N=8 matter coupled AdS_3 supergravities

H. Nicolai; H. Samtleben

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N=8 matter coupled AdS_3 supergravities

H. Nicolai, H. Samtleben

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

Pith's verdict: A competent, checkable extension of the embedding-tensor program to N=8 matter-coupled AdS3 supergravity, with a real holographic payoff; the stress-test objection about the L=I vacuum does not land.

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arXiv IDhep-th/0106153 (version 1)

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CitationH. Nicolai, H. Samtleben. "N=8 matter coupled AdS_3 supergravities." arXiv:hep-th/0106153 (2001). https://pith.science/paper/WLPNTXOD

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Quick take on Nicolai–Samtleben, hep-th/0106153. The new content is real. They take the embedding-tensor machinery from their N=16 papers and apply it to the Marcus–Schwarz N=8 theories with SO(8,n)/(SO(8)×SO(n)) scalars: gauged Lagrangian, supersymmetry variations, the projection condition (14)/(15) on the embedding tensor, and explicit solutions — SO(p,4−p)×SO(q,4−q), the twisted SO(4)×SO(4) of (18), and some noncompact exceptional cases. The payoff is the α=0 case: local SO(4)=SO(3)_L×SO(3)_R with a global SO(4), which reproduces the symmetries and massless spectrum of D=6 N=(2,0) on AdS3×S3. That match is why the paper matters, and the field content (20) and the identification of the nonpropagating fields with the (3,1)_S+(1,3)_S multiplets look right. What's solid: the explicit embedding tensors (17)–(18), the projection condition, and the holographic identification. These are checkable and I don't see errors. Soft spots. The completeness step — that (15) encodes the full set of T-tensor identities needed for SUSY — is asserted as 'lengthy but straightforward' rather than shown. The authors acknowledge this; it's a real but modest gap, the kind this subfield routinely defers. The 'obviously solved by' for (18) is a hand-wave, though the algebra checks out. And the claim that the L=I vacuum has isometry D1(2,1;α)_L×D1(2,1;α)_R is asserted rather than derived. On the stress-test note: it computes A1 at L=I as Γ^{1234}+αΓ^{5678} and argues this isn't ∝δ, so only half the SUSY survives. The algebra is right, but the inference isn't. The paper's claim is a one-parameter family of supergroups, so an α-dependent gravitino mass matrix is exactly what you'd expect; requiring A1∝δ would force α=1 and kill the family. Both eigenspaces give valid AdS Killing spinors, so all 8 supercharges survive for α≠−1. The stress test is applying the wrong criterion. The fair version of the concern is that the isometry claim is asserted, not that it's wrong. Verdict: competent, well-targeted extension, not a paradigm shift. The holographic match is the real content. Worth referee time; the derivation gap is addressable and the checkable pieces hold up.

Referee report

Major objections (2)

[Eq. (18) and the paragraph following it; Eq. (9); Eq. (21)] The claim that the twisted SO(4) embedding (18) yields a maximally supersymmetric ground state at L=I with isometry D1(2,1;α)_L x D1(2,1;α)_R is load-bearing for the holographic match, but it does not appear to follow from the stated formulas. At L=I one has T=Θ, and with θ=0 Eq. (9) gives A1^{AB} = -(1/48)Γ^{IJKL AB}Θ_{IJ,KL}. For Θ = ε^{ijkl}+α ε^{i'j'k'l'} this is proportional to Γ^{1234}+αΓ^{5678}, which is not proportional to δ^{AB} for general α, including α=0. Maximal supersymmetry of the AdS3 vacuum requires A1^{AB}|_{L=I} ∝ δ^{AB} in the gravitino variation δψ^A_μ = D_μ ε^A + ig A1^{AB}γ_μ ε^B. As written, the L=I background appears to preserve only the chirality-projected subset of supersymmetries, which would contradict the claimed N=(4,4) isometry (21) and weaken the identification with the six-dimensional theory. The authors should either exhibit the explicit Killing spinors and the residual supersymmetry at L=I, or correct the isometry/supersymmetry claim. This is the sharpest checkable point in the paper and should be resolved before publication.
[Eq. (14)-(15) and the paragraph 'Group theory and T-identities'] The equivalence between local supersymmetry of the Lagrangian (8) and the projection condition (14)/(15) is the central structural result, but the completeness direction is deferred to a sentence stating that 'it may be verified by lengthy but straightforward computation that (15) ... indeed encodes the full set of identities which are required for supersymmetry.' The derivation shows that the nonvanishing components of T are expressible via A1,2,3 and that the surviving pieces sit in the last term of (11); it does not show that no further independent identities are required. Since the existence claim for all the gaugings rests on this equivalence, the authors should at least outline the argument that the listed identities are complete (e.g. by counting the independent components of the supersymmetry variation of (8) against the components of (15)), or state explicitly which identities are verified and which are assumed.

