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Consistent SO(6) Reduction Of Type IIB Supergravity on S^5

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M. Cvetic, H. Lu, C.N. Pope, A. Sadrzadeh, T.A. Tran

High Energy Physics - Theory · Submitted 13 Mar 2000 · 23 references

Pith's verdict: The explicit non-linear S^5 Ansatz for gravity plus the self-dual five-form is the real deliverable, and the self-duality argument is sound; the unchecked Einstein equation is a genuine but acknowledged gap.

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arXiv IDhep-th/0003103 (version 2)
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CitationA. Sadrzadeh, C.N. Pope, H. Lu, M. Cvetic, T.A. Tran. "Consistent SO(6) Reduction Of Type IIB Supergravity on S^5." arXiv:hep-th/0003103 (2000). https://pith.science/paper/KGXKJWFY
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The paper

The paper claims

A ten-dimensional theory on a five-sphere reduces exactly to a five-dimensional theory of gravity, fifteen SO(6) gauge fields, and twenty scalars, and the self-duality of the 5-form is what makes the reduction work.

The reading

Kaluza-Klein reductions — curling extra dimensions into a sphere and reading off the lower-dimensional physics — are usually only trustworthy order by order, because the non-linear terms of the higher-dimensional theory tend to generate extra fields that were set to zero. This paper writes down a closed, non-linear reduction Ansatz for type IIB supergravity truncated to its metric and self-dual 5-form, compactified on $S^5$, and argues that it lands exactly on five-dimensional gravity coupled to the fifteen $SO(6)$ Yang-Mills fields and the twenty scalars of the coset $SL(6,R)/SO(6)$. The mechanism the authors highlight is self-duality: with an unconstrained 5-form the field equation would impose the unacceptable constraint $F^{ij}_{(2)}=0$-like conditions on the gauge fields, and only the self-dual combination cancels that against the Bianchi identity to reproduce the correct Yang-Mills equations. The paper then asks which higher-dimensional theories could possibly admit such a sphere reduction at all, and answers that only $(D,n)=(11,4)$, $(11,7)$ and $(10,5)$ survive, with the eleven-dimensional cases additionally requiring the $F_{(4)}^{2}$-type Chern-Simons term and the ten-dimensional case requiring self-duality.

What carries the argument

The load-bearing object is the pair of Ansätze for $d s^2_{10}$ and the self-dual 5-form $H_{(5)}=G_{(5)}+{}^*G_{(5)}$, built from a unimodular symmetric tensor $T_{ij}$ (the twenty scalars), antisymmetric 1-forms $A^{ij}_{(1)}$ (the fifteen gauge fields), the constrained coordinates $mu^i$ on $S^5$, and the warping factor $Delta = T_{ij}mu^i mu^j$. The gauge fields enter through the covariant derivative $D mu^i = d mu^i + g A^{ij}_{(1)} mu^j$, so the scalars are charged under the gauge group and cannot be kept while the gauge fields are dropped. The mechanism that carries the consistency argument is the split of $dH_{(5)}=0$ into structures $X^{ij}_{(4)} wedge D mu^i wedge D mu^j$, which yield the Yang-Mills equations, and $X^{ij}_{(5)} wedge mu^i D mu^j$, which yield the scalar equations with a trace subtraction forced by $mu^i D mu^i = 0$; the Schouten over-antisymmetrisation identity does the bookkeeping.

Extended reading notes

Core claim

The central claim is that the metric-plus-self-dual-5-form sector of type IIB supergravity admits a complete, non-linear, consistent reduction on $S^5$, with the Ansatz given explicitly in equations (1)–(5) of the paper, and that substituting it into the ten-dimensional equations $dH_{(5)}=0$ and $R_{MN} = H_{MPQRS}H_{N}{}^{PQRS}/96$ reproduces the five-dimensional equations of motion (7) and the Lagrangian (8), which the authors identify with the gravity, fifteen $SO(6)$ gauge fields and twenty $SL(6,R)/SO(6)$ scalars subsector of $N=8$, $D=5$ gauged supergravity. A second claim is a consistency filter on sphere reductions in general: starting from gravity plus an $n$-form, a reduction on $S^n$ that keeps the full $SO(n+1)$ gauge fields and the $SL(n+1,R)/SO(n+1)$ scalars is possible only for $(D,n) = (11,4), (11,7), (10,5)$, and in each case extra structure is mandatory — the $FFA$ term in eleven dimensions, self-duality in ten. The authors are explicit that their consistency testing has so far covered the 5-form equations only, and that a full check of the ten-dimensional Einstein equations is deferred to future work; they offer agreement with previously proven truncations, notably the $SU(2)imes U(1)$ case of [8], as supporting evidence.

Load-bearing premise

The load-bearing premise is that verifying the five-form equations of motion, together with agreement with three previously proven truncations, is enough to certify consistency; the authors state in section 2 that the ten-dimensional Einstein equations have not yet been checked and are left to future work.

