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Chern-Simons Supergravities with Off-Shell Local Superalgebras ∗ Ricardo Troncoso and Jorge Zanelli † Centro de Estudios Cient´ıficos de Santiago, Casilla 16443, Santiago 9, Chile and Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile. The final answer to these questions is beyond the scope Anewfamilyofsupergravitytheoriesinodddimensionsis of this paper, howeverone can note a difference between presented. TheLagrangian densitiesareChern-Simonsforms YM and GR which might turn out to be an important fortheconnectionofasupersymmetricextensionoftheanti- clue: YM theory is defined on a fiber bundle, with the deSitteralgebra. Thesuperalgebras are thesupersymmetric connectionasthe dynamicalobject,whereasthe dynam- extensions of the AdS algebra for each dimension, thus com- ical fields of GR cannot be interpreted as components of pleting the analysis of van Holten and Van Proeyen, which 9 a connection. Therefore, gravitation does not lend itself 9 was valid for N = 1 and for D = 2,3,4,mod 8. The Chern- naturally for a fiber bundle interpretation. 9 Simons form of the Lagrangian ensures invariance under the The closest one could get to a connection formulation 1 gaugesupergroupbyconstructionand,inparticular,underlo- forGRisthe Palatiniformalism,with the Hilbertaction cal supersymmetry. Thus, unlike standard supergravity, the n local supersymmetry algebra closes off-shell and without re- a J quiringauxiliaryfields. TheLagrangianisexplicitlygivenfor I[ω,e]= ǫabcdRab∧ea∧eb, (1) D = 5, 7 and 11. In all cases the dynamical field content Z 0 3 iinticnliu(dψesµi)t,haenvdieslobmeiene(xetaµr)a,bthoesosnpiicn“cmonantteecrt”iofine(ldωsµabw),hiNchgvraarvy- weahiesreaRloacbal=ordtωhaobn+oωrmca∧alωfbcraismteh.eTcuhrivsaatcutrioentwisos-foomrmet,imaneds 1 from one dimension to another. The superalgebras fall into v threefamilies: osp(m|N)forD=2,3,4,mod8,osp(N|m)for claimedto describe agaugetheoryforlocaltranslations. 3 D=6,7,8,mod8,andsu(m−2,2|N)forD=5mod4,with However, in our view this is a mistake. If ω and e were 0 m = 2[D/2]. The possible connection between the D = 11 the components of the Poincaré connection, under local 0 case and M-Theory is also discussed. translations they should transform as 2 Abstract 0 δωab =0, δea =Dλa =dλa+ωa∧λb. (2) 9 b 9 Invariance of (1) under (2)would require the torsion-free / I. INTRODUCTION h condition, t p- Agoodpartoftheresultspresentedinthislecturewere Ta =dea+ωa∧eb =0. (3) alsodiscussedin[2]andalsopresentedattheJanuary’98 b e h meetinginBariloche[3]–wherethedetailedconstruction Thisconditionisanequationofmotionfortheaction(1). v: ofthesuperalgebracanbefound–,butitwasatthemeet- Thismeansthattheinvarianceoftheaction(1)under(2) i ingcoveredbytheseproceedingswheretheseresultswere couldnotresultfromthetransformationpropertiesofthe X first presented. fieldsalone,butitwouldbeapropertyoftheirdynamics r Threeofthefourfundamentalforcesofnaturearecon- aswell. The torsion-freecondition, being oneofthe field a sistently described by Yang-Mills (YM) quantum theo- equations, implies that local translational invariance is ries. Gravity, the fourth fundamental interaction, resists at best an on-shell symmetry, which wouldprobably not quantization in spite of several decades of intensive re- survive quantization. search in this direction. This is intriguing in view of the The contradiction stems from the identification be- factthatGeneralRelativity(GR)andYMtheorieshave tween localtranslations in the base manifold (diffeomor- adeepgeometricalfoundation: thegaugeprinciple. How phisms) cometwotheoriesconstructedonalmostthesamemath- ematical basis produce such radically different physical xµ →xµ =xµ+ζµ(x), (4) ′ behaviours? What is the obstruction for the application of the methods of YM quantum field theory to gravity? –which is a genuine invariance of the action (1)–, and local translations in the tangent space (2). Since the invariance of the Hilbert action under gen- eralcoordinatetransformations(4)isreflectedintheclo- ∗TalkpresentedattheSixthMeetingonQuantumMechan- sureofthefirst-classhamiltonianconstraintsintheDirac formalism, one could try to push the analogy between ics of Fundamental Systems: Black Holes and the Structure of the Universe,Santiago, August 1997. the Hamiltonian constraints Hµ and the generators of a †John Simon Guggenheim fellow gauge algebra. However, the fact that the constraint algebra requires structure functions, which depend on the 1 dynamical fields, is another indication that the genera- II. SUPERGRAVITY tors of diffeomorphism invariance of the theory do not form a Lie algebra but an open algebra (see, e. g., [4]). For some time it was hoped that the nonrenormaliz- More precisely, the subalgebra of spatial diffeomor- ability of GR could be cured by supersymmetry. How- phismsisagenuineLiealgebrainthesensethatitsstruc- ever, the initial glamour of supergravity (SUGRA) as tureconstantsareindependentofthedynamicalfieldsof a mechanism for taming the wild ultraviolet divergences gravitation, of pure gravity,was eventually spoiled by the realization thatittoowouldleadtoanonrenormalizableanswer[10]. [H ,H ]∼H δ −H δ . (5) i j′ j′ i i′ j Again, one can see that SUGRA is not a gauge theory | | either in the sense of a fiber bundle, and that the lo- In contrast, the generators of timelike diffeomorphisms calsymmetryalgebraclosesnaturallyonly onshell. The form an open algebra, algebra can be made to close off shell at the cost of in- [H ,H ]∼gijH δ . (6) troducingauxiliaryfields,buttheyarenotguaranteedto ⊥ ⊥′ j′ |i exist for all D and N [11]. This comment is particularly relevant in a CHern- Whether the lack of fiber bundle structure is the ul- Simonstheory,wherespatialdiffeomorphismsarealways timate reason for the nonrenormalizability of gravity re- part of the true gauge symmetries of the theory. The mains to be proven. However, it is certainly true that if generators of timelike displacements (H ), on the other GR could be formulated as a gauge theory, the chances hand, are combinations of the internal g⊥auge generators for proving its renormalizability would clearly grow. and the generatorsof spatialdiffeomorphism, and there- In three spacetime dimensions both GR and SUGRA fore do not generate independent symmetries [5]. define renormalizable quantum theories. It is strongly suggestivethatpreciselyin2+1dimensionsboththeories Higher D The minimal requirements for a consistent canalsobeformulatedasgaugetheoriesonafiberbundle theorywhichincludesgravityinanydimensionare: gen- [12]. Itmightseemthattheexactsolvabilitymiraclewas eral covariance and second order field equations for the due to the absence of propagating degrees of freedom in metric. For D > 4 the most general action for gravity three-dimensionalgravity,butthe powercountingrenor- satisfying this criterion is a polynomial of degree [D/2] malizabilityargumentrestsonthe fiberbundle structure inthe curvature,firstdiscussedby Lanczosfor D =5 [6] of the Chern-Simons form of those systems. and, in general, by Lovelock [7,8]. There are other known examples of gravitation theo- riesinodddimensionswhicharegenuine(off-shell)gauge First order theory theories for the anti-de Sitter (AdS) or Poincaré groups If the theory contains spinors that couple to gravity, [13–16]. These theories, as well as their supersymmetric it is necessary to decouple the affine and metric proper- extensions have propagating degrees of freedom [5] and ties of spacetime. A metric formulation is sufficient for areCSsystemsforthecorrespondinggroupsasshownin spinless point particles and fields because they only cou- [17]. ple to the symmetric part of the affine connection, while a spinning particle can “feel” the torsion of spacetime. A. From Rigid Supersymmetry to Supergravity Thus,itisreasonabletolookforaformulationofgravity in which the spin connection (ωab) and the vielbein (ea) µ µ are dynamically independent fields, with curvature and Rigid SUSY can be understood as an extension of the torsion standing on a similar footing. Thus, the most Poincaréalgebrabyincludingsuperchargeswhicharethe generalgravitationalLagrangianwouldbe ofthe general “square roots” of the generators of rigid translations, form L=L(ω,e) [9]. {Q¯,Q}∼Γ·P. Thebasicstrategytogeneralizethisidea Allowing an independent spin connection in four di- tolocalSUSYwastosubstitutethemomentumPµ =i∂µ mensions does not modify the standard picture in prac- bythegeneratorsofdiffeomorphisms,H,andrelatethem ticebecauseanyoccurrenceoftorsionintheactionleaves tothesuperchargesby{Q¯,Q}∼Γ·H. Theresultingthe- the classical dynamics essentially intact. In higher di- ory has on-shell local supersymmetry algebra [18]. mensions, however, theories that include torsion can be An alternative point of view –which is the one we ad- dynamically quite different from their torsion-free coun- vocate here– would be to construct the supersymmetry terparts. onthe tangentspaceandnotonthe basemanifold. This As we shall see below, the dynamical independence of approach is more natural if one recalls that spinors pro- ωab andea alsoallowsdefining these gravitationtheories videabasisofirreduciblerepresentationsforSO(N),and in 2n+ 1 dimensions on a fiber bundle structure as a notfor GL(N). Thus, spinorsare naturallydefined rela- Yang-Millstheory,afeaturethatisnotsharedbyGeneral tive to a local frame on the tangent space rather than in Relativity except in three dimensions. the coordinatebasis. Thebasicpointistoreproducethe 2+1 “miracle” in higher dimensions. This idea has been successfully applied by Chamseddine in five dimensions 2 [14],andby us for pure gravity[15,16]andin supergrav- [21], the counting of degrees of freedom in CS theories is ity[2,17]. TheSUGRAconstructionhasbeencarriedout completelydifferentfromtheoneforthesameconnection for spacetimes whose tangent space has AdS symmetry one-forms in a YM theory. [2], and for its Poincarécontraction in [17]. In [17], a family of theories in odd dimensions, invariant under the supertranslation algebra whose bosonic III. LANCZOS–LOVELOCK GRAVITY sector contains the Poincaré generators was presented. The anticommutator of the supersymmetry generators A. Lagrangian gives a translation plus a tensor “central” extension, For D > 4, assumption (ii) is an unnecessary restric- {Qα,Q¯ }=−i(Γa)αP −i(Γabcde)αZ , (7) β β a β abcde tion on the available theories of gravitation. In fact, as The commutators of Q,Q¯ and Z with the Lorentz gen- mentioned above, the most general action for gravity – generallycovariantandwithsecondorderfieldequations erators can be read off from their tensorial character. forthemetric–istheLanczos-LovelockLagrangian(LL). All the remaining commutators vanish. This algebra is TheLLLagrangianinaD-dimensionalRiemannianman- thecontinuationtoallodd-dimensionalspacetimesofthe ifold can be defined in at least four ways: D = 10 superalgebra of van Holten and Van Proeyen (a)Asthemostgeneralinvariantconstructedfromthe [19], and yields supersymmetric theories with off-shell metric and curvature leading to second order field equa- Poincaré superalgebra. The existence of these theories tions for the metric [6–8]. suggeststhatthereshouldbesimilarsupergravitiesbased (b) As the most general D-form invariant under local on the AdS algebra. Lorentz transformations, constructed with the vielbein, the spin connection, and their exterior derivatives, without using the Hogde dual (∗) [22]. B. Assumptions of Standard Supergravity (c) As a linear combination of the dimensional con- tinuation of all the Euler classes of dimension 2p < D. Three implicit assumptions are usually made in the [8,23] construction of standard SUGRA: (d) As the most general low energy effective gravita- (i)The fermionicandbosonic fields inthe Lagrangian tional theory that can be obtained from string theory shouldcomeincombinationssuchthattheirpropagating [24]. degrees of freedom are equal in number. This is usually Definition (a) was historically the first. It is appropri- achieved by adding to the graviton and the gravitini a ateforthemetricformulationandassumesvanishingtor- number of lower spin fields (s < 3/2) [18]. This match- sion. Definition(b)isslightlymoregeneralthanthefirst ing, however, is not necessarily true in AdS space, nor and allows for a coordinate-independent first-order for- in Minkowski space if a different representation of the mulation,andevenallowstorsion-dependenttermsinthe Poincaré group (e.g., the adjoint representation) is used action [9]. As a consequence of (b), the field configura- [20]. tionsthatextremizetheactionobeyfirstorderequations Theothertwoassumptionsconcernthepurelygravita- for ω and e. Assertion (c) gives directly the Lanczos– tionalsector. TheyareasoldasGeneralRelativityitself Lovelocksolution as a polynomial of degree [D/2] in the andaredictatedbyeconomy: (ii)gravitonsaredescribed curvature of the form by the Hilbert action (plus a possible cosmological constant),and,(iii)thespinconnectionandthevielbeinare [D/2] not independent fields but are related through the tor- I = α Lp, (8) G p Z sion equation. The fact that the supergravitygenerators Xp=0 do not form a closedoff-shell algebracan be tracedback to these asumptions. where αp are arbitrary constants and1 The procedure behind (i) is tightly linked to the idea thatthe fieldsshouldbe inavectorrepresentationofthe LpG =ǫa1···aDRa1a2···Ra2p−1a2pea2p+1···eaD, (9) Poincaré group [20] and that the kinetic terms and cou- wherewedgeproductofformsisunderstoodthroughout. plings are such that the counting of degrees of freedom Statement (d) reflects the empirical observation that works like in a minimally coupled gauge theory. This the vanishing of the superstring β-function in D = 10 assumption comes from the interpretation of supersym- gives rise to an effective Lagrangianof the form (9) [24]. metric states as represented by the in- and out- plane waves in an asymptotically free, weakly interacting theory in a minkowskian background. These conditions are not necessarily met by a CS theory in an asymptotically AdS background. Apart from the difference in back- 1Forevenandodddimensionsthesameexpression(9)canbe ground,whichrequiresacarefultreatmentoftheunitary used,butforoddD,Chern-SimonsformsfortheLorentzcon- irreduciblerepresentationsofthe asymptoticsymmetries nectioncouldalsobeincluded(thispointisdiscussedbelow). 3 Inevendimensions,thelastterminthesumistheEu- C. The vanishing of Classical Torsion lercharacter,whichdoesnotcontributetothe equations ofmotion. However,in the quantumtheory,this termin Obviously Ta = 0 solves (12). However, for D > 4 the partition function would assign different weights to this equation does not imply vanishing torsion in gen- nonhomeomorphic geometries. eral. In fact, there are choices of the coefficients α and p The large number of dimensionful constants αp in the configurations of ωab, ea such that Ta is completely ar- LLtheorycontrastswiththetwoconstantsoftheEHac- bitrary. On the other hand, as already mentioned, the tion (G and Λ) [25,26,15]. This feature could be seem as torsion-freepostulateis atbestagooddescriptionofthe an indication that renormalizability would be even more classical dynamics only. Thus, an off-shell treatment of remote for the LL theorythan inordinarygravity. How- gravityshouldallowfordynamicaltorsioneveninfourdi- ever,this is not necessarily so. There are some very spe- mensions. In the first order formulation, the theory has cialchoicesofαp suchthatthe theorybecomes invariant second class constraints due to the presence of a large underalargergaugegroupinoddspacetimedimensions, number of “coordinates” which are actually “momenta” which could actually improve renormalizability [12,16]. [30],thuscomplicatingthedynamicalanalysisofthethe- ory. On the other hand, if torsion is assumed to vanish, ω B. Equations could be solved as a function of e 1 and its first deriva- − tives,butthis wouldrestrictthe validityofthe approach Consider the Lovelock action (8), viewed as a func- to nonsingular configurations for which det(ea) 6= 0. µ tional of the spin connection and the vielbein, In this framework, the theory has no second class con- straintsandthenumberofdegreesoffreedomisthesame ILL =ILL ωab,ea . (10) as in the Einstein-Hilbert theory, namely D(D−3) [23]. 2 (cid:2) (cid:3) Varying with respect to the vielbein, the generalized Einstein equations are obtained, D. Dynamics and Degrees of Freedom n 1 − α (D−2p)ǫ Ra1a2···Ra2p−1a2p × Imposing Ta = 0 from the start, the action is Xp=0 p a1···aD I=ILL[ea,ω(e)] and varying respect to e, the “1.5 order ea2p+1···eaD−1 =0. (11) formalism” [18] is obtained, Varying with respect to the spin connection, the torsion δI δILLδea+ δILLδωbcδea. (13) equations are found, = δea δωbc δea n 1 Assuming δILL = 0 the equations of motion consist − δωbc α p(D−2p)ǫ Ra3a4···Ra2p−1a2p × of the Einstein equations (11), defined on a restricted Xp=0 p aba3···aD configuration space. ea2p+1···eaD−1TaD =0. (12) For D ≤ 4, Ta = 0 is the unique solution of eqn.(12). In those dimensions, the different variational principles The presence of the arbitrary coefficients α in (first-,second-and1.5-thorder)areclassicallyequivalent p the action implies that static, spherically symmetric in the absence of sources. On the contrary, for D > 4, Schwarzschild-like solutions possess a large number of Ta = 0 is not logically necessary and is therefore unjus- horizons [27], and time-dependent solutions have an un- tified. predictable evolution [23,28]. However, as shown below, The LL–Lagrangians (9) include the Einstein-Hilbert for a particular choice of the constants α the dynamics (EH) theory as a particular case, but they are dynami- p is significantly better behaved. cally very different in general. The classical solutions of Additional terms containing torsion explicitly can be the LL theory are not perturbatively related to those of included in the action. It can be shown, however, that the Einstein theory. For instance, it was observed that the presence of torsional terms in the Lagrangian does the time evolution of the classical solutions in the LL not change the degrees of freedom of gravity in four di- theory starting from a generic initial state can be un- mensions. Indeed, the matter-free theory with torsion predictable, whereas the EH theory defines a well-posed terms is indistinguishable (at least classically) from GR, Cauchy problem. [29]. However,inhigherdimensions,thesituationiscom- It can also be seen that even for some simple minisu- pletely different [9]. perspacemodels,thedynamicscouldbecomequitemessy because the equations of motion are not deterministic in the classical sense, due to the vanishing of some eigen- values ofthe Hessianmatrix oncriticalsurfacesin phase space [23,28]. 4 E. Choice of Coefficients The resulting Lagrangian is the Euler-CS form. Its exterior derivative is the Euler form in 2n dimensions, Atleastforsomesimpleminisuperspacegeometriesthe dLAdS =κǫ RA1A2···RA2n−1A2n (18) indeterminate classical evolution can be avoided if the G2n−1 A1···A2n coefficients are chosen so that the Lagrangian is based =κE2n, on the connection for the AdS group, where RAB = dWAB +WAWCB is the AdS curvature, C n−1 which contains the Riemann and torsion tensors, (D−2p) 1 , D =2n−1  − (cid:18) p (cid:19) αplD−2p = (cid:18)np (cid:19), D =2n. RAB =(cid:20) Rab−+Tbl12/elaeb Ta0/l (cid:21). (19)  (14) Theconstantκis quantized[16](inthefollowingwewill set κ=l=1). This corresponds to the Born-Infeld theory in even di- In general, a Chern-Simons Lagrangian in 2n−1 dimensions [26], and to the AdS Chern-Simons theory in mensions is defined by the condition that its exterior odd dimensions [15,13,14], derivative be an invariant homogeneous polynomial of degree n in the curvature, that is, a characteristic class. Inthe caseabove,(??)defines the CSformforthe Euler 1. D=2n: Born-Infeld Gravity class 2n-form. A generic CS Lagrangian in 2n−1 dimensions for a In even dimensions the choice (14) gives rise to a La- Lie algebra g can be defined by grangian of the form dLg =hFni, (20) 2n 1 ea1ea2 eaD−1eaD − L=κǫ (Ra1a2 + )···(RaD−1aD + ). a1···aD l2 l2 whereh istandsforamultilinearfunctionintheLiealge- brag,invariantundercyclicpermutationssuchasTr,for (15) an ordinaryLie algebra,or STr, in the case of a superal- ThisisthePfaffianofthetwo–formRab+ 1eaeb,and,in gebra. In the caseabove,the only nonvanishingbrackets l2 in the algebra are this sense it can be written in the Born-Infeld-like form, J ,···,J =ǫ . (21) 1 A1A2 AD−1AD A1···AD L=κ det(Rab+ eaeb). (16) (cid:10) (cid:11) r l2 The combinations Rab+ 1eaeb are the components of l2 3. D=2n−1: Poincaré Gauge Gravity theAdScurvature(c.f.(19)below). Thisseemstosuggest thatthe systemmightbe naturallydescribedintermsof Starting from the AdS theory (??) in odd dimensions, anAdSconnection[31]. However,thisisnotthecase: In a Wigner- Ino¨nu¨ contraction deforms the AdS algebra even dimensions, the Lagrangian (15) is invariant under into the Poincaré one. The same result is also obtained local Lorentz transformations and not under the entire choosing α =δn. Then, the Lagrangian(8) becomes: AdS group. As will be shown below, it is possible, in p p odd dimensions, to construct gauge invariant theories of LP =ǫ Ra1a2···RaD−2aD−1eaD. (22) gravity under the full AdS group. G a1···aD Inthiswaythelocalsymmetrygroupof(8)isextended fromLorentz(SO(D−1,1))toPoincaré(ISO(D−1,1)). 2. D=2n−1: AdS Gauge Gravity Analogously to the anti-de Sitter case, one can see that the action depends on the Poincaré connection: A = Theodd-dimensionalcasewasdiscussedin[13,14],and eaP + 1ωabJ . It is straightforward to verify the in- lateralsoin[15]. Considerthe action(8)withthe choice a 2 ab variance of the action under local translations, given by (14) for D =2n−1. The constant parameter l hasdimensions oflengthandits purpose isto renderthe δea =Dλa, δωab =0, (23) action dimensionless. This also allows the interpretation of ω and e as components of the AdS connection [26], Here D stands for covariant derivative in the Lorentz A= 1ωabJ +eaJ = 1WABJ , where connection. If λ is the Lie algebra-valuedzero-form,λ= 2 ab aD+1 2 AB λaP , the transformations (23) are read from the gen- a ωab ea/l eral gauge transformation for the connection, δA=∇λ, WAB = , A,B =1,...D+1. (17) (cid:20) −eb/l 0 (cid:21) where ∇ is the covariant derivative in the Poincaré connection. 5 Moreover,theLagrangian(22)isaChern-Simonsform. which would be inconsistent with the transformation of Indeed, with the curvature for the Poincaréalgebra, F= the fields under local translations (2). Thus, the spin dA +A∧A =21RabJab+TaPa, LPG satisfies connection and the vielbein –the soldering between the base manifold and the tangent space– cannot be identi- dLP = Fn+1 , (24) fiedasthecompensatingfieldsforlocalLorentzrotations G (cid:10) (cid:11) and translations, respectively. where the only nonvanishing components in the bracket Inourconstructionω andeareassumedtobedynam- are ically independent and thus torsion necessarily contains propagating degrees of freedom, represented by the con- Ja1a2,···,JaD−2aD−1,PaD =ǫa1···aD. (25) torsiontensorkµab :=ωµab−ω¯µab(e,...),whereω¯ isthespin (cid:10) (cid:11) connection which solves the (algebraic) torsion equation Thus, the Chern character for the Poincaré group is in terms of the remaining fields. written in terms of the Riemman curvature and the tor- The generalization of the Lovelock theory to include sion as torsion explicitly can be obtained assuming definition (b). This is a cumbersome problem due to the lack of F3 =ǫ Ra1a2···RaD−2aD−1TaD. (26) a1···aD a simple algorithmto classify all possible invariants con- (cid:10) (cid:11) structedfromea,RabandTa. InRef.[9]auseful“recipe” Thesimplestexampleofthisisordinarygravityin2+1 to generate all those invariants is given. dimensions, where the Einstein-Hilbert action with cosmological constant is a genuine gauge theory of the AdS group, while for zero cosmological constant it is invari- A. The Two Families of AdS Theories ant under local Poincaré transformations. Although this gauge invariance of 2+1 gravity is not always empha- sized, it lies at the heart of the proof of integrability of Similarly to the theory discussed in section III, the the theory [12]. torsional additions to the Lagrangianbring in a number ofarbitrarydimensionfulcoefficientsβ ,analogoustothe k α ’s. Also in this case, one can try choosing the β’s in p IV. ADS GAUGE GRAVITY suchawayastoenlargethelocalLorentzinvarianceinto anAdSgaugesymmetry. Ifnoadditionalstructure(e.g., inversemetric,Hodge-∗,etc.) isassumed,AdSinvariants As shown above, the LL action assumes spacetime can only be produced in dimensions 4k and 4k−1. to be a Riemannian, torsion-free, manifold. That as- The proof of this claim is as follows: invariance under sumption is justified a posteriori by the observationthat Ta = 0 is always a solution of the classical equations, AdS requires that the D-form be at least Lorentz invariant. Then,inorderforthesescalarstobeinvariantunder and means that e and ω are not dynamically indepen- AdS as well, it is necessary and sufficient that they be dent. This is the essence of the second order or metric expressible in terms of the AdS connection (17). As is approachtoGR,inwhichdistanceandparalleltransport well-known (see, e.g., [33]), in even dimensions, the only arenotindependentnotions,butarerelatedthroughthe D-forminvariantunderSO(N)constructedaccordingto Christoffelsymbol. Thereisnofundamentaljustification the recipe mentionedaboveare2 the Eulercharacter(for for this assumptionand this was the issue of the historic N = D), and the Chern characters (for any N). Thus, discussion between Einstein and Cartan [32]. In four dimensions, the equation Ta = 0 is algebraic the only AdS invariant D-forms are the Euler class, and linear conbinations of products of the type and could in principle be solvedfor ω in terms of the re- mainingfields. However,forD >5,CSgravityhasmore P =c ···c , (28) degrees of freedom than those encountered in the corre- r1···rs r1 rs sponding second order formulation [5]. This means that with 2(r +r +···+r )=D, where 1 2 s theCSgravityactionhaspropagatingdegreesoffreedom for the spin connection. This is a compelling argument c =Tr(Fr), (29) r toseriouslyconsiderthepossibilityofintroducingtorsion terms in the Lagrangianfrom the start. defines the r-th Chern character of SO(N). Now, since Another consequence of imposing a dynamical depen- the curvature two-formF in the vectorialrepresentation dencebetweenωandethroughthetorsion-freecondition is that it spoils the possibility of interpreting the local translational invariance as a gauge symmetry of the action. Consider the action of the Poincaré group on the 2For simplicity we will not always distinguish between dif- fields as given by (23); taking Ta ≡0 implies ferent signatures. Thus, if no confussion can occur, the AdS group in D dimensions will also be denoted as SO(D+1). δωab = δωabδec 6=0, (27) The de Sitter case can be obtained replacing αp by (−1)pαp δec in (14). 6 is antisymmetric in its indices, the exponents {r } are concentrateontheconstructionofthepuregravitysector j necessarily even, and therefore (28) vanishes unless D is asagaugetheorywhichisparity-odd. Thisconstruction a multiple of four. Thus, one arrives at the following was discussed in [35], and also briefly in [2,3]. lemmas: Lemma: 1 For D = 4k, the only D-forms built from ea, Rab and Ta, invariant under the AdS group, are the B. Even dimensions Chern characters for SO(D+1). Lemma: 2ForD =4k+2,therearenoAdS-invariant In D =4, the the only local Lorentz-invariant4-forms D-forms constructed from ea, Rab and Ta. constructed with the recipe just described are [9]: In view of this, it is clear why attempts to construct E =ǫ RabRcd gravitation theories with local AdS invariance in even 4 abcd dimensions have been unsuccessful [31,34]. L =ǫ Rabeced EH abcd boSuinndcaerythteerfmorsmisnP4rk1···drsimaernesicolnosse–dw,htihcehydaorenoatt cboenst- LC =ǫabcdeaebeced C =RabR tribute totheclassicalequations,butcouldassigndiffer- 2 cd ent weights to configurations with nontrivial torsion in LT1 =Rabeaeb the quantum theory. In other words, they can be locally L =TaT . T2 a expressed as P =dLAdS (W). (30) The first three terms are even under parity and the r1···rs {r}4k−1 rest are odd. Of these, E and C are topological invari- 4 2 Thus,foreachcollection{r},the(4k−1)-formLAdS antdensities (closedforms): the Eulercharacterandthe r 4k 1 defines a Lagrangian for the AdS group in 4k −{ }1 d−i- second Chern character for SO(4),respectively. The re- mensions. It takes direct computation to see that these mainingfourtermsdefinethemostgeneralgravityaction Lagrangians involve torsion explicitly. These results are in four dimensions, summarized in the following I = [αL +βL +γL +ρL ]. (31) EH C T1 T2 Z Theorem: Therearetwofamiliesofgravitationalfirst M4 orderLagrangiansforeandω,invariantunderlocalAdS Itcanalsobeseen,thatbychoosingγ =−ρ,thelasttwo transformations: terms are combined into a topological invariant density a: Euler-Chern-SimonsforminD =2n−1,whose ex- (theNieh-Yanform). Thus,withthischoicetheoddpart terior derivative is the Euler character in dimension 2n, of the action becomes a boundary term. Furthermore, which do not involve torsion explicitly, and C , L andL canbe combinedinto the secondChern 2 T1 T2 character of the AdS group, b: Pontryagin-Chern-Simons forms in D = 4k − 1, RaRb +2(TaT −2Rabe e )=RA RB. (32) whoseexteriorderivativesaretheCherncharactersin4k b a a a b B A dimensions, which involves torsion explicitly. ThisistheonlyAdSinvariantconstructedwithea,ωab It must be stressed that locally AdS-invariant gravity andtheirexteriorderivativesalone,confirmingthatthere theories only exist in odd dimensions. They aregenuine arenolocallyAdSinvariantgravitiesinfourdimensions. gauge systems, whose action comes from topological in- Ingeneral,theonlyAdS-invariantfunctionalsinhigher variants in one dimension above. These topological in- dimensionscanbe writtenintermsoftheAdScurvature variants can be written as the trace of a homogeneous as [9] polynomialofdegreen inthe AdS curvature. Obviously, for dimensions 4k−1 both a- and b-families exist. The I˜ = C ···C , (33) mostgeneralLagrangianof this sortis a linear combina- r1···rs ZM r1 rs tion of the two families. or linear combinations thereof, where C =Tr[(RA)r] is An important difference between these two families is r B ther-thCherncharacterfortheAdSgroup. Forexample, thatunderaparitytransformationthe firstisevenwhile the second is odd 3. The parity invariant family has en D =8 the Chern characters for the AdS group are beenextensivelystudiedin[13–15,26]. Inwhatfollowswe Tr[(RA )4]=C , B 4 (34) Tr[(RAB)2]∧Tr[(RAB)2]=(C2)2. 3Parityisunderstoodhereasaninversionofonecoordinate, Similar Chern classes are also found for D = 4k. (As binostthaninceththeetaEnugleenrtcshpaarcaectaenrdisininthvearbiaansetmunadneifrolpda.riTtyh,uws,hfioler oaldrdea,dwyhimchenistitohneedc,asI˜er1i·n··r4skv+an2isdhiemseinfsoionnes.o)f the r’s is theLorentzCherncharactersandthetorsionaltermsarepar- Thus, there are no AdS-invariant gauge theories in ity violating. even dimensions. 7 C. Odd dimensions V. EXACT SOLUTIONS The simplest example is found in three spacetime di- As stressed here, the local symmetry of odd- mensions where there are two locally AdS-invariant La- dimensionalgravitycanbeextendedfromLorentztoAdS grangians, namely, the Einstein-Hilbert with cosmologi- byanappropriatechoiceofthefreecoefficientsintheac- cal constant, tion. TheresultingLagrangians(withorwithouttorsion terms), are Chern-Simons D-forms defined in terms of 1 LAdS =ǫ [Rabec+ eaebec], (35) the AdS connection A, whose components include the G3 abc 3l2 vielbein and the spin connection [see eqn. (17)]. This impliesthatthefieldequations(11,12)obtainedbyvary- and the “exotic” Lagrangian ingthevielbeinandthespinconnectionrespectively,can LATd3S =L∗3(ω)+2eaTa, (36) be written in an AdS-covariant form <Fn 1J >=0, (39) where − AB 2 where F= 1RABJ is the AdS curvature with RAB L ≡ωadωb + ωaωbωc, (37) 2 AB ∗3 b a 3 b c a given by (19) and JAB are the AdS generators. It is easily checked that any locally AdS spacetime is istheLorentzChern-Simonsform. Notethatin(36),the a solution of (39). Apart from anti-de Sitter space it- localAdSsymmetryfixestherelativecoefficientbetween self, some interesting spacetimes with this feature are L∗3(ω), and the torsion term eaTa. The most general the topological black holes of Ref. [37], and some “black action for gravitationin D =3, which is invariant under branes”withconstantcurvatureworldsheet[38]. Forany SO(4) is thereforealinear combinationαLAGd3S+βLATd3S. D, there is also a unique static, spherically symmetric, For D = 4k−1, the number of possible exotic forms asymptotically AdS black hole solution [15], as well as grows as the partitions of k, in correspondence with their topological extensions which have nontrivial event the number of composite Chern invariants of the form horizons [39]. P{r} = jCrj. The most general Lagrangian in 4k−1 Exact solutions of the form AdS4 × SD−4 have also dimensioQns takes the form αLAdS +β LAdS , been found [40] 4 as well as alternative four-dimensional G4k 1 r T r 4k 1 where dLAdS = P , with− r {=} 4k.{ }Th−ese cosmologicalmodels. T r 4k 1 r j j Lagrangians{h}ave−proper{d}ynamicsPand, unlike the even All ofthe above geometriescanbe extended into solu- tionsofthe gravitationalBorn-Infeldtheory(16)ineven dimensionalcases,they are notboundary terms. For ex- dimensions. Friedmann-Robertson-Walker like cosmolo- ample, in seven dimensions one finds [35,36] gies have been shown to exist in even dimensions [26], and it could be expected that similar solutions exists in LAdS =β [Ra Rb +2(TaT −Rabe e )]LAdS odd dimensions as well. T 7 2,2 b a a a b T 3 +β [L (ω)+2(TaT +Rabe e )Tae +4T Ra Rb ec], 4 ∗7 a a b a a b c VI. CHERN-SIMONS SUPERGRAVITIES where L is the Lorentz-CS (2n-1)-form, ∗2n 1 − dL (ω)=Tr[(Ra)n]. (38) Wenowconsiderthesupersymmetricextensionsofthe ∗2n 1 b − locallyAdStheoriesdefinedabove. Theideaistoenlarge Summarizing: TherequirementoflocalAdSsymme- the AdS algebra incorporating SUSY generators. The try is rather strong and has the following consequences: closure of the algebra (Jacobi identity) forces the addi- tion of further bososnic generators as well [19]. In order • Locally AdS invariant theories of gravity exist in to accomodate spinors in a natural way, it is useful to odd dimensions only. cast the AdS generators in the spinor representation of SO(D+1). In particular, one can write, • ForD =4k−1therearetwofamilies: oneinvolving only the curvature and the vielbein (Euler Chern- −1 Simons form), and the other involving torsion ex- dLATd4Sk−1 = 24kTr[(RABΓAB)2k]. (40) plicitly in the Lagrangian. These families are even and odd under space reflections, respectively. • ForD =4k+1onlytheEuler-Chern-Simonsforms 4The de-Sitter case (Λ > 0) was discussed in [41] for the exist. These ar parity even and don’t involve tor- torsion-free theory. Changing the sign in the cosmological sion explicitly. constant has deep consequences. In fact, the solutions are radically different,andlocally supersymmetricextensionsfor positive cosmological constant don’t exist in general. 8 which is a particular form of (20) where hi has been re- D S-Algebra Conjugation Matrix Internal Metric placed by the ordinary trace over spinor indices in this 8k−1 osp(N|m) CT =C uT =−u representation. 8k+3 osp(m|N) CT =−C uT =u Other possibilities of the form Fn−p hFpi, are not 4k+1 su(m|N) C† =C u† =u necessary to reproduce the minima(cid:10)l supe(cid:11)rsymmetric extensions of AdS containing the Hilbert action. In the In each of these cases, m = 2[D/2] and the connection supergravity theories discussed below, the gravitational takes the form sector is given by ± 1 LAdS − 1LAdS . The ± sign 2n G2n 1 2 T 2n 1 1 1 corresponds to the two choic−es of inequiv−alent represen- A= ωabJ +eaJ + b[r]Z + 2 ab a r! [r] tations of Γ’s, which in turn reflect the two chiral repre- 1 1 sentationsinD+1. Asinthethree-dimensionalcase,the (ψ¯iQ −Q¯iψ )+ a Mij. (42) i i ij 2 2 supersymmetricextensionsofL oranyoftheexoticLa- G grangianssuchasLT,requireusingbothchiralities,thus The generators Jab,Ja span the AdS algebra and the doubling the algebras. Here we choosethe + sign,which Qi’s generate (extended) supersymmetry transforma- α gives the minimal superextension [35]. tions. TheQ’stransforminavectorrepresentationunder The bosonic theory (40) is our starting point. The theactionofM andasspinorsundertheLorentzgroup. ij idea now is to construct its supersymmetric extension. Finally, the Z’s complete the extension of AdS into the For this, we need to express the adjoint representation largeralgebrasso(m), sp(m)orsu(m),and[r] denotes a in terms ofthe Dirac matrices of the appropriatedimen- set of r antisymmetrized Lorentz indices. sthioenD. iTrahcisaligsebarlwa,a{yIs,pΓoas,siΓbaleb,.b..e}c,apursoevitdheeagbenaesirsatfoorrsthoef whIenre(4C2)aψn¯id=uψajrTeCguijvien(ψ¯iin=thψej†tCabuljei afobrovDe.=T4hkes+e 1a)l-, spaceofsquarematrices. Theadvantageofthisapproach gebrasadmit(m+N)×(m+N)matrix representations is that it gives an explicit representation of the algebra [31],where the J andZ haveentriesinthe m×m block, and writing the Lagrangiansis straightforward. the M ’s inthe N×N block,while the fermionicgener- ij ThesupersymmetricextensionsoftheAdSalgebrasin ators Q have entries in the complementary off-diagonal D = 2,3, 4,mod8, werestudied by vanHoltenandVan blocks. Proeyenin[19]. TheyaddedoneMajoranasupersymme- Under a gauge transformation, A transforms by δA= trygeneratortotheAdSalgebraandfoundalltheN =1 ∇λ,where∇isthecovariantderivativeforthesamecon- extensionsdemandingclosureofthefullsuperalgebra. In nection A. In particular, under a supersymmetry trans- spite of the fact that the algebra for N = 1 AdS super- formation, λ=ǫ¯iQ −Q¯iǫ , and i i gravity in eleven dimensions was conjectured in 1978 to be osp(32|1) by Cremer, Julia and Scherk [42], and this ǫkψ¯ −ψkǫ¯ Dǫ δ A= k k j , (43) wasconfirmedin[19],nobodyconstructedasupergravity ǫ (cid:20) −Dǫ¯i ¯ǫiψj −ψ¯iǫj (cid:21) action for this algebra in the intervening twenty years. Onereasonforthelackofinterestintheproblemmight where D is the covariant derivative on the bosonic con- have been the fact that the osp(32|1) algebra contains nection,Dǫj =(d+12[eaΓa+21ωabΓab+r1!b[r]Γ[r]])ǫj−aijǫi. generatorswhichareLorentztensorsofrankhigherthan two.In the past, supergravityalgebras were traditionally B. D=5 Supergravity limited to generators which are Lorentz tensors up to second rank. This constraint was based on the observation that elementary particle states of spin higher than Inthiscase,asineverydimensionD =4k+1,thereis two would be inconsistent [43]. However, this does not no torsional Lagrangians LT due to the vanishing of the rule out the relevance ofthose tensor generatorsin theo- Pontrjagin4k+2-formsfor the Riemann cirvature. This ries of extended objects [44]. In fact, it is quite common factimplies thatthe localsupersymmetric extensionwill nowadays to find algebras like the M−brane superalge- be of the form L=LG+···. bra [45,46], As shown in the previous table, the appropriate AdS superalgebrainfive dimensionsissu(2,2|N),whosegen- {Q,Q¯}∼ΓaP +ΓabZ +ΓabcdeZ . (41) erators are K,J ,J ,Qα,Q¯ ,Mij, with a,b = 1,...,5 a ab abcde a ab β and i,j = 1,...,N. The connection is A= bK +eaJ + a 1ωabJ +a Mij +ψ¯iQ −Q¯jψ , so that in the adjoint 2 ab ij i j representation A. Superalgebra and Connection Ωα ψα A= β j , (44) The smallest superalgebra containing the AdS alge- (cid:20)−ψ¯βi Aij (cid:21) bra in the bosonic sector is found followingthe same ap- with Ωα = 1(ibI +eaΓ +ωabΓ )α, Ai = i δib+ai, proachasin[19],butliftingtherestrictionofN =1[35]. β 2 2 a ab β j N j j The result, for odd D >3 is (see [3] for details) and ψ¯βi = ψ†αjGαβ. Here G is the Dirac conjugate (e. g., G=iΓ0). The curvature is 9 R¯α Dψα ea → 1ea F= β j (45) α (cid:20) −Dψ¯i F¯i (cid:21) ωab → ωab β j b → 1 b 3α where ψ → 1 ψ (53) i √α i Dψα =dψα+Ωαψβ −Aiψα, ψi → 1 ψi j j β j j i √α R¯α =Rα−ψαψ¯i, (46) aij → aij. β β i β F¯i =Fi−ψ¯iψβ. Then,ifthegravitationalconstantisalsorescaledasκ→ j j β j ακ, in the limit α→∞ the action becomes that in [17], Here Fi =dAi +AiAk+ i dbδi is the su(N) curvature, plus a su(N) CS form, j j k j N j andRα =dΩα+ΩαΩσ is the u(2,2)curvature. Interms β β σ β of the standard (2n−1)-dimensional fields, Rβα can be I = 1 [ǫabcdeRabRcdee−RabRabb− (54) written as 8Z Rα = idbδα+ 1 TaΓ +(Rab+eaeb)Γ α. (47) 2Rab(ψiΓabDψi+DψiΓabψi)+Lsu(N)]. β 4 β 2 a ab β (cid:2) (cid:3) The rescaling (53) induces a contraction of the su- In six dimensions the only invariant form is per AdS algebra su(m|N) into [super Poincaré]⊗su(N), where the second factor is an automorfism. P =iStr F3 , (48) (cid:2) (cid:3) which in this case reads C. D=7 Supergravity P =Tr R3 −Tr F3 (49) +3 D(cid:2)ψ¯((cid:3)R¯+F¯)(cid:2)Dψ(cid:3)−ψ¯(R2−F2+[R−F](ψ)2)ψ , The smallest AdS superalgebra in seven dimensions is osp(2|8). The connection (42) is A =1ωabJ +eaJ + (cid:2) (cid:3) 2 ab a where (ψ)2 = ψ¯ψ. The resulting five-dimensional C-S Q¯iψi+ 21aijMij, where Mij are the generators of sp(2). density can de descompossed as a sum a a gravitational In the representation given above, the bracket h i is the part, a b-dependent piece, a su(N) gauge part, and a supertraceand,intermsofthe componentfieldsappear- fermionic term, ing in the connection, the CS form is L=LAGdS+Lb+Lsu(N)+LF, (50) Lo7sp(2|8)(A)=2−4LAGd7S(ω,e)− 21LATd7S(ω,e) with −L7∗sp(2)(a)+LF(ψ,ω,e,a). (55) LAdS = 1ǫ (RabRcdee+ 2Rabecedee+ 1eaebecedee) G 8 abcde 3 5 Here the fermionic Lagrangianis L =−( 1 − 1 )(db)2b+ 3(TaT −Rabe e − 1RabR )b b N2 42 4 a a b 2 ab L =4ψ¯j(R2δi +Rfi+(f2)i)Dψ +3bfifj F j j j i N j i +4(ψ¯iψ )[(ψ¯jψ )(ψ¯kDψ )−ψ¯j(Rδk+fk)Dψ ] j k i i i k L =−(aidajdak+ 3aiajakdal + 3aiajakal am) −2(ψ¯iDψ )[ψ¯j(Rδk+fk)ψ +Dψ¯jDψ ], su(N) j k i 2 j k l i 5 j k l m i j i i k i LF = 23 ψ¯(R¯+F¯)Dψ− 12(ψ)2(ψ¯Dψ) . wherefji =daij+aikakj,andR= 14(Rab+eaeb)Γab+21TaΓa are the sp(2) and so(8) curvatures,respectively. The su- (cid:2) (cid:3) (51) persymmetry transformations (43) read The action is invariant under local gauge transforma- δea = 1ǫ¯iΓaψ δωab =−1ǫ¯iΓabψ tions, which contain the local SUSY transformations 2 i 2 i δea = −1(ǫiΓaψ −ψiΓaǫ ) δψi =Dǫi δaij =ǫ¯iψj −ψ¯iǫj. 2 i i δωab = 1(ǫiΓabψ −ψiΓabǫ ) Standard seven-dimensional supergravity is an N = 2 4 i i theory (its maximal extension is N=4), whose gravi- δb = i(ǫiψ −ψiǫ ) i i (52) tational sector is given by the Einstein-Hilbert action δψ = Dǫ i i with cosmological constant and with an osp(2|8) invari- δψi = Dǫi ant background [47,48]. In the case presented here, the δai = i(ǫiψ −ψiǫ ). extension to larger N is straighforward: the index i is j j j allowed to run from 2 to 2s, and the Lagrangianis a CS As in 2+1dimensions, the Poincarésupergravitythe- form for osp(2s|8). ory is recovered contracting the super AdS group. Con- sider the following rescaling of the fields 10