp factor $(sin xi)^{1/12} X^{1/8}$ multiplying the six-dimensional line element, a $Delta^{3/8}$ factor on $ds_6^2$, a $dxi^2$ term, and a round-three-sphere part built from $h^i = sigma^i - g A^i_{(1)}$; together with ansaetze for $hat F_{(4)}$, $hat F_{(3)}$, $hat F_{(2)}$ and $e^{hatvarphi}$ written in terms of the six-dimensional field strengths, Hodge duals, $X$, $Delta$, and a quantity $U$. The mechanism carrying the argument is direct verification: the ansatz is inserted into the ten-dimensional form equations and the ten-dimensional Einstein equation, and the claim is that the resulting expressions vanish identically once the six-dimensional equations of motion hold. The paper also displays the ten-dimensional Hodge duals of the form fields, which are needed for that substitution, and stresses that the check is done at the level of equations of motion rather than by integrating a Lagrangian over the internal space.

Extended reading notes

Core claim

The central claim is that massive type IIA supergravity on a warped $S^4$ reduces consistently to Romans' $F(4)$ gauged supergravity in six dimensions, with the full non-linear ansatz given explicitly rather than to leading order in fluctuations. The internal space is foliated by three-spheres, with left-invariant one-forms $h^i = sigma^i - g A^i_{(1)}$ that carry the six-dimensional SU(2) gauge fields, and a latitude coordinate $xi$ whose warp factor $(sin xi)^{1/12}$ multiplies the whole ten-dimensional metric. A scalar $X$, related to the six-dimensional dilaton by $X = e^{-varphi/(2sqrt{2})}$, and a combination $Delta = X cos^2 xi + X^{-3} sin^2 xi$ control how the four-sphere deforms when the six-dimensional fields are excited. The authors report that substituting the ansatz into the ten-dimensional equations of motion and Bianchi identities reproduces exactly the six-dimensional equations of motion following from the Romans Lagrangian, with the mass-coupling relation $m = (2/3)^{1/2} g$. Setting $X = 1$ and the gauge fields to zero recovers the warped $AdS_6 times S^4$ solution known from the near-horizon limit of localised D4-D8 branes, and the authors also exhibit an AdS black hole supported by one SU(2) component and note a second, non-supersymmetric $AdS_6$ vacuum at $X = 3^{-1/4}$ where the four-sphere is distorted.

Load-bearing premise

The consistency claim rests on substituting the full non-linear ansatz into the ten-dimensional Einstein equation, a step the paper calls a more exacting task and reports only as a stated outcome, so the matching of the $U$, $Delta$ and $X$-dependent terms is asserted rather than displayed term by term.

Editorial extensions

If this is right

  • Any solution of the six-dimensional Romans theory, including non-abelian ones, acquires a ten-dimensional massive IIA interpretation by substitution into the ansatz.
  • The warped $AdS_6 times S^4$ near-horizon geometry of localised D4-D8 branes becomes the $X = 1$, $A^i_{(1)} = 0$ special case of the reduction, giving the AdS/CFT entry a concrete ten-dimensional embedding.
  • The AdS black hole supported by a single SU(2) Yang-Mills component oxidises to a ten-dimensional configuration, which the authors expect to describe a rotating D4-D8 system with angular frequency set by the black-hole charge.
  • The second stationary point of the scalar potential, $X = 3^{-1/4}$, gives a non-supersymmetric $AdS_6$ whose internal four-sphere is distorted by a $Delta$ that is no longer constant.
  • Because the reduction is consistent in the equation-of-motion sense, massive Kaluza-Klein fields can be dropped when computing boundary correlation functions in the dual five-dimensional theory.

Reading between the lines

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

  • The paper's own remark that the action-level reduction fails, because the warp factor vanishes at $xi = 0$ and the integral over the internal coordinate diverges, suggests the internal space should really be read as a hemisphere with a boundary; that boundary may be where the dual five-dimensional description ceases to be a conventional field theory and becomes a defect or interface system.
  • Since the dilaton diverges at the same edge, the ten-dimensional supergravity used to verify consistency is itself leaving its regime of validity there, so the reduction is best trusted away from the equator; a reader could test whether the claimed matching of equations survives when the ansatz is expanded near $xi = 0$.
  • The rotating black hole interpretation invites a check against known spinning brane solutions: if the oxidation produces a metric with the expected angular momentum parameter, the reduction gains an independent, solution-level confirmation beyond the equation-by-equation substitution.
  • The distorted non-supersymmetric $AdS_6$ vacuum raises a stability question the letter does not pursue: whether the $Delta$-deformed four-sphere signals an instability of that vacuum, which would be visible as a tachyonic mode in the six-dimensional spectrum.
Original abstract

We obtain the complete non-linear Kaluza-Klein ansatz for the reduction of the bosonic sector of massive type IIA supergravity to the Romans F(4) gauged supergravity in six dimensions. The latter arises as a consistent warped S^4 reduction.

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, 2,261 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We obtain the complete non-linear Kaluza-Klein ansatz for the reduction of the bosonic sector of massive type IIA supergravity to the Romans F(4) gauged supergravity in six dimensions. The latter arises as a consistent warped S^4 reduction.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We obtain the complete non-linear Kaluza-Klein ansatz for the reduction of the bosonic sector of massive type IIA supergravity to the Romans F(4) gauged supergravity in six dimensions. contribution 0.76 0.95 0.84 asserted 1.00
s1 The latter arises as a consistent warped S^4 reduction. result 0.68 0.67 0.16 asserted 1.00

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

What was examined, and what was not

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  • references and citing sentences
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  • 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
How later papers used it

81 later papers cite it

Evidence: substantial. 75 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 30 of the 158 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

Uses its method or tool 4 of 7 papers with this label
How the literature uses it

70 arXiv papers cite this in 143 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 · 95 sentences without a confident label

Builds on it 4 papers
Uses its method or tool 7 papers
Uses its data or values 1 paper
  • The fixed point solution [ref] has metric (in the Einstein frame and the conventions of [ref] ) s^2_IIA=Δ{ s^2_AdS_4+3/2^2 +Ξs^2_ℂℙ^2+Ωη^2 } where $=L^2(3+cos2)^1/2(5+cos2)^1/8$, $Ξ=6sin^2/(3+cos2)$, $Ω=9sin^2/(5+cos2)$ and $∈[0, π]$.

    Observables in the Guarino-Jafferis-Varela/CS-SYM duality · Guarino-Jafferis-Varela solution and duality to SYM-CS · 0.88

Compares with it 3 papers
Cites it as support 1 paper
Cites it as background 24 papers
  • The bosonic part of Lagrangian is [ref] [equation] In order to describe the AdS$_6$ black hole solutions of this theory conveniently, we make the following ansatz [equation] The charged rotating AdS$_6$ in the asymptotic static frame is [We take the static frame instead of other $ψ_i$ coordinates as these are appropriate for computing the thermodynamic quantities.

    Probing decoupled Throats of AdS$_{D}$ Black Holes in $D=6,7$ · AdS$_6$ black hole · 0.88

  • Let us recall the main features of the consistent truncation of massive IIA supergravity constructed in [ref] .

    Searching for Coleman-de Luccia bubbles in AdS compactifications · Romans supergravity · 0.91

  • The $F(4)$ gauged supergravity theory was obtained from a warped $S^4$ reduction of massive IIA supergravity in [ref] , and more recently from a reduction of IIB supergravity over an $S^2$ and a Riemann surface in [ref] .

    Gauged Six-Dimensional Supergravity from Warped IIB Reductions · Generalized reduction ansatz · 0.87

Discussion

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