Minor objections (5)

[Eq. (25) and the paragraph on C4(4)xSL(2,R)] The coupling ratio g1/g2 = -1/2 is quoted from formula (25) of [2]. Since this is a published result the citation is adequate, but the reader would benefit from a one-line statement of why the same dimension-counting formula applies to the N=8 case, given that the H0 embedding here differs from the maximal theory.
[Eq. (20) and surrounding text] The decomposition of the physical fields into n copies of (2,2) plus the stated multiplicities of (1,1), (2,1), (1,2) is asserted rather than derived. A short accounting of how the 8n bosonic and 8n fermionic degrees of freedom split under SO(3)_L x SO(3)_R would make the spectrum match with [9,10] verifiable at a glance.
[Eq. (1) and the 'Admissible gauge groups' section] The paper states that the subgroups (1) solve (14) but does not show the embedding tensor for the noncompact SO(p,4-p) factors explicitly; only the compact case (17)-(18) is written out. Given that the noncompact gaugings are among the paper's advertised results, at least one explicit noncompact embedding tensor would strengthen the presentation.
[Abstract and introduction] The abstract's claim to 'reproduce the symmetries and massless spectrum' of the six-dimensional theory is stronger than what is demonstrated in the body, where the spectrum matching is argued at the level of representation content (20) and the isometry identification depends on the unresolved vacuum-supersymmetry point above. The abstract should be softened to match the level of verification actually provided.
[References] The paper relies on [4] for notations and on [1,2] for the gauging method; a brief restatement of the conventions for the SO(8,n) generators and the T-tensor would make the paper more self-contained for readers not already familiar with the maximal case.

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The paper

The paper claims

Three-dimensional N=8 supergravity can be gauged, and a single tensor condition decides which gauge groups survive.

The reading

This paper asks which gauge groups can be switched on in three-dimensional $N=8$ supergravity coupled to $n$ matter multiplets, whose scalars live on the coset $SO(8,n)/(SO(8)×SO(n))$. The authors deform the ungauged Marcus–Schwarz theory by the Noether procedure, adding a Chern–Simons term for the gauge fields instead of a Yang–Mills term, because in three dimensions vectors and scalars are dual and the vectors carry no propagating degrees of freedom. The question of whether local supersymmetry survives is then claimed to collapse into one equation: a projection condition on the embedding tensor that records how the gauge group sits inside $SO(8,n)$. Solving it produces families of compact and noncompact gauge groups, and the special case with local $SO(4)=SO(3)_L×SO(3)_R$ is argued to reproduce the symmetries and massless spectrum of $D=6$, $N=(2,0)$ supergravity compactified on $AdS_3×S^3$. If the claim holds, half-maximal AdS$_3$ supergravities become a classification problem settled by a linear-algebra test rather than a case-by-case search.

What carries the argument

The load-bearing object is the embedding tensor $Θ_{MN}$, a $G_0$-invariant symmetric tensor in the product of two adjoint representations of $SO(8,n)$ that encodes how the gauge group $G_0$ sits inside the global symmetry $G$. It enters the covariantized currents $L^{-1}(∂_μ+gΘ^{MN}B_{μM}t_N)L$, the Chern–Simons term, and through the rotated $T$-tensor $T_{A|B}=Θ^{MN}V_{MA}V_{NB}$ all of the Yukawa tensors $A_1,A_2,A_3$ and the potential $W$. The machinery that carries the argument is the decomposition of $Θ$ into irreducible $SO(8,n)$ pieces, matched against the $SO(8)×SO(n)$ content of the Yukawa tensors, which shows that every component of $T$ except the singlet and the last irreducible piece must vanish. That yields the projection condition $Θ_{AB}=θη_{AB}+P_{AB}{}^{CD}Θ_{CD}$, which is then solved group by group.