Editorial extensions

If this is right

  • Any solution of the five-dimensional theory built from these scalars necessarily carries non-zero $SO(6)$ Yang-Mills fields, because the scalars act as sources in the gauge-field equations; one cannot study the scalar sector alone, unlike in a toroidal reduction.
  • The $S^5$ reduction provides an explicit embedding of the gravity, fifteen-gauge-field and twenty-scalar sector of $N=8$, $D=5$ gauged supergravity into type IIB, which is the sector that couples to the D3-brane and is therefore the natural setting for AdS/CFT computations of the dual four-dimensional gauge theory.
  • The dilaton and axion of type IIB can be carried along unchanged, adding an $SL(2,R)$-invariant Lagrangian to the five-dimensional theory without entering the scalar potential, so the truncation extends beyond the pure metric-plus-5-form starting point.
  • The reduction reproduces, under successive truncations, the diagonal-scalar embedding, the maximal abelian $U(1)^3$ embedding, and the $N=4$ $SU(2)imes U(1)$ embedding whose consistency was already established, giving three independent consistency checks.
  • The general consistency argument turns a technical requirement into a selection principle: demanding a consistent sphere reduction that keeps all massless fields isolates $D=11$ and $D=10$ and forces the $FFA$ term and self-duality respectively, with $D=11$ emerging as an upper bound on the dimension.

Reading between the lines

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

  • If the deferred Einstein-equation check were to fail, the correct reading would be that the 5-form Bianchi identity and the Yang-Mills equations close among themselves but the metric sector does not — a failure mode distinguishable from a mere calculational gap, and one worth testing directly rather than inferring from the previously verified truncations.
  • The observation that the $F wedge F$ term in $D=5$ acts as a source for the Yang-Mills fields themselves, whereas in $D=7$ and $D=4$ it sources the 2-form potentials and the pseudoscalars, suggests a structural reason why the five-dimensional case is the simplest of the three: there is nowhere else for the current to go, so the truncation closes with fewer fields.
  • The filter that picks out $(11,4)$, $(11,7)$ and $(10,5)$ rests on the dilaton-coupling invariant taking the value $Delta=4$, which is preserved under toroidal reduction; a reader could test the filter's robustness by checking whether any theory with a non-minimally coupled $n$-form, or with several $n$-forms, evades the enumeration.
  • Because the truncation is non-supersymmetric at the ten-dimensional level while its five-dimensional image sits inside a supersymmetric theory, the construction offers a case study in how much of a supersymmetric structure survives a truncation that keeps only bosonic fields.
Original abstract

Type IIB supergravity can be consistently truncated to the metric and the self-dual 5-form. We obtain the complete non-linear Kaluza-Klein S^5 reduction Ansatz for this theory, giving rise to gravity coupled to the fifteen Yang-Mills gauge fields of SO(6) and the twenty scalars of the coset SL(6,R)/SO(6). This provides a consistent embedding of this subsector of N=8, D=5 gauged supergravity in type IIB in D=10. We demonstrate that the self-duality of the 5-form plays a crucial role in the consistency of the reduction. We also discuss certain necessary conditions for a theory of gravity and an antisymmetric tensor in an arbitrary dimension D to admit a consistent sphere reduction, keeping all the massless fields. We find that it is only possible for D=11, with a 4-form field, and D=10, with a 5-form. Furthermore, in D=11 the full bosonic structure of eleven-dimensional supergravity is required, while in D=10 the 5-form must be self-dual. It is remarkable that just from the consistency requirement alone one would discover D=11 and type IIB supergravities, and that D=11 is an upper bound on the dimension.

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, 6,376 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.46 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.85 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Type IIB supergravity can be consistently truncated to the metric and the self-dual 5-form. We obtain the complete non-linear Kaluza-Klein S^5 reduction Ansatz for this theory, giving rise to gravity coupled to the fifteen Yang-Mills gauge fields of SO(6) and the twenty scalars of the coset SL(6,R)/SO(6). This provides a consistent embedding of this subsector of N=8, D=5 gauged supergravity in type IIB in D=10. We demonstrate that the self-duality of the 5-form plays a crucial role in the consistency of the reduction. We also discuss certain necessary conditions for a theory of gravity and an antisymmetric tensor in an arbitrary dimension D to admit a consistent sphere reduction, keeping all the massless fields. We find that it is only possible for D=11, with a 4-form field, and D=10, with a 5-form. Furthermore, in D=11 the full bosonic structure of eleven-dimensional supergravity is required, while in D=10 the 5-form must be self-dual. It is remarkable that just from the consistency requirement alone one would discover D=11 and type IIB supergravities, and that D=11 is an upper bound on the dimension.

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 Type IIB supergravity can be consistently truncated to the metric and the self-dual 5-form. background 0.53 0.28 0.24 asserted 0.98
s1 We obtain the complete non-linear Kaluza-Klein S^5 reduction Ansatz for this theory, giving rise to gravity coupled to the fifteen Yang-Mills gauge fields of SO(6) and the twenty scalars of the coset SL(6,R)/SO(6). contribution 0.73 0.95 0.55 asserted 1.00
s2 This provides a consistent embedding of this subsector of N=8, D=5 gauged supergravity in type IIB in D=10. interpretation 0.72 0.81 0.27 asserted 1.00
s3 We demonstrate that the self-duality of the 5-form plays a crucial role in the consistency of the reduction. result 0.84 0.89 0.12 asserted 1.00
s4 We also discuss certain necessary conditions for a theory of gravity and an antisymmetric tensor in an arbitrary dimension D to admit a consistent sphere reduction, keeping all the massless fields. contribution 0.97 0.76 0.07 asserted 0.67
s5 We find that it is only possible for D=11, with a 4-form field, and D=10, with a 5-form. result 1.00 0.90 0.14 asserted 1.00
s6 Furthermore, in D=11 the full bosonic structure of eleven-dimensional supergravity is required, while in D=10 the 5-form must be self-dual. result 0.97 0.73 0.12 asserted 1.00
s7 It is remarkable that just from the consistency requirement alone one would discover D=11 and type IIB supergravities, and that D=11 is an upper bound on the dimension. interpretation 0.97 0.77 0.26 asserted 0.87

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

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

72 later papers cite it

Evidence: substantial. 68 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 112 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

Builds on it 1 of 2 papers with this label
Uses its method or tool 3 of 8 papers with this label
How the literature uses it

71 arXiv papers cite this in 113 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 · 69 sentences without a confident label

Builds on it 2 papers
  • Nonetheless, extending the arguments of [ref] , it was shown in[See [ref] for related earlier work.] [ref] that RN black holes in $AdS_5×S^5$ have instabilities involving a combination of off-diagonal gauge fields and neutral scalars, when interpreted within STU consistent truncation [ref] of type IIB supergravity.

    The ordered phase of charged N=4 SYM plasma · Introduction and summary · 0.88

  • We have extended the analysis of [ref] , and constructed a more complete class of black hole solutions that are consistent truncations of the Kaluza-Klein reduction of type IIB supergravity on $S^5$ [ref] , and dual to an anisotropically deformed $N=4$ Yang-Mills plasma.

    Phase transitions of an anisotropic N=4 super Yang-Mills plasma via holography · Discussion · 1.00

Uses its method or tool 9 papers
Compares with it 1 paper
Cites it as support 1 paper
  • Of course, the well-known consistent KK reductions of $D=11$ supergravity on $S^4$ [ref] and $S^7$ [ref] , or of IIB supergravity[No complete reduction ansatz of IIB supergravity on $S^5$ has been yet constructed.] on $S^5$ (see, e.g., [ref] ), related to the maximally supersymmetric solutions $AdS_7 ×S^4$, $AdS_4 ×S^7$ and $AdS_5 ×S^5$, respectively, are also examples supporting the conjecture.

    D=5 SU(2)xU(1) Gauged Supergravity from D=11 Supergravity · Introduction · 0.97

Cites it as background 29 papers
  • In [ref] (GIKS) the authors argued that the extremal limit is never achieved[Identical phenomenon occurs in other holographic models as well [ref] .]: there is a critical temperature, more precisely the critical value of $T/μ$, below which the charged SYM plasma is unstable to fluctuations of the off-diagonal gauge fields and the neutral scalars of the bulk gravitational description, when interpreted within STU consistent truncation [ref] of type IIB supergravity.

    On the relevance of GIKS instability of charged N=4 SYM plasma · Introduction · 0.86

  • The action for $SO(6)$ gauged supergravity is given by [ref] [equation] where $R$ is the Ricci scalar, $V$ is the potential associated to the scalar fields described by the symmetric $SO(6)$ tensors $T_ij$ with unit determinant, $(i,j,k,⋯)$ denote the $SO(6)$ vector indices, $(a, b,⋯)$ are the spacetime indices, $ε^abcde$ denotes the spacetime Levi-Civita tensor with $^01234 ≡- 1/√(-g)$, $ε_i_1 ⋯i_6$ is the Levi-Civita tensor with $ε_i_1 ⋯i_6≡1$, and [equation] Using the holographic dictionary, we can relate the 5-dimensional Newton's constant with the rank $N$ of the gauge group $SU(N)$ of the dual $N=4$ SYM theory and with the $_5$ radius $L$ as[In detail, matching the low energy limit of string theory with type IIB supergravity in the Einstein frame one finds that the 10-dimensional Newton constant is $G_10=8π^6 g_s^2ℓ_s^8$ where $g_s$ and $ℓ_s$ are the string coupling and string leng

    New phases of $\mathcal{N}=4$ SYM at finite chemical potential · Truncating $SO(6)$ down to $U(1)^3$ gauged supergravity · 0.80

  • The $S^5$ truncation is a well-known example [ref] , however, we notice that some of the key details of this truncation are often only briefly discussed in the literature.

    Type IIA embeddings of $D=5$ minimal gauged supergravity via Non-Abelian T-duality · Truncations of IIB supergravity to $D=5$ minimal gauged supergravity · 0.90

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