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(Non-)Abelian Gauged Supergravities in Nine Dimensions PDF

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UG-02-41 hep-th/0209205 (Non-)Abelian Gauged Supergravities in Nine Dimensions E. Bergshoe(cid:11), T. de Wit, U. Gran, R. Linares and D. Roest Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands. E-mail: {e.a.bergshoeff, t.c.de.wit, u.gran, r.linares, d.roest}@phys.rug.nl ABSTRACT We construct (cid:12)ve di(cid:11)erent two{parameter massive deformations of the unique nine{ dimensional N = 2 supergravity. All of these deformations have a higher{dimensional origin via Scherk{Schwarz reduction and correspond to gauged supergravities. The gauge groups we encounter are SO(2), SO(1;1)+, R, R+ and the two{dimensional non{Abelian Lie group CR1, which consists of scalings and translations in one dimension. We make a systematic search for half-supersymmetric domain walls and non-supersym- metricdeSitterspacesolutions. Furthermore,wediscussinwhichsensethesupergravitieswe have constructed can be considered as low-energy limits of compacti(cid:12)ed superstring theory. 1 Introduction It iswell-known thatthelow-energy limit ofsuperstring theory and/orM{theoryisdescribed by a supergravity theory in the same spacetime dimension and with the same number of supersymmetries. Thus, M{theoryleadstoD=11supergravityandTypeIIA/IIBsuperstring theory leads to D=10 IIA/IIB supergravity. The same applies to the compacti(cid:12)cations of these theories to lower dimensions. The other way round is less clear: not every supergravity theory has necessarily a string or M{theory origin. A well-known example of a supergravity theory whose role in string the- ory was unclear until a few years ago is the D=10massive supergravity theory of Romans [1]. It was pointed out by Romans that the D=10 (massless) IIA supergravity theory [2,3] can be deformed into a massive supergravity with mass parameter m . The role of this mas- R sive supergravity within string theory has become clear only after the introduction of the D{branes, in particular the D8{brane [4]. An interesting feature of the massive supergravity of Romans is that the Lagrangian possesses a dilaton potential proportional to m2 which R acts as an e(cid:11)ective cosmological constant. Due to this scalar potential the massive super- gravity, unlike the massless case, does not allow a maximally supersymmetric Minkowski spacetime as a vacuum solution. Instead, the scalar potential leads to the possibility of a half-supersymmetric domain wall solution interpolating between di(cid:11)erent values of the cos- mological constant. Such a solution indeed exists [5,6] and is identi(cid:12)ed as the D8-brane of [4]. The massive supergravity of [1] is not a gauged supergravity and, at the (cid:12)eld theory level, has no D=11origin1. The only candidate symmetry of the Lagrangianto begauged is a rigid R+ symmetry (see Table 2). However, the Ramond-Ramond gauge vector has a nontrivial weight under this R+ symmetry and this leads to inconsistencies with the supersymmetry algebra. There does exist another massive deformation of D=10 IIA supergravity, with mass parameter m , which is a gauged supergravity and does have a D=11 origin [8,9]. However, 11 it can only be de(cid:12)ned at the level of the equations of motion. The Ramond-Ramond gauge vector has weight zero with respect to the R+ group that is gauged (see Table 2) and in this case there are no inconsistencies with the supersymmetry algebra. The role of this second massive deformation within string theory is not (yet) clear. An interesting feature of the theory is that it allows for a (non-supersymmetric) de Sitter space solution [9]. The possible physical signi(cid:12)cance of this de Sitter space solution has been discussed in [10,11]. A common feature of the D=10 massive supergravity of [1] and the D=10 gauged su- pergravity of [8,9] is that there is a dilaton potential which is proportional to the square of the mass parameter, m2 and m2 , respectively. Due to this scalar potential the D=10 R 11 Minkowski spacetime is no longer a maximally supersymmetric vacuum solution of the the- ory. Instead one can look for half-supersymmetric vacuum solutions. A natural class of half- supersymmetric solutions that makes use of the scalar potential is the set of domain-wall solutions, like the D8{brane mentioned above. Recently, domain wall solutions of lower- dimensional supergravities have attracted attention in view of their relevance for a super- symmetric Randall-Sundrum scenario [12,13], the domain-wall/QFT correspondence [14,15] 1We assume that we are not using the existence of extra Killing vectors, like in [7]. 1 and applications to cosmology [16,17]. In all these applications the properties of the domain walls play a crucial role and these properties are determined by the details of the scalar potential. Motivated by this we studied in a previous paper general domain wall solutions in D=9 dimensions [18]2. We took D=9 because on the one hand this case shares some of the complexities of the lower-dimensional cases, on the other hand the scalar potential for this case is simple enough to study the corresponding domain-wall solutions in full detail. The supergravity theory we considered in [18] was obtained by a generalized Scherk-Schwarz reduction of D=10 IIB supergravity. This is not the most general possibility in D=9. The aim of this paper is to make a systematic search for massive deformations of the unique D=9, N=2 supergravity theory. All deformations we (cid:12)nd correspond to gauged supergravities. Such supergravities have a gauge symmetry which reduces, for constant values of the gauge parameter, to a nontrivial rigid symmetry. The hope is that the D=9 case will teach us something about the more complicated situation in D < 9 dimensions. In the (cid:12)rst part of this paper we will present in two steps the D=9 gauged supergravities we have found. In a (cid:12)rst step we will present seven massive deformations with a single mass parameter m, all giving rise to gauged supergravities. All of them are obtained by generalized dimensional reduction [22] from a higher-dimensional theory (11D, IIA or IIB supergravity). The consistency of these 9D gauged supergravities is guaranteed by their higher{dimensional origin. The gauge groups we encounter are either3 SO(2), SO(1;1)+, R (all of which are subgroups of SL(2;R) with Cartan-Killing metric diag(1,1), diag(1,{1) and diag(1,0), respectively [23,24]), R+ or the two{dimensional non{Abelian Lie group CR1. The latter consists of so-called ‘collinear transformations’ [25] (scalings and translations) in one dimension and forms a non{semi{simple Lie group. In a second step we will consider combinations of these seven massive deformations. The closure of the supersymmetry algebra will be guaranteed based on a linearity argument but it turns out that non-linear restrictions enter via the back door. Satisfying these restrictions leaves us with (cid:12)ve di(cid:11)erent two{parameter deformations (rather than the seven{parameter deformation that one could have if there were no non-linear restrictions). These are the most general gauged supergravities we construct in this paper. In the second part of this paper we make a systematic search for vacuum solutions of the supergravities we have obtained. The existence of such vacuum solutions is needed in order to de(cid:12)ne the spectrum of the theory as fluctuations around this vacuum. Our search includes half-supersymmetric domain wall solutions and non-supersymmetric de Sitter spaces. Throughout the paper we will reduce (cid:12)eld equations rather than Lagrangians. The reason for this is the fact that (some of) the rigid symmetries we employ for Scherk{Schwarz reduction scale the Lagrangian. As was noted by [9] and is illustrated by the SS reduction of a simple toy model in Appendix B, Scherk{Schwarz reduction with a symmetry that scales the Lagrangian can only be performed at the level of the (cid:12)eld equations. Reduction of the 2For earlier discussions of domain wall solutions in D=9 dimensions, see [19,20]. For a more recent discussion, see [21]. 3ThroughoutthepaperwewillusethenotationSO(1;1)+ ratherthan(theisomorphic)R+ forthescaling symmetry that is a subgroup of SL(2;R). The di(cid:11)erent notation is used to emphasize the di(cid:11)erent origin. 2 Lagrangian itself gives rise to the wrong equations. In fact, the reduced (cid:12)eld equations can not be obtained as Euler{Lagrange equations of any Lagrangian. This paper is organized as follows. In Section 2 we briefly review the situation in D=11 dimensions where no massive deformation has been constructed so far. This case is needed for the discussion of the D=10 and D=9 cases. In Section 3 we discuss the two di(cid:11)erent massