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Properties of some conformal field theories with M-theory duals

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Jerome P. Gauntlett, Eoin O Colgain, Oscar Varela

High Energy Physics - Theory · Submitted 20 Nov 2006 · 36 references

Pith's verdict: A solid, well-checked extension of the η/s = 1/4π universality to the full class of supersymmetric AdS5 M-theory solutions, with the usual scope caveats honestly stated.

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arXiv IDhep-th/0611219 (version 2)
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CitationEoin O Colgain, Jerome P. Gauntlett, Oscar Varela. "Properties of some conformal field theories with M-theory duals." arXiv:hep-th/0611219 (2006). https://pith.science/paper/VT6ZBMJ4
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The paper

The paper claims

Every supersymmetric AdS$_5$ solution of eleven-dimensional supergravity reduces to a five-dimensional gauged supergravity, which pins the shear-viscosity-to-entropy ratio of the dual plasma at $1/4π$ even when an R-charge chemical potent

The reading

The paper asks what can be said about the four-dimensional $N=1$ superconformal field theories that are dual to the broadest known family of supersymmetric $AdS_5$ solutions of eleven-dimensional supergravity. For several explicit members of that family it computes the central charge of the dual theory from the quantised four-form flux, and finds two different behaviours: an $N^3$ scaling for Kähler–Einstein and $S^2×S^2$ bases, and an $N^2$ scaling for the $S^2×T^2$ case that reproduces the known central charge of the $Y^{p,q}$ Sasaki–Einstein duals. It then identifies which wrapped membranes on the internal six-manifold are supersymmetric, using a generalised calibration built from the Killing spinors, and reads off the conformal dimensions of the corresponding chiral primary operators. The headline result is that the most general solution in the class admits a consistent Kaluza–Klein truncation to minimal five-dimensional gauged supergravity, so that black-hole solutions of that five-dimensional theory lift back to exact eleven-dimensional ones. That lets the authors import known black-hole hydrodynamics and conclude that the shear-viscosity-to-entropy ratio is $1/4π$ and the speed of sound is $1/√3$ for these plasmas at non-zero R-charge chemical potential.

What carries the argument

The load-bearing device is the consistent truncation ansatz itself: the internal metric $ds^2(M_6)=g_{4ij}dx^idx^j+e^{-6λ}sec^2ζ⁡dy^2+⁡(1/9)cos^2ζ⁡(dψ+ρ+A)^2$ together with the four-form $L^{-3}G_{(4)}=d⁡C̄+⁡(1/3)dy[(∗_5F)+⁡(1/3)AF]$, where $C̄=C_0+⁡(1/3)A(e^{3λ}Y')$ and $C_0$ is the undeformed flux. The shift $dψ→dψ+A$ is what gauges the R-symmetry, and the two-form $Y'=-sin⁡ζ⁡J+K_1∧K_2$, built as a spinor bilinear, does double duty: it satisfies the generalised calibration condition $d(e^{3λ}Y')=-i⁡K̃_2⁡G_{(4)}/L^3$, so a membrane wrapping a cycle it calibrates is supersymmetric, and it also controls how the gauge field enters the three-form potential, so the same object fixes both the spectrum of wrapped-membrane operators and the R-charge normalisation. Flux quantisation, $N_{Σ_4}=(1/(2π)^3l_{11}^3)∫_{Σ_4}G_{(4)}∈ℤ$, supplies the integer charges in terms of which the central charges are expressed.

Extended reading notes

Core claim

The central claim is that the most general supersymmetric $AdS_5$ solution of $D=11$ supergravity classified earlier by Gauntlett, Martelli, Sparks and Waldram admits a consistent Kaluza–Klein truncation to minimal $D=5$ gauged supergravity. Concretely, the authors write a metric ansatz in which the internal coordinate $ψ$ is shifted by a five-dimensional gauge field $A$, propose a four-form flux ansatz built from the undeformed flux plus terms linear in $F=dA$, and show that substituting both into the eleven-dimensional equations of motion reduces them exactly to the Einstein–Maxwell–Chern–Simons equations of minimal gauged supergravity, with the bosonic Lagrangian $R+12-⁡(1/12)F^2$ plus a Chern–Simons term. They argue that $A$ corresponds to the abelian R-symmetry of the dual $N=1$ superconformal field theory, and use the truncation to fix the normalisation of the R-charge of wrapped membranes, recovering the relation $Δ=3R/2$ and thereby justifying their earlier formula for the conformal dimension as a warped two-cycle volume. Because the truncation is consistent, any charged black hole of the five-dimensional theory uplifts to a genuine eleven-dimensional solution, and existing calculations of the dual hydrodynamics carry over: the shear-viscosity-to-entropy ratio is $1/4π$ and the speed of sound is $1/√3$, now at non-zero R-charge chemical potential.

Load-bearing premise

The claim that the black-hole solutions carry R-charge rests on identifying the Kaluza–Klein gauge field with the R-symmetry of the dual field theory; the paper notes that the R-symmetry is generally a mixture of the $∂_ψ$ isometry and gauge transformations of the eleven-dimensional three-form, and works this out explicitly only for the $S^2×T^2$ case, arguing the rest by analogy.

Editorial extensions

If this is right

  • The universal ratio $η/s=1/4π$ extends from $N=4$ super-Yang–Mills and the $Y^{p,q}$ gauge theories to the whole class of $N=1$ superconformal field theories with supersymmetric $AdS_5×_wM_6$ M-theory duals, including those whose central charge scales as $N^3$ rather than $N^2$.
  • Chiral primary operators dual to wrapped membranes have conformal dimensions fixed by a warped two-cycle volume, $Δ=τ_{M2}L^3∫_{Σ_2}e^{3λ}vol_{M_6}(Σ_2)$, with the relation $Δ=3R/2$ derived rather than assumed.
  • For the $S^2×T^2$ case the central charge reproduces $c=π^3N^2/(4\,\mathrm{Vol}(Y^{p,q}))$, and the wrapped-membrane dimensions match earlier type IIB results for three-cycles and for the long BPS mesonic operators, providing a consistency check across the duality chain.
  • The speed of sound in these plasmas is $1/√3$, the value of a conformal fluid, independently of the R-charge chemical potential.
  • Any solution of minimal five-dimensional gauged supergravity — including rotating or charged black holes — generates an exact eleven-dimensional solution based on any of these internal manifolds.

Reading between the lines

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

  • If a consistent truncation to minimal gauged supergravity exists for every supersymmetric $AdS_5$ M-theory solution, one might conjecture the same for supersymmetric $AdS_n$ solutions more generally, with the truncation retaining at least part of the R-symmetry; the paper floats this but leaves it as a conjecture.
  • The $N^3$ central-charge scaling for Kähler–Einstein and $S^2×S^2$ bases is the scaling of the six-dimensional $(2,0)$ theory on M5-branes, which suggests the dual field theories in those cases are not conventional gauge theories; identifying what they are is the obvious next question the paper does not answer.
  • Because the truncation is consistent at the level of bosons and fermions, one could in principle test the R-symmetry identification by computing R-charges of wrapped membranes directly from the eleven-dimensional geometry and comparing with the gauge-field normalisation, rather than relying on the analogy to the $Y^{p,q}$ case.
  • The flux-quantisation analysis is carried out only for the $c=0$ slice of the Kähler–Einstein and $S^2×S^2$ families; extending it to $c≠0$ would test whether the $N^3$ scaling and the operator dimensions are generic or an artifact of that slice.
Original abstract

By studying classes of supersymmetric solutions of D=11 supergravity with AdS_5 factors, we determine some properties of the dual four-dimensional N=1 superconformal field theories. For some explicit solutions we calculate the central charges and also the conformal dimensions of certain chiral primary operators arising from wrapped membranes. For the most general class of solutions we show that there is a consistent Kaluza-Klein truncation to minimal D=5 gauged supergravity. This latter result allows us to study some aspects of the dual strongly coupled thermal plasma with a non-zero R-charge chemical potential and, in particular, we show that the ratio of the shear viscosity to the entropy density has the universal value of 1/4 pi.

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, 3,716 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.10 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

By studying classes of supersymmetric solutions of D=11 supergravity with AdS_5 factors, we determine some properties of the dual four-dimensional N=1 superconformal field theories. For some explicit solutions we calculate the central charges and also the conformal dimensions of certain chiral primary operators arising from wrapped membranes. For the most general class of solutions we show that there is a consistent Kaluza-Klein truncation to minimal D=5 gauged supergravity. This latter result allows us to study some aspects of the dual strongly coupled thermal plasma with a non-zero R-charge chemical potential and, in particular, we show that the ratio of the shear viscosity to the entropy density has the universal value of 1/4 pi.

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 By studying classes of supersymmetric solutions of D=11 supergravity with AdS_5 factors, we determine some properties of the dual four-dimensional N=1 superconformal field theories. contribution 0.83 0.95 0.17 asserted 1.00
s1 For some explicit solutions we calculate the central charges and also the conformal dimensions of certain chiral primary operators arising from wrapped membranes. result 0.70 0.95 0.08 asserted 1.00
s2 For the most general class of solutions we show that there is a consistent Kaluza-Klein truncation to minimal D=5 gauged supergravity. result 0.93 0.94 0.17 asserted 0.98
s3 This latter result allows us to study some aspects of the dual strongly coupled thermal plasma with a non-zero R-charge chemical potential and, in particular, we show that the ratio of the shear viscosity to the entropy density has the universal value of 1/4 pi. result 0.97 0.93 0.40 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

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

49 later papers cite it

Evidence: substantial. 36 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 102 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 3 of 4 papers with this label
Compares with it 2 of 4 papers with this label · 1 by the same authors
  • Extremizing [ref] we get ^(3)= ϵ_3^3 ν_M5^3(-5+3 √(3)) N^3 { 1/9 , 1/3, 2 } ,for $ℙ^2$, $ℙ^1×ℙ^1$ and dP$_3$, respectively, which agrees with (2.16) in [ref] for $ϵ_3 ν_M5 =3$.[$N_C_N$ in [ref] can be identified with $M_d+1=-∑M_a$, so that $N_there=-h N_C_N/M = h ∑M_a/M$ where $(h,M)$ are defined in [ref] and they have value $(3,9),(4,8)$ and $(2,6)$ for $^2$, $^1×^1$ and dP$_3$, respectively.]

    Equivariant volume extremization and holography (2023) · The extremal function for geometries with $_2$ symmetry · 0.99

By the same authors

  • Found that its central charge precisely coincides with the result computed from the explicit solution in this paper.

    Equivariant localization for AdS/CFT (2023) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

49 arXiv papers cite this in 102 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 · 70 sentences without a confident label

Builds on it 3 papers
Uses its method or tool 4 papers
  • The holographic central charge is extracted from the warped volume of the internal space [ref] , [equation] where $vol_M_6$ is the volume form of $ds^2_M_6$ in [ref].

    Holographic Duals of Argyres-Douglas Theories · Holographic Central Charge, Flavor Central Charge, and Probe M2-Branes · 0.93

  • We can follow [ref] , where given the form [ref] of the $D=11$ metric it was shown that the central charge can be expressed as [equation] where $ℓ_p$ is the eleven-dimensional Planck length.

    M5-branes wrapped on a spindle · Central charge · 0.97

  • We use that there is a general truncation of the GMSW classification to five-dimensional minimal gauged supergravity as given in [ref] .

    On the Class $\mathcal{S}$ Origin of Spindle Solutions · Canonical N=1 form · 0.95

Compares with it 4 papers
  • The `defect central charge' was computed in [ref] using the standard formula for the central charge $c_4d$ of standalone $4d$ $=2$ SCFTs at large $N$ holographically dual to AdS$_5$ solutions in M-theory [ref] [equation] which applies to $11d$ metrics of the form [equation] This formula had been used to find the holographic central charge dual to electrostatic solutions with compact internal space engineering irregular punctures [ref] .

    Holographic Weyl Anomalies for 4d Defects in 6d SCFTs · Electrostatic solutions · 0.93

  • Extremizing [ref] we get ^(3)= ϵ_3^3 ν_M5^3(-5+3 √(3)) N^3 { 1/9 , 1/3, 2 } ,for $ℙ^2$, $ℙ^1×ℙ^1$ and dP$_3$, respectively, which agrees with (2.16) in [ref] for $ϵ_3 ν_M5 =3$.[$N_C_N$ in [ref] can be identified with $M_d+1=-∑M_a$, so that $N_there=-h N_C_N/M = h ∑M_a/M$ where $(h,M)$ are defined in [ref] and they have value $(3,9),(4,8)$ and $(2,6)$ for $^2$, $^1×^1$ and dP$_3$, respectively.]

    Equivariant volume extremization and holography · The extremal function for geometries with $_2$ symmetry · 0.99

  • We write [equation] and find that the central charge can be written as [equation] This precisely coincides with the result computed from the explicit solution in [ref] .

    Equivariant localization for AdS/CFT · B4 S2 Sigmag · 1.00

Cites it as support 2 papers
  • As shown in [ref] this class includes all such theories with a dual IIB or M-theory description, as all these theories admit a consistent truncation to $D=5$ minimal gauged supergravity.

    Magnetic Brane Solutions in AdS · Discussion · 0.96

  • The rationale for using 5d gauged supergravity to study these models is based on the fact that, as shown in [ref] and [ref] , the minimal 5d $𝒩=2$ gauged supergravity theory is a consistent truncation of 11d or type IIB supergravity.

    AdS$_5$ Holography and Higher-Derivative Supergravity · Examples · 0.99

Cites it as background 6 papers
  • The class of warped $AdS_5 ×_w M_6$ solutions we consider in this paper were initially analysed in [ref] and some of the properties of the $𝒩=1$ SCFT duals were examined in [ref] .

    NR $CFT_3$ duals in M-theory · Preliminaries · 0.93

  • A few years ago it was shown that the most general supersymmetric $AdS_5$ solutions of $D=11$ supergravity [ref] (of which Sasaki-Einstein $Y^p,q$ [ref] belong) permits a consistent reduction to minimal $N=2$ $D=5$ gauged supergravity [ref] .

    3D gauged supergravity from wrapped M5-branes with AdS/CMT applications · Introduction · 0.85

  • Recall that any solution of this theory can be uplifted to type IIB supergravity using the general class of $X_5$ [ref] associated with supersymmetric $AdS_5×X_5$ solutions of type IIB supergravity that are dual to $N=1$ SCFTs in $d=4$ [ref] or to $D=11$ using the general class of $N_6$ [ref] associated with supersymmetric $AdS_5×N_6$ solutions of $D=11$ supergravity also dual to $N=1$ SCFTs in $d=4$ [ref] .

    Magnetic and electric AdS solutions in string- and M-theory · $U(1)^3⊂SO(6)$ gauged supergravity · 0.89

Discussion

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