Extended reading notes

Core claim

The central claim is that the gauged $D=3$, $N=8$ theories with scalar manifold $SO(8,n)/(SO(8)×SO(n))$ exist, and that their existence is governed by a single $SO(8,n)$-covariant constraint. The authors build the Lagrangian by covariantizing the currents of the ungauged $\sigma$ model with an embedding tensor $Θ_{MN}$ and adding a Chern–Simons term of order $g$; the Yukawa couplings $A_1,A_2,A_3$ and the scalar potential $W$ are then written as linear and quadratic functions of the associated $T$-tensor $T_{A|B}=Θ^{MN}V_{MA}V_{NB}$. Requiring the supersymmetry variations to leave the Lagrangian invariant is claimed to be equivalent to the statement that $T$ (equivalently $Θ$) lies in one particular irreducible representation of $SO(8,n)$, written $Θ_{AB}=θη_{AB}+P_{AB}{}^{CD}Θ_{CD}$. The solutions exhibited include $SO(p,4-p)×SO(q,4-q)$ with $p,q∈{0,2}$ and independent coupling constants, compact $SO(4)×SO(4)$ with a relative twist of $-1$ between the two $SO(3)$ factors inside each $SO(4)$, and noncompact exceptional cases such as $C_{4(4)}×SL(2,R)$ and $Sp(m,2)×SO(3)$ at fixed ratios of coupling constants. The special case $α=0$, giving local $SO(4)=SO(3)_L×SO(3)_R$ with a surviving global $SO(4)$, is argued to reproduce the field content and symmetries of $D=6$, $N=(2,0)$ supergravity with $n$ tensor multiplets on $AdS_3×S^3$.

Load-bearing premise

The load-bearing premise is that the identities the $T$-tensor is required to satisfy, which the authors say can be checked by a lengthy but straightforward computation, are complete, so that the projection condition on the embedding tensor really is equivalent to invariance of the Lagrangian under the stated supersymmetry variations.

Editorial extensions

If this is right

Every gauge group compatible with local $N=8$ supersymmetry in three dimensions can be found by solving one linear projection condition on the embedding tensor, rather than by checking the Lagrangian case by case.
The compact gauging $SO(4)=SO(3)_L×SO(3)_R$ gives a three-dimensional theory whose spectrum matches the massless sector of $D=6$, $N=(2,0)$ supergravity on $AdS_3×S^3$, so the six-dimensional compactification has a purely three-dimensional description.
The nonpropagating fields of that theory combine into $(3,1)_S+(1,3)_S$ short multiplets of $SU(2|1,1)_L×SU(2|1,1)_R$, while the matter falls into $n$ copies of the $(2,2)_S$ multiplet, fixing the multiplet structure of the dual two-dimensional conformal theory.
Turning off the second $SO(4)$ coupling leaves a topological Chern–Simons supergravity, connecting the construction to earlier three-dimensional Chern–Simons supergravity actions.
Noncompact gaugings such as $C_{4(4)}×SL(2,R)$ and $Sp(m,2)×SO(3)$ require fixed ratios of coupling constants, so the projection condition also predicts relations among coupling constants that a generic group-theoretic ansatz would not.
The same technique is expected to extend downward to theories with less than maximal supersymmetry, since the pattern relies only on the coset structure and the embedding tensor.

Reading between the lines

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

The projection condition functions as a filter on group theory rather than on dynamics: it says that among all subgroups of $SO(8,n)$, only those whose embedding tensor lands in one irreducible representation can be gauged. That suggests the classification of three-dimensional gauged supergravities is finite in a strong sense, and one could try to enumerate all solutions of the projection conditio
The failure of $SO(3,1)≅SL(2,C)$ to be gauged, because the epsilon tensor would need a factor of $i$ and hence an imaginary coupling, is a signature that the criterion is sensitive to the real form of the gauge group and not just its complexification. This hints that the space of allowed real forms is itself constrained by a reality condition on the embedding tensor, which would be worth stating a
The absence of a higher-dimensional origin for the exceptional noncompact gaugings such as $C_{4(4)}×SL(2,R)$ is the most conspicuous loose end. If these theories are to be more than algebraic curiosities, one would look for them as consistent truncations of ten- or eleven-dimensional supergravity on backgrounds with fewer isometries than $S^3×S^3$, or as limits of brane configurations in the spir
The matching with the six-dimensional spectrum is a check on representation content, not on dynamics. A sharper test would be to compare correlation functions or the chiral ring of the dual two-dimensional theory against quantities computed from the three-dimensional Lagrangian, which would probe the interacting part of the correspondence rather than the multiplet count.

Original abstract

Following the recent construction of maximal (N=16) gauged supergravity in three dimensions, we derive gauged D=3, N=8 supergravities in three dimensions as deformations of the corresponding ungauged theories with scalar manifolds SO(8,n)/(SO(8)x SO(n)). As a special case, we recover the N=(4,4) theories with local SO(4) = SO(3)_L x SO(3)_R, which reproduce the symmetries and massless spectrum of D=6, N=(2,0) supergravity compactified on AdS_3 x S^3.

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Following the recent construction of maximal (N=16) gauged supergravity in three dimensions, we derive gauged D=3, N=8 supergravities in three dimensions as deformations of the corresponding ungauged theories with scalar manifolds SO(8,n)/(SO(8)x SO(n)). As a special case, we recover the N=(4,4) theories with local SO(4) = SO(3)_L x SO(3)_R, which reproduce the symmetries and massless spectrum of D=6, N=(2,0) supergravity compactified on AdS_3 x S^3.

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s0	Following the recent construction of maximal (N=16) gauged supergravity in three dimensions, we derive gauged D=3, N=8 supergravities in three dimensions as deformations of the corresponding ungauged theories with scalar manifolds SO(8,n)/(SO(8)x SO(n)).	contribution 0.95	0.94	0.66	asserted 1.00
s1	As a special case, we recover the N=(4,4) theories with local SO(4) = SO(3)_L x SO(3)_R, which reproduce the symmetries and massless spectrum of D=6, N=(2,0) supergravity compactified on AdS_3 x S^3.	result 0.88	0.80	0.19	asserted 1.00

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References

Works cited (15)

[2]

H. Nicolai and H. Samtleben, Compact and noncompact gaug ed maximal supergravities in three dimensions, JHEP 0104 (2001) 022, [hep-th/0103032]. 10

View on Pith Pith review arXiv 2001
[1]

H. Nicolai and H. Samtleben, Maximal gauged supergravit y in three dimensions, Phys. Rev. Lett. 86 (2001) 1686–1689, [ hep-th/0010076]

View on Pith Pith review arXiv 2001
[3]

N. S. Deger, A. Kaya, E. Sezgin, and P. Sundell, Matter cou pled AdS3 supergravities and their black strings, Nucl. Phys. B573 (2 000) 275–290, [hep-th/9908089]

View on Pith Pith review arXiv
[4]

N. Marcus and J. H. Schwarz, Three-dimensional supergra vity theories, Nucl. Phys. B228 (1983) 145–162

View on Pith 1983
[5]

L. J. Romans, Selfduality for interacting ﬁelds: Covari ant ﬁeld equations for six-dimensional chiral supergravities, Nucl. Phys. B276 ( 1986) 71–92

View on Pith 1986
[6]

F. Riccioni, Tensor multiplets in six-dimensional (2, 0) supergravity, Phys. Lett. B422 (1998) 126–134, [ hep-th/9712176]

View on Pith Pith review arXiv 1998
[7]

J. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005, [ hep-th/9804085]

View on Pith Pith review arXiv 1998
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A. Giveon, D. Kutasov, and N. Seiberg, Comments on string theory on AdS3, Adv. Theor. Math. Phys. 2 (1998) 733–780, [ hep-th/9806194]

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S. Deger, A. Kaya, E. Sezgin, and P. Sundell, Spectrum of D = 6,N = 4b supergravity on AdS3×S3, Nucl. Phys. B536 (1998) 110–140, [hep-th/9804166]

View on Pith Pith review arXiv 1998
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J. de Boer, Six-dimensional supergravity on S3×AdS3 and 2d conformal ﬁeld theory, Nucl. Phys. B548 (1999) 139–166, [ hep-th/9806104]

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A. Ach´ ucarro and P. K. Townsend, A Chern-Simons actionfor three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B180 (1986) 89–92

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M. G¨ unaydin, G. Sierra, and P. K. Townsend, The unitarysupermultiplets of d = 3 Anti-de Sitter andd = 2 conformal superalgebras, Nucl. Phys. B274 (1986) 429–447

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H. J. Boonstra, B. Peeters, and K. Skenderis, Brane inte rsections, anti-de Sitter spacetimes and dual superconformal theories, Nucl. Phys. B533 (1998) 127–162, [hep-th/9803231]

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J. de Boer, A. Pasquinucci, and K. Skenderis, AdS/CFT du alities involving large 2dN = 4 superconformal symmetry, Adv. Theor. Math. Phys. 3 (1999) 577–614, [hep-th/9904073]

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[15]

B. de Wit, A. K. Tollst´ en, and H. Nicolai, Locally supersymmetricD = 3 nonlinear sigma models, Nucl. Phys. B392 (1993) 3–38, [hep-th/9208074]. 11

View on Pith Pith review arXiv 1993

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How the literature uses it

33 arXiv papers cite this in 87 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 · 50 sentences without a confident label

Builds on it 1 paper

This paper is a continuation of the work presented [ref] , which considered three-dimensional $=8$ gauged supergravity with $n=4$ vector multiplets, first discussed in [ref] .

Janus and RG interfaces in three-dimensional gauged supergravity II: General $\alpha$ · Introduction · 0.98

Uses its method or tool 6 papers

In this work we used the three-dimensional gauged supergravity models of [ref] to obtain information about superconformal gauge theories in three dimensions for an arbitrary number $N≤8$ of supersymmetries.

The Superconformal Gaugings in Three Dimensions · Discussion · 0.83
Here we are focussing on the supersymmetries in three dimensions, and for this we make extensive use of [ref] .

Supersymmetry and Superstrata in Three Dimensions · Introduction · 0.91
Recasting it in the framework of [ref] enables the application of specific supergravity techniques which, when combined with tools from algebraic geometry [ref] , allow for a systematic classification of AdS$_3$ flux vacua.

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Compares with it 3 papers

Precisely the same issue was encountered in the construction of gauged supergravity in three dimensions [ref] .

Multiple Membranes from Gauged Supergravity · Introduction · 0.84
The corresponding expression in [ref] (equation (2.6)) is “non-canonical" in that the sign of the $A ∧A∧A$ term is reversed relative to our “canonical choice.” Earlier references, like [ref] , have the canonical form of the Cherns-Simons term, ([ref]).

Microstate Geometries from Gauged Supergravity in Three Dimensions · Consistency of gauge field actions, Chern-Simons term and conventions · 0.96
One should note that we have replaced the potential $W$ of [ref] by $-V$ as we wish to avoid confusion with the superpotential that will be defined later.

Supersymmetry and Superstrata in Three Dimensions · The $8$ theory · 0.84

Cites it as background 20 papers

In this section, we review the $=8$ gauged supergravity first constructed in [ref] mainly following the conventions of [ref] .

Janus and RG interfaces in three-dimensional gauged supergravity II: General $\alpha$ · Three-dimensional $=8$ gauged supergravity · 0.95
We start by reviewing $N=8$ gauged supergravity in $D=3$ [ref] .

Multiple Membranes from Gauged Supergravity · The Lagrangian and the embedding tensor · 0.95
A thorough survey of this literature would take us far afield, but in this paper we have drawn heavily upon [ref] .

Supersymmetry and Superstrata in Three Dimensions · Introduction · 0.86

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [2]

H. Nicolai and H. Samtleben, Compact and noncompact gaug ed maximal supergravities in three dimensions, JHEP 0104 (2001) 022, [hep-th/0103032]. 10

View on Pith Pith review arXiv 2001

[2] [1]

H. Nicolai and H. Samtleben, Maximal gauged supergravit y in three dimensions, Phys. Rev. Lett. 86 (2001) 1686–1689, [ hep-th/0010076]

View on Pith Pith review arXiv 2001

[3] [3]

N. S. Deger, A. Kaya, E. Sezgin, and P. Sundell, Matter cou pled AdS3 supergravities and their black strings, Nucl. Phys. B573 (2 000) 275–290, [hep-th/9908089]

View on Pith Pith review arXiv

[4] [4]

N. Marcus and J. H. Schwarz, Three-dimensional supergra vity theories, Nucl. Phys. B228 (1983) 145–162

View on Pith 1983

[5] [5]

L. J. Romans, Selfduality for interacting ﬁelds: Covari ant ﬁeld equations for six-dimensional chiral supergravities, Nucl. Phys. B276 ( 1986) 71–92

View on Pith 1986

[6] [6]

F. Riccioni, Tensor multiplets in six-dimensional (2, 0) supergravity, Phys. Lett. B422 (1998) 126–134, [ hep-th/9712176]

View on Pith Pith review arXiv 1998

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