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Matter Coupled F(4) Gauged Supergravity Lagrangian

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Laura Andrianopoli, Riccardo D'Auria, Silvia Vaula`

High Energy Physics - Theory · Submitted 18 Apr 2001 · 15 references

Pith's verdict: The reference action for matter-coupled F(4) supergravity, sound within its stated scope; the one thing worth pressing a referee on is the unshown Fierz verification behind the ρ(2ψ) term.

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arXiv IDhep-th/0104155 (version 2)
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CitationLaura Andrianopoli, Riccardo D'Auria, Silvia Vaula`. "Matter Coupled F(4) Gauged Supergravity Lagrangian." arXiv:hep-th/0104155 (2001). https://pith.science/paper/T3VUC2R5
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The paper

The paper claims

The paper constructs the first explicit Lagrangian for six-dimensional F(4) supergravity coupled to any number of vector multiplets, in both ungauged and gauged form.

The reading

Six-dimensional $N=2$ supergravity with the exceptional $F(4)$ symmetry is the natural holographic partner of a five-dimensional superconformal field theory, but before this paper only the pure theory with no matter was known in closed Lagrangian form. The authors write down the full action for the theory coupled to an arbitrary number $n$ of vector multiplets, whose $4n$ scalars live on the coset $SO(4,n)/[SO(4)times SO(n)]$, first without gauging and then with a compact gauge group $SU(2)times G$ where $G$ is an $n$-dimensional subgroup of $SO(n)$. The route is the superspace Bianchi-identity programme: define curvatures for every field, solve the identities, read off the supersymmetry transformation laws, then assemble a superspace six-form whose restriction to ordinary spacetime gives the Lagrangian. The payoff the authors point to is that the AdS$_6$/CFT$_5$ correspondence can now be pushed past the level of multiplet structure to actual masses and couplings, since the potential, the fermion mass matrices, and the linearised field equations are all in hand.

What carries the argument

The load-bearing device is the free differential algebra extending the $F(4)$ Maurer-Cartan equations to include the two-form $B$ and the singlet vector $A^0$, in which the combination $dA^0 - m B$ appears from the outset. That single combination is what later becomes the massive two-form and the $g=3m$ condition. On top of it, the authors solve the superspace Bianchi identities for the matter-coupled curvatures, and the solution requires a two-gravitino term $rho^{(2psi)}$ in the gravitino curvature of Eq. (A.7) in order for the three-gravitino sector to close. The gauging then enters as fermionic shifts $S_{AB}$, $N_{AB}$, $M^I_{AB}$ proportional to $g$, $g'$ and $m$, built from

Extended reading notes

Core claim

The paper's claim is that the complete $D=6$, $N=2$ $F(4)$ supergravity Lagrangian coupled to $n$ vector multiplets exists and can be written explicitly, up to four-fermion terms. Two features carry the result. First, the gauging: the $F(4)$ algebra forces the $SU(2)$ gauge coupling $g$ and the antisymmetric-tensor mass $m$ to appear together, and the closure of the Maurer-Cartan equations requires $g=3m$; at that ratio the potential has a supersymmetric AdS$_6$ extremum with radius fixed by $m$, and the singlet vector $A^0_mu$ is eaten by the two-form $B_{mu nu}$, which becomes massive. Second, the matter coupling: the $4+n$ vectors are rotated into the $SU(2)$ gravitino frame by the coset representative $L^Lambda{}_Sigma$, producing dressed field strengths $T_{[AB]}$, $T_{(AB)}$, $T_I$, and the scalar potential in Eq. (5.9), rewritten in Eq. (5.21) as a sum of squares of the fermionic shifts plus an $m^2 e^{-6sigma} N_{00}$ term. The authors check the potential by a supersymmetry Ward identity, finding a cancellation that requires the non-trivial relation $D^t_I C^I_t = 0$, and they extract the linearised mass spectrum, which reproduces the values obtained independently from conformal dimensions of the five-dimensional singleton multiplets.

Load-bearing premise

The consistency of the transformation laws rests on the claim that once the three-gravitino sector of the Bianchi identities closes, all lower sectors close automatically on shell, while the three-fermion terms of the form $psichi chi$, $psilambda lambda$ and $psilambda chi$ are explicitly left uncomputed.

Editorial extensions

If this is right

  • The complete Kaluza-Klein spectrum of the warped $AdS_6$ reduction can in principle be computed, including the flavour quantum numbers under $G$ that the earlier multiplet-level analysis could not fix.
  • The relation $g = 3m$ between the gauge coupling and the AdS radius is derived from the algebra rather than imposed, so the AdS$_6$ vacuum radius is fixed by the gauging.
  • The two-form $B_{mu nu}$ acquires mass $m$ through the Higgs mechanism, with $A^0_mu$ supplying its longitudinal degree of freedom; the Lagrangian can be rewritten with no reference to $A^0_mu$ at all.
  • The gradient flow equations of Eqs. (5.26)-(5.32), relating derivatives of the fermionic shifts to the spin-one-half mass matrices, give a starting point for studying renormalisation group flows in the dual five-dimensional theory.
  • The linearised masses $m^2_sigma = -6$, $m^2_{q^{I0}} = -4$, $m^2_{q^{Ir}} = -6$, $m_psi = 2$, $m_lambda = m_chi = 1$, $m^2_B = 2$ in units of $R_{AdS}^{-2}$ match the values read off from the conformal dimensions listed in Table 1.

Reading between the lines

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

  • Because the gauged group must be $SU(2)times G$ with $dim G = n$ and $G subset SO(n)$, the construction only covers cosets of a special size; a reader wanting to describe a dual flavour group of different dimension would need a different truncation or a different scalar manifold.
  • The potential written as squares of the shifts plus a negative $m^2$ term has the shape of a superpotential-derived potential, and the gradient flow equations reinforce that reading; this suggests the stability of the $g=3m$ vacuum could be argued structurally rather than by expanding to quadratic order.
  • The omitted three-fermion terms of the form $psichi chi$, $psilambda lambda$, $psilambda chi$ would matter for any use of this action beyond the classical bosonic spectrum, for instance in counting BPS states or in loop computations, so the
  • The matching of the mass spectrum to the singleton-derived conformal dimensions is a consistency check rather than a derivation within this paper; an independent Kaluza-Klein computation on the warped background would turn it into a genuine test.
Original abstract

We construct the so far unknown Lagrangian of D=6, N=2 F(4) Supergravity coupled to an arbitrary number of vector multiplets whose scalars span the coset manifold SO(4,n)/{SO(4) x SO(n)}. This is done first in the ungauged case and then extended to the compact gauging of SU(2) x G, where SU(2) is the R-symmetry diagonal subgroup of SU(2)_L x SU(2)_R = SO(4) and G is a compact subgroup of SO(n), n being the number of vector multiplets, and such that dim G = n. The knowledge of the Lagrangian allows in principle to refine the AdS_6/CFT_5 correspondence already discussed, as far as supersymmetric multiplets are concerned, in a previous related paper. With respect to the latter we also give a more exaustive treatment of the construction of the theory at the level of superspace Bianchi identities and in particular of the scalar potential.

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, 3,924 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.61 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.86 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct the so far unknown Lagrangian of D=6, N=2 F(4) Supergravity coupled to an arbitrary number of vector multiplets whose scalars span the coset manifold SO(4,n)/{SO(4) x SO(n)}. This is done first in the ungauged case and then extended to the compact gauging of SU(2) x G, where SU(2) is the R-symmetry diagonal subgroup of SU(2)_L x SU(2)_R = SO(4) and G is a compact subgroup of SO(n), n being the number of vector multiplets, and such that dim G = n. The knowledge of the Lagrangian allows in principle to refine the AdS_6/CFT_5 correspondence already discussed, as far as supersymmetric multiplets are concerned, in a previous related paper. With respect to the latter we also give a more exaustive treatment of the construction of the theory at the level of superspace Bianchi identities and in particular of the scalar potential.

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 construct the so far unknown Lagrangian of D=6, N=2 F(4) Supergravity coupled to an arbitrary number of vector multiplets whose scalars span the coset manifold SO(4,n)/{SO(4) x SO(n)}. contribution 0.97 0.96 0.78 asserted 1.00
s1 This is done first in the ungauged case and then extended to the compact gauging of SU(2) x G, where SU(2) is the R-symmetry diagonal subgroup of SU(2)_L x SU(2)_R = SO(4) and G is a compact subgroup of SO(n), n being the number of vector multiplets, and such that dim G = n. method 0.90 0.94 0.18 asserted 1.00
s2 The knowledge of the Lagrangian allows in principle to refine the AdS_6/CFT_5 correspondence already discussed, as far as supersymmetric multiplets are concerned, in a previous related paper. interpretation 0.91 0.22 0.12 hedged possibility 0.97
s3 With respect to the latter we also give a more exaustive treatment of the construction of the theory at the level of superspace Bianchi identities and in particular of the scalar potential. contribution 0.95 0.91 0.09 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

46 later papers cite it

Evidence: substantial. 45 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 119 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 1 of 2 papers with this label

  • Studied the consistent truncation of matter coupled F(4) gauged supergravity from this paper on a Riemann surface to construct black hole solutions asymptotic to the AdS_4 vacuum.

    Spindle black holes and theories of class $\mathcal{F}$ (2024) · 0.83 · paraphrased: the licence does not allow quoting

How the literature uses it

40 arXiv papers cite this in 112 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 · 72 sentences without a confident label

Builds on it 2 papers
Uses its method or tool 2 papers
Uses its data or values 1 paper
Compares with it 3 papers
Cites it as background 26 papers

Discussion

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