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.Comment: 10 pages, LaTeX2e, minor changes in text, references added, version
  to appear in Phys. Rev. Let

A. Achúcarro

B. de Wit

B. Julia

C. M. Hull

D. Z. Freedman

E. Cremmer

H. Lu

H. Nicolai

H. Samtleben

K. Koepsell

M. Cvetič

M. Günaydin

N. Marcus

N. S. Deger

O. Aharony

R. Dijkgraaf

English

arXiv

We construct maximally supersymmetric gauged N = 16 supergravity in three dimensions, thereby obtaining an entirely new class of anti-de Sitter supergravities. These models apparently cannot be derived from any known higher-dimensional theory and point to the existence of a new type of super gravity beyond D = 11. One of their noteworthy features is a non-Abelian generalization of the duality between scalar and vector fields in three dimensions. Among the possible gauge groups, SO(8) x SO(8) is distinguished as the maximal compact gauge group, but there are also more exotic possibilities such as F4(-20) x G(2)

Nicolai, H.

Samtleben, H.

MPG.PuRe

Maximal gauged supergravity in three dimensions

VOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 20011686Maximal Gauged Supergravity in Three DimensionsH. NicolaiMax-Planck-Institut für Gravitationsphysik, Mühlenberg 1, D-14476 Golm, GermanyH. SamtlebenLPT-ENS,* 24 Rue Lhomond, F-75231 Paris Cedex 05, France(Received 11 October 2000; revised manuscript received 15 December 2000)We construct maximally supersymmetric gauged N  16 supergravity in three dimensions, therebyobtaining an entirely new class of anti–de Sitter supergravities. These models apparently cannot bederived from any known higher-dimensional theory and point to the existence of a new type of super-gravity beyond D  11. One of their noteworthy features is a non-Abelian generalization of the dualitybetween scalar and vector fields in three dimensions. Among the possible gauge groups, SO8 3 SO8is distinguished as the maximal compact gauge group, but there are also more exotic possibilities suchas F4220 3 G2.DOI: 10.1103/PhysRevLett.86.1686 PACS numbers: 11.10.Kk, 04.50.+h, 04.65.+e, 11.15.–qGauged supergravities have attracted strong interestagain recently in the context of the conjectured dualitybetween anti–de Sitter (AdS) supergravities and super-conformal quantum field theories on the AdS boundary[1]. Since the construction of maximal N  8 gauged su-pergravity in four dimensions [2] various maximal gaugedsupergravities in dimensions D . 4 have been shown toexist [3]. Much less is known, however, about gaugedsupergravities in dimensions D , 4—despite the fact thatthe theories admitting AdS3 ground states are expectedto be of particular interest for the AdS/conformal fieldtheory (AdS/CFT) duality due to the rich and rather wellunderstood structure of two-dimensional superconformalfield theories. A better understanding of the supergravityside is of vital importance for any further test of thisduality.In this Letter, we construct maximally supersymmetricgauged N  16 supergravity in three dimensions. Thisis the most complex and most symmetric of all theoriesof this type known so far (and may remain so in viewof the appearance of the maximal exceptional Lie groupE88 in it). It gives rise to an entirely new class of three-dimensional models, none of which is contained in orderivable by any known mechanism from the known su-pergravity or superstring theories in higher dimensions.This is all the more remarkable in view of the fact thatthese models constitute continuous deformations of a the-ory (ungauged N  16 supergravity [4,5]) that is obtainedby a torus compactification of D  11 supergravity [6].Their existence not only hints at a new type of supergrav-ity beyond D  11 supergravity but also indicates thatthe framework of theories considered so far in attemptsto unify string theories at the nonperturbative level may beinadequate. Likewise, the boundary theory associated withour SO8 3 SO8 gauged supergravity is expected to en-joy an exceptional status among the superconformal fieldtheories. The AdS/CFT correspondence indeed suggests a0031-90070186(9)1686(4)$15.00duality with a sector of maximally supersymmetric Yang-Mills theory on the two-dimensional AdS boundary —but this is just matrix string theory, which itself has beenproposed as a candidate for M theory [7].Topological gauged supergravities in three dimensionswere constructed in [8]; these theories are supersymmet-ric extensions of Chern-Simons (CS) theories with p,qsupersymmetry and gauge groups Op 3 Oq but haveno propagating matter degrees of freedom. Matter coupledgauged supergravities can, of course, be obtained by directdimensional reduction of gauged supergravities in D $ 4to three dimensions and below [9]; see also [10] and [11]for different constructions. However, no such theories pre-serving the maximal number of supersymmetries in threedimensions have been known until now.The scarcity of results is in part explained by the differ-ent status of vector fields in three dimensions in compari-son with their higher dimensional relatives, namely, theiron-shell equivalence with scalar fields via the duality rela-tion emnr≠rwm  ≠mBnm. This relation plays a specialrole in the derivation of maximal N  16 supergravity inthree dimensions [4,5]: in order to expose its rigid E88symmetry, all vector fields obtained by dimensional re-duction of D  11 supergravity [6] on an 8-torus mustbe dualized into scalar fields. Thus an immediate (but,as we will see, only apparent) obstacle towards the gaug-ing of the maximally extended three-dimensional super-gravity is the a priori absence of vector fields that couldbe used for such a gauging. Let us therefore emphasizethe main features by which our construction differs fromprevious ones:(i) The theory makes simultaneous use of both scalarfields and their dual vectors, which appear via a CS term inorder of the gauge coupling constant g and therefore do notcarry additional physical degrees of freedom. Their equa-tions of motion yield a non-Abelian generalization of thestandard duality relation between scalar and vector fields,© 2001 The American Physical SocietyVOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001such that a direct elimination of the latter is no longer pos-sible (and would lead to a nonlocal theory in terms of the128 physical scalar fields of N  16 supergravity).(ii) Several consistent choices are possible for the gaugegroup G0 , E88, although G0  SO8 3 SO8 is dis-tinguished as the maximal compact subgroup of E88 thatcan be made local. The noncompact gauge groups allowedby our construction do not arise as analytic continuations ofa compact subgroup unlike the known noncompact gaug-ings in higher dimensions [12].(iii) Conventional Kaluza-Klein compactification ofD  11 (orD  10) supergravity cannot give rise to thesegauged theories because it would yield Yang-Mills-typetheories rather than non-Abelian CS theories and thereare no 8-manifolds (or 7-manifolds) whose isometrygroups would coincide with any of the possible gaugegroups G0.(iv) Although the rigid E88 symmetry is broken, suchthat the gauged theory is only invariant under local G0 3SO16, the construction makes essential use of propertiesof the exceptional Lie group E88. In particular, the checkof local supersymmetry requires “T identities” based onE88 analogous to the ones derived in [2].The physical fields of ungauged maximal N  16supergravity in three dimensions constitute an irreduciblesupermultiplet with 128 bosons and 128 fermions trans-forming as inequivalent fundamental spinors of SO16(see [5] whose conventions and notation we follow).In addition, the theory contains the dreibein ema and16 gravitino fields cIm. The scalar fields are describedby an element V of the noncompact coset spaceE88SO16 in the fundamental 248-dimensional repre-sentation [4]. The equivalence of the fundamental and theadjoint representations of E88 is expressed by the relationV 21tAV  VABtB . (1)Indices A,B, . . . here label E88 generators tA (see [13,14]for our E88 conventions and some useful formulas). Wesplit the generators tA into 120 compact ones XIJ  and128 noncompact ones YA.N  16 supergravity is invariant under a hidden rigidE88 symmetry as well as a local SO16 symmetry. Thesupersymmetry variations are [5]dema  ie¯IgacIm, V 21dV  GIA Ax¯AeIYA,dcIm  DmeI , dxA i2gmeIGIA APAm.(2)The scalar fields couple to the fermions via the currentsV 21≠mV  12QIJm XIJ 1 PAmYA, (3)where the composite SO16 connection QIJm enters thecovariant derivative Dm in (2) and (9) below.A central role in our construction is played by the su-percovariant duality relationemnrBnrm 1 2V mIJcIncJr 2 2iGIA AV mAcIngrxA  2V mAPmA 2 GIA AcmIxA 2 i2V mIJGIJA BxAgmxB, (4)whose compatibility condition gives rise to the equationsof motion for the scalar fields. At the same time (4)serves to define the dual vector fields Bmm with (Abelian)field strengths Bmnm. Although we are still concernedwith Abelian vector fields at this point, we introduce spe-cial labels m,n, . . . for the generators of the gauge groupG0 , E88 already here. The gauge group is properlydescribed by means of its (symmetric) embedding tensorQAB , such thatBmmtm  BmAQAB tB . (5)We emphasize that the number of vector fields is not yetspecified since it will depend on the choice of the gaugegroup G0. Although the duality (4) holds only on-shell, weshall see that the gauged theory provides a natural off-shellframework which accommodates both the scalars and theirdual vectors.Under local supersymmetry the vectors transform as [14]dBmm  22V mIJeIcIm 1 iGIA AV mAeIgmxA. (6)To gauge the theory, we now assume the existence ofa suitable subgroup G0 , E88 with embedding tensorQAB . The possible choices for G0 are determined be-low by demanding consistency and full supersymmetry ofthe gauged theory. The first step is the covariantization ofderivatives in (3) according toV 21DmV  V 21≠mV 1 gBmmV 21tmV P AmYA 1 12QIJmXIJ , (7)with gauge coupling constant g, where we now use pro-jected indices m, n, . . . for the gauge group. The non-Abelian field strength readsBmnm  ≠mBnm 2 ≠nBmm 1 gfmnpBmnBnp . (8)The modified currents defined in (7) lead to a firstg-dependent modification of the ungauged Lagrangian [5]L 0  2 14eR 1 14eP mAP Am 1 12elmncIlDmc In2i2exAgmDmxA 212exAgmgncImGIA AP An .(9)Next we observe that the non-Abelian extension of theduality relation (4) is obtained from this Lagrangian if weadd the following CS term for the vector fields Bmm:L 1g  2 14gemnrBmm≠nBrm 1 13gfmnpBnnBrp .(10)This is very different from the situation in higher di-mensions where the vector fields appear already in the1687VOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001ungauged theory via an Abelian kinetic term. By contrast,we find here that the CS term (10) induces a non-Abelianversion of the duality relation (4) between scalar and vectorfields; the latter is simply obtained by replacing theAbelian field strengths by (8), and the current PAm by theone defined in (7), with no further modifications. Thisshows how the gauged theory brings in the dual vectorfields, but without introducing new propagating degrees offreedom. However, because of the explicit appearance ofthe gauge fields on the right-hand side of the non-Abelianduality relation it is no longer possible to trade the vectorfields for scalar fields and thereby eliminate them, unlikein [15].The next steps in the construction are standard [2]:the O g terms in the currents necessitate extra bilinearfermionic terms in the LagrangianL 2g  12geAIJ1 cImgmncJn 1 igeAI A2 x AgmcIm112geAA B3 xAxB, (11)as well as the following modifications of the fermionicvariations:dgcIm  igAIJ1 gmeJ , dgxA  gAIA2 eI . (12)Here the SO16 tensors A1,2,3 depend on the scalar fieldsV in a way that remains to be specified and introduceYukawa-type couplings between the scalars and the fer-mions beyond the derivative couplings generated by (3).At O g2, supersymmetry demands that we add the scalarpotentialL 3g  18g2eAIJ1 AIJ1 2 12AI A2 AI A2  . (13)The check of local supersymmetry requires several iden-tities that these tensors must obey. We list here only themost important ones. They areV mAGIA AAJ A2  V mIKAJK1 1 V mJKAIK1 ,GIA AAJ A2  V C IJQCDV DA , (14)dABAII1 1 GIA AAA B3 GIB B 16V CAQCDV DB ,with GIA AAIA2  0. In addition we have the differentialrelationsDmAIJ1  PmAGIA AAJ A2 ,DmAI A2  12PmAGIA BAA B3 1 GJA AAIJ1  (15)212PmAGIB AVAAQABV BB ,and the quadratic identitiesAIK1 AKJ1 212AI A2 AJ A2 116dIJAKL1 AKL1 212AK A2 AK A2 ,3AIJ1 AJ A2 2 AI B2 AA B3 116GIA AGJA B3AJK1 AK B2 (16)2 AJC2 AB C3 .Remarkably, the SO16 representations of the tensorsA1,2,3 combine into a representation of E88 accordingto 135 © 1820 © 1920  3875, with the possibility of1688an extra singlet in the trace parts of AIJ1 and AA B3 . Apartfrom the extra singlet, this is analogous to the situationin D  4 and D  5, where the Yukawa couplings aregiven by tensors transforming in the 912 of E77 [16] andin the 351 of E66 [17], respectively.Modulo higher order fermionic terms we can now es-tablish that the full LagrangianL  L 0 1 L 1g 1 L 2g 1 L 3g (17)is indeed supersymmetric provided the tensors A1,2,3 solveEqs. (14)–(16). In the limit g ! 0 one reobtains the un-gauged theory of Ref. [5]. Furthermore, the N  16 su-peralgebra closes in the usual way, except for an extranon-Abelian G0-gauge transformation with parameterLm  2V mIJeI1eJ2 1 iBmmeI1gmeI2 . (18)The main task is now to ascertain that the relations(14)–(16) do admit nontrivial solutions and to determinethe possible gauge groupsG0. Any subgroupG0 , E88 ischaracterized by its embedding tensor QAB [cf. (5)] whichdecomposes asQAB  uhAB 1 Q3875AB 1 Q27 000AB , (19)with the (indefinite) E88 Cartan-Killing metric hAB , inaccordance with the symmetric E88 tensor product248 3 248sym  1 1 3875 1 27 000 . (20)Guided by [2] we define the T tensor byTAj B  V CAV DBQCD (21)and note that Eqs. (14) define the SO16 tensors A1,2,3 asfunctions of TAj B [unlike the cubic expressions in [2] and[17], the T tensor (21) is quadratic in V due to (1)]. Asthe tensors A1,2,3 combine into the 387511 representa-tion(s) of E88, Eqs. (14)–(16) can only be consistent ifthe 27 000 is absent in (21), which in turn requires QAB tosatisfy the conditionQ27 000AB  0 () QAB  uhAB 1 Q3875AB . (22)If (22) is satisfied, the tensors A1,2,3 which solveEqs. (14)–(16) are given byAIJ1  2udIJ 217V C IKV DKJQ3875CD ,AIA2  217GJA AV C IJV DAQ3875CD , (23)AA B3  2ud A B 1148GIJKLA BV C IJV DKLQ3875CD ,as can be shown by a rather lengthy computation whichrelies in particular on the form of the projectors onto theirreducible parts of its tensor products [13]. Thus, (22)encodes the complete set of consistency equations for thegauged theory, thereby reducing the differential and bi-linear identities for the a priori unknown tensors A1,2,3 toa single linear algebraic equation for the embedding tensorQAB . The formulation of this equation as a projection con-dition allows one to select the admissible gauge subgroupsVOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001of E88 by purely group theoretical arguments. We do notaim for a complete classification of the solutions of (22)here, but rather discuss the most important examples.The maximal compact subgroup of E88 satisfying thecondition (22) isSO16 . G0  SO8 3 SO8 , (24)where the ratio of coupling constants of the two factorsis g1g2  21 such that the trace part u of QAB van-ishes. The embedding tensor for SO8 3 SO8 is ex-plicitly given byQIJ,KL  dIKA´LJ , QIJ,A  0  QA,B ,with ´IJ  diag,2 , (25)with a 16 ! 8 1 8 split of the indices I , J. This tensortransforms in the 135 under SO16 and automatically sat-isfies (22), because there is no 135 in the 27 000 of E88.By contrast, no SO9 subgroup of SO16 satisfies (22), inagreement with the fact that there is no S8 compactificationof D  11 supergravity to three dimensions. Likewise, itmay be shown that apart from (24) none of its maximalcompact subgroups of type SOp 3 SO16 2 p satis-fies (22).Among the maximal subgroups of E88, we have foundthree possible noncompact gauge groupsG0  E725 3 SU2 with gSU2gE7  23 ,G0  E6214 3 SU3 with gSU3gE6  22 , (26)G0  F4220 3 G2 with gG2gF4  232 ,all of which have nonvanishing trace u. A further and“extremal” solution to (22) is G0  E88 with QAB uhAB .An immediate question concerns the possible existenceof AdS3 ground states with additional symmetries. Thefact that d  11 supergravity does not appear to admitcompactifications of this type would seem to indicate thatthe potential (13) does not possess fully supersymmetricstationary points. Inspection of the supersymmetry varia-tions (2) and (12) reveals that a 16, 0 invariant AdS3ground state can only exist if AIJ1 ~ dIJ, i.e., if the trace ofQAB does not vanish. The SO8 3 SO8 gauged theory(24) has a maximally symmetric ground state characterizedby V  248, which by (23) impliesAIJ1  217´IJ , AI A2  0, AA B3  0 , (27)with ´IJ from (25). The Lagrangian (17) then reduces tothe 8, 8 CS theory of [8] and reproduces the vacuumfound there. On the other hand, the truncation of thegauge group SO8 3 SO8 to its diagonal subgroup ispresumably related to the theory obtained by direct reduc-tion of the D  4 gauged theory, which no longer exhibitsan SO8 invariant AdS but rather a domain wall groundstate [9]. Pursuing an analysis of the full theory along thelines of [18] should allow one to understand in detailthe presumed relation to the matrix string theory [7] onthe boundary.Finally, dimensional reduction of the gauged theories(17) yields maximal gauged supergravities in two dimen-sions as well. In this reduction, the vector fields Bmmgive rise not only to two-dimensional vectors, but also tothe dual scalar fields at the bottom of the infinite towerof dual potentials generated by the E99 symmetry ofN  16,D  2 supergravity. It would be most interestingto see whether our construction can be extended to yieldgauged supergravities in two dimensions with infinite-dimensional gauge groups.A detailed account of our results will be presented else-where. This work was supported in part by the EuropeanUnion under Contract No. HPRN-CT-2000-00122.*UMR 8549: Unité Mixte du Centre National de laRecherche Scientifique, et de l’Ecole Normale Supérieure.[1] O. Aharony et al., Phys. Rep. 323, 183 (2000).[2] B. de Wit and H. Nicolai, Nucl. Phys. B208, 323 (1982).[3] Supergravities in Diverse Dimensions, edited by A. Salamand E. Sezgin (World Scientific, Singapore, 1989).[4] B. Julia, in Unified Field Theories in More Than 4 Dimen-sions Including Exact Solutions, edited by V. D. Sabbataand E. Schmutzer (World Scientific, Singapore, 1983),pp. 215–236.[5] N. Marcus and J. Schwarz, Nucl. Phys. B228, 145 (1983).[6] E. Cremmer, B. Julia, and J. Scherk, Phys. Lett. 76B, 409(1978).[7] R. Dijkgraaf, E. Verlinde, and H. Verlinde, Nucl. Phys.B500, 43 (1997).[8] A. Achúcarro and P. K. Townsend, Phys. Lett. B 180, 89(1986).[9] H. Lu, C. N. Pope, and P. K. Townsend, Phys. Lett. B 391,39 (1997).[10] M. Cveticˇ, H. Lu, and C. N. Pope, Phys. Rev. D 62, 064028(2000).[11] N. S. Deger, A. Kaya, E. Sezgin, and P. Sundell, Nucl.Phys. B573, 275 (2000).[12] C. M. Hull, Phys. Lett. 142B, 39 (1984).[13] K. Koepsell, H. Nicolai, and H. Samtleben, J. High EnergyPhys. 4, 023 (1999).[14] K. Koepsell, H. Nicolai, and H. Samtleben, Classical Quan-tum Gravity 17, 3689 (2000).[15] E. Cremmer, B. Julia, H. Lu, and C. N. Pope, Nucl. Phys.B523, 73 (1998).[16] B. de Wit and H. Nicolai, Nucl. Phys. B231, 506 (1984).[17] M. Günaydin, L. J. Romans, and N. P. Warner, Nucl. Phys.B272, 598 (1986).[18] D. Z. 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Maximal gauged supergravity in three dimensions

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