Maximal gauged supergravity in three dimensions

H. Nicolai; H. Samtleben

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Maximal gauged supergravity in three dimensions

H. Nicolai, H. Samtleben

High Energy Physics - Theory · Submitted 11 Oct 2000 · 19 references

Pith's verdict: A genuinely new class of maximal D=3 gauged supergravities, with the group theory checkable but the supersymmetry proof deferred to an unidentified companion paper.

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arXiv IDhep-th/0010076 (version 2)

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CitationH. Nicolai, H. Samtleben. "Maximal gauged supergravity in three dimensions." arXiv:hep-th/0010076 (2000). https://pith.science/paper/A52KXDRM

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Editor's note

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Short version: the construction is real and the group theory is checkable; the supersymmetry proof is not in the paper. What's new: they keep the 128 scalars and their dual vectors simultaneously, couple the vectors through a Chern-Simons term (10), and get a nonabelian version of the D=3 scalar-vector duality. That is the trick that gets around the usual objection that maximal D=3 supergravity has no vectors to gauge. Everything then reduces to one linear algebraic condition on the embedding tensor, Theta_27000 = 0, which is a clean statement you can check with E8 representation theory. The gauge groups they list -- SO(8)xSO(8), E7(-5)xSU(2), E6(-14)xSU(3), F4(-20)xG2, and E8(8) itself -- come out of that condition rather than being put in by hand. The coupling ratios are outputs, not inputs. That is the right shape for this kind of paper. Where it is soft: the actual verification that the A-tensors (23) solve the identities (14)-(16) is a lengthy calculation deferred to a companion paper with no preprint number. The identity list is explicitly partial (they say they list only the most important ones), yet (22) is claimed to be the complete set of consistency equations. That is the step that turns a necessary condition into a classification, and it is asserted. I would want to see the sufficiency check before treating the group list as final. One correction to a common worry: the O(g) ansatz (11)-(12) is not obviously incomplete. The available SO(16)-covariant fermion bilinears are essentially fixed by index structure, so extra tensor structures are not the issue. The issue is the unshown computation. Also, the abstract says a new class of AdS supergravities, but the body shows SO(8)xSO(8) has no N=16 invariant AdS3 vacuum -- the maximally symmetric point is the (8,8) CS vacuum of [8]. The authors are upfront about this in the text; the abstract just runs ahead of it. Same with not derivable from any known higher-dimensional theory: that is a plausibility argument (no 8-manifold with these isometries, CS rather than Yang-Mills), not a theorem. Who should read it: anyone working on AdS3/CFT2, on gauged supergravities, or on E8 structure in D=3. Worth a serious referee, mainly to get the deferred computation on record.

Referee report

Significance

If correct, this is the first construction of maximal gauged supergravity in three dimensions, and the first class of maximal AdS supergravities not obtainable by conventional Kaluza-Klein reduction of D=11 or D=10 supergravity. The nonabelian scalar-vector duality induced by the Chern-Simons term is a genuinely three-dimensional mechanism with no higher-dimensional analogue, and the embedding-tensor condition (22) is a clean group-theoretic classification tool. The paper deserves credit for making the construction explicit enough to be checkable: the coupling ratios in (24)-(26) are outputs of (22) rather than inputs, and the observation that no SO(9) subgroup of SO(16) satisfies (22), in agreement with the absence of an S^8 compactification of D=11 supergravity, is a useful internal consistency check of the kind that makes the framework falsifiable. The main caveat is that this is a letter: the detailed verification of the identities (14)-(16) and of the necessity of (22) is deferred to a companion paper.

Major objections (3)

[Abstract and pp. 8-9 (AdS ground states)] The abstract advertises 'an entirely new class of AdS supergravities', but the paper's own analysis shows that the distinguished SO(8)×SO(8) gauging cannot have an N=16 supersymmetric AdS3 ground state, because A1^{IJ} ∝ δ^{IJ} requires a nonvanishing trace θ, which vanishes for (25). The noncompact gaugings in (26) do have θ ≠ 0 and are therefore the only candidates for N=16 AdS3 vacua, but their potentials are not analyzed and no such vacuum is exhibited. The letter should either exhibit an N=16 stationary point for at least one of the gaugings in (26), or state explicitly that the maximally supersymmetric AdS3 ground states of the new models remain to be constructed. As written, the abstract's framing outruns what is shown.
[Eqs. (14)-(16), (22)-(23)] The paper states of the identities (14)-(16) that it lists 'only the most important ones', and then asserts that (22) 'encodes the complete set of consistency equations'. The gap between these two statements is exactly what a reader needs bridged: the sufficiency check that the explicit tensors (23) solve all components of the differential identities (15) and the quadratic identities (16), and the necessity argument that a nonvanishing Θ_27000 cannot be compensated by additional A-tensor structures, are both deferred to 'a detailed account ... presented elsewhere'. I agree with the authors that the O(g) ansatz (11)-(12) is exhaustive by SO(16) representation theory, so the concern is not missing tensor structures; it is that the classification (24)-(26) rests on a verification not reproducible from the letter. The authors should state the publication status of the companion and identify which of (15) and (16) are the nontrivial checks, so that the reader can distinguish a verified classification from a conjectured one.
[Abstract and p. 3] The claim that these models 'are not derivable from any known higher-dimensional theory' is supported by two arguments: that conventional Kaluza-Klein reduction would give Yang-Mills rather than Chern-Simons vectors, and that there is no 8-manifold (or 7-manifold) whose isometry group matches any of the gauge groups G0. Both arguments rule out standard Kaluza-Klein compactification, but neither excludes more general higher-dimensional origins such as non-geometric, brane-based, or Scherk-Schwarz-like constructions. The claim should be scoped accordingly, for example to 'not obtainable by conventional Kaluza-Klein compactification of D=11 or D=10 supergravity', or the additional assumptions needed to reach the stronger conclusion should be stated.

Minor objections (7)

[Eq. (25)] The embedding tensor Θ_{IJ,KL} = δ_{I[K} ε_{L]J} should be written with explicit antisymmetrization in the SO(16) generator indices I, J, since the displayed expression is manifestly antisymmetric in K, L but not in I, J. A short parenthetical clarifying the convention would remove the ambiguity.
[Eq. (26)] The relative coupling ratios g_SU(2)/g_E7 = -3, g_SU(3)/g_E6 = -2, and g_G2/g_F4 = -3/2 are ambiguous up to an overall sign unless the sign convention for the Chern-Simons levels is stated. The paper should specify which factor's level is taken positive, particularly since the negative ratios are one of the notable features of the construction.
[Abstract vs. Eq. (26)] The abstract writes 'F_4 x G_2' while the text specifies the real forms F4(-20) × G2. The abstract should match the text, since the real form is part of the result.
[Reference [11]] Reference [11] contains a doubled period in 'Nucl.. Phys. B573, 275 (2000)'.
[p. 8] The assertion that 'there are no 8-manifolds (or 7-manifolds) whose isometry groups would coincide with any of the possible gauge groups G0' is stated without citation. Either a reference to the relevant classification should be supplied or the statement should be softened to 'we are not aware of'.
[Eq. (4)] The supercovariant duality relation (4) mixes ψψ, ψχ, and χχ terms with explicit γ-matrix and ǫ-tensor conventions. A sentence fixing the sign conventions for the ǫ-tensor and the chirality of the spinors would help the reader reproduce (4) from the ungauged results of [5].
[p. 9] The relation between the truncation of SO(8)×SO(8) to its diagonal subgroup and the D=4 reduction of [9] is described as 'presumably related'. This is appropriately hedged, but it would be useful to state explicitly that this is an open question rather than a result of the letter.

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The paper

The paper claims

The authors construct the first maximally supersymmetric gauged supergravity in three dimensions, arguing that its gauge groups and its scalar–vector duality cannot arise from any known higher-dimensional theory.

The reading

Three-dimensional supergravity with the maximal amount of supersymmetry ($N=16$) has been known since the 1980s as a torus reduction of eleven-dimensional supergravity, and its 128 scalar fields live on the coset $E_{8(8)}/SO(16)$. Gauging such a theory — turning a subgroup of $E_{8(8)}$ into a local symmetry, which is what produces an anti-de Sitter ground state and a candidate boundary conformal theory — looked blocked, because in three dimensions vectors are dual to scalars and the ungauged theory had traded all its vectors away. The authors show the block is only apparent: adding a Chern–Simons term of order $g$ for the dual vector fields keeps them non-propagating while making the scalar–vector duality non-abelian, so both the scalars and their duals are needed and neither can be eliminated. They then carry out the standard gauging program — modified currents, fermion bilinears, shifted supersymmetry variations, a scalar potential — and reduce the entire consistency requirement to one linear algebraic condition on the embedding tensor of the gauge group. Solving it yields $SO(8)times SO(8)$ as the maximal compact option and several non-compact ones, none of which can come from Kaluza–Klein reduction of eleven- or ten-dimensional supergravity.

What carries the argument

The load-bearing device is the embedding tensor $Theta_{AB}$, which specifies how the gauge generators sit inside $E_{8(8)}$ and hence how many vector fields there are, combined with the $T$-tensor $T_{A|B} = V^C{}_A V^D{}_B Theta_{CD}$ built from the coset representative $V$. Because $V^{-1} t_A V = V_A{}^B t_B$, this $T$-tensor is quadratic in $V$ rather than cubic as in the four- and five-dimensional analogues, and the three Yukawa tensors $A_1, A_2, A_3$ — which sit in $135 oplus 1820 oplus 1920 = 3875$ of $SO(16)$, plus possible trace singlets — are read off from it by equation (23). The other ingredient is the Chern–Simons term $-frac{1}{4} g, epsilon^{mu nu rho} B_mu{}^m (partial_nu B_{rho m} + tfrac{1}{3} g f_{mnp} B_nu{}^n B_rho{}^p)$, which is what supplies the vectors needed for gauging without adding propagating degrees of freedom and which induces the non-abelian version of the three-dimensional scalar–vector duality.

Extended reading notes

Core claim

The paper's central claim is that the full Lagrangian $L = L^{(0)} + L^{(1)}_g + L^{(2)}_g + L^{(3)}_g$ of equation (17) is locally $N=16$ supersymmetric precisely when the embedding tensor $Theta_{AB}$ of the gauge group $G_0 subset E_{8(8)}$ has no piece in the $27000$ representation, i.e. $Theta^{27000}_{AB}=0$. When that holds, the three $SO(16)$ Yukawa tensors $A_1, A_2, A_3$ that the deformed supersymmetry variations and potential require are given explicitly in equation (23) as bilinears in the scalar coset representative $V$ and the $3875$ part of $Theta$, and the differential and quadratic identities (14)–(16) are satisfied. This turns an apparently open-ended search over gauge groups into a group-theoretic projection condition, and the solutions listed are $SO(8)times SO(8)$ (maximal compact, with the two coupling constants in ratio $-1$), $E_{7(-5)}times SU(2)$, $E_{6(-14)}times SU(3)$, $F_{4(-20)}times G_2$, and the extremal case $G_0 = E_{8(8)}$ with $Theta_{AB} = theta, eta_{AB}$. The authors stress that these are continuous deformations of a theory that does come from eleven dimensions, yet are not themselves obtainable from it.

Load-bearing premise

The construction assumes that the order-$g$ deformation is exhausted by the three $SO(16)$ tensors $A_1, A_2, A_3$ with the index structure of equations (11)–(12), so that the identities (14)–(16) are the complete set required for local supersymmetry and $Theta^{27000}=0$ is both necessary and sufficient; the paper lists only what it calls the most important of these identities before asserting that equation (22) encodes the complete set of consistency equations.

Editorial extensions

If this is right

A new class of three-dimensional anti-de Sitter supergravities exists whose gauge groups — $SO(8)times SO(8)$, $E_{7(-5)}times SU(2)$, $E_{6(-14)}times SU(3)$, $F_{4(-20)}times G_2$, $E_{8(8)}$ — cannot be realized as isometry groups of any eight- or seven-manifold used in Kaluza–Klein reduction.
The duality between a scalar and a vector in three dimensions becomes non-abelian: the vector fields appear explicitly on the right-hand side of the duality relation, so the vectors can no longer be integrated out to give a local theory in the 128 physical scalars alone.
The $SO(8)times SO(8)$ theory cannot have a fully supersymmetric $AdS_3$ ground state, because that would require $A_1^{IJ} propto delta^{IJ}$, i.e. a non-vanishing trace $theta$ of the embedding tensor; its maximally symmetric vacuum instead reduces to the $(8,8)$ Chern–Simons theory already known.
Dimensional reduction of these models gives maximal gauged supergravities in two dimensions, where the three-dimensional vectors generate both two-dimensional vectors and the dual scalars at the bottom of the tower associated with the $E_{9(9)}$ symmetry.
The construction suggests that the space of supergravity theories is larger than what eleven-dimensional supergravity and its compactifications can generate.

Reading between the lines

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

If the $SO(8)times SO(8)$ model is dual to maximally supersymmetric Yang–Mills on the two-dimensional boundary, the boundary theory is matrix string theory, so the correspondence would connect a supergravity with no higher-dimensional origin to a candidate microscopic definition of M theory — a test that runs in the opposite direction from usual compactification logic.
The pattern suggests a systematic classification program: enumerate all solutions of $Theta^{27000}=0$ inside $E_{8(8)}$, including non-maximal and non-semisimple subgroups, and check which admit supersymmetric or partially supersymmetric vacua; the paper explicitly declines a complete classification.
Because the potential's stationary points are not analyzed here, the natural next computation is a critical-point search along the lines used for the four-dimensional $N=8$ theory, which would tell us whether any of the non-compact gaugings have stable $AdS_3$ vacua with reduced supersymmetry.
The claim that no eight-manifold has an isometry group matching these $G_0$ could be turned into a sharper no-go statement by checking the maximal compact subgroups $SO(p)times SO(16-p)$ systematically rather than case by case.

Original abstract

We construct maximally supersymmetric gauged N=16 supergravity in three dimensions, thereby obtaining an entirely new class of AdS supergravities. These models are not derivable from any known higher-dimensional theory, indicating the existence of a new type of supergravity beyond D=11. They are expected to be of special importance also for the conjectured AdS/CFT correspondence. One of their noteworthy features is a nonabelian generalization of the duality between scalar and vector fields in three dimensions. Among the possible gauge groups, SO(8)xSO(8) is distinguished as the maximal compact gauge group, but there are also more exotic possibilities such as F_4 x G_2.

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, 4,271 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.61 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.21 · compares with prior results 0.73 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct maximally supersymmetric gauged N=16 supergravity in three dimensions, thereby obtaining an entirely new class of AdS supergravities. These models are not derivable from any known higher-dimensional theory, indicating the existence of a new type of supergravity beyond D=11. They are expected to be of special importance also for the conjectured AdS/CFT correspondence. One of their noteworthy features is a nonabelian generalization of the duality between scalar and vector fields in three dimensions. Among the possible gauge groups, SO(8)xSO(8) is distinguished as the maximal compact gauge group, but there are also more exotic possibilities such as F_4 x G_2.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	We construct maximally supersymmetric gauged N=16 supergravity in three dimensions, thereby obtaining an entirely new class of AdS supergravities.	contribution 0.99	0.96	0.86	asserted 1.00
s1	These models are not derivable from any known higher-dimensional theory, indicating the existence of a new type of supergravity beyond D=11.	interpretation 0.84	0.66	0.25	hedged possibility 0.58
s2	They are expected to be of special importance also for the conjectured AdS/CFT correspondence.	interpretation 1.00	0.08	0.06	hedged possibility 0.94
s3	One of their noteworthy features is a nonabelian generalization of the duality between scalar and vector fields in three dimensions.	result 0.87	0.70	0.05	asserted 0.98
s4	Among the possible gauge groups, SO(8)xSO(8) is distinguished as the maximal compact gauge group, but there are also more exotic possibilities such as F_4 x G_2.	result 0.95	0.69	0.06	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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

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S sections
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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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How the literature uses it

69 arXiv papers cite this in 133 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 · 80 sentences without a confident label

Uses its method or tool 11 papers

Using the embedding tensor formalism [ref] , we will find the most general gauging of a group $G_0 ⊂SO(5,5)$ by employing a suitable combination of the $16$ vector fields in the theory.

The Gaugings of Maximal D=6 Supergravity · Gauging $G_0 ⊂SO(5,5)$ · 0.96
In the following, we will employ a covariant formalism in which the possible gaugings will be encoded into an object called embedding tensor, that can be characterized group-theoretically [ref] .

Introductory Lectures on Extended Supergravities and Gaugings · Gauging Supergravities · 0.83
The derivation of $ℒ_4d$ can be carried out within the framework of the embedding tensor [ref] and we will present it here.

On new maximal supergravity and its BPS domain-walls · Motivation and outlook · 0.83

Compares with it 4 papers

The tensors $A_1$ and $A_2$ are given in terms of the $Θ$'s and of $θ$: [equation] which, using [ref]–[ref] and setting $θ= 4$, are written as in [ref].] [equation] one gets [equation] which is the potential found in [ref] .

Teleparallel formulation of $E_{8(8)}$ Exceptional Field Theory · Comparison with the supergravity potential · 0.92
Precisely the same issue was encountered in the construction of gauged supergravity in three dimensions [ref] .

Multiple Membranes from Gauged Supergravity · Introduction · 0.84
Our results are in complete agreement with the classification of gauged supergravities in nine dimensions [ref] and in any dimension from seven to three [ref] that has been found using supersymmetry.

The E_{11} origin of all maximal supergravities · Conclusions · 0.98

Cites it as background 24 papers

The bosonic Lagrangian of three-dimensional maximal gauged supergravity is [ref] [equation] with $= (_μ^α)$ and the Chern-Simons term [equation] Since there is no kinetic term for them, the gauge fields $_μ^$ do not contain propagating degrees of freedom.

E10 and Gauged Maximal Supergravity · The Lagrangian · 0.80
In Section 2 we first introduce the maximal gauged supergravity theory in three dimensions, following [ref] .

A Note on E11 and Three-dimensional Gauged Supergravity · Introduction · 0.84
The maximal (32 supercharges) gauged $D=2+1$ supergravity of Nicolai and Samtleben [ref] so far has been mostly regarded as an exotic curiosity, as to this date there is no known way to embed it into M theory.

From Binary Error Correcting Codes to a Relation Between Maximal D=4 and D=3 Supergravities · Discussion · 0.94

Discussion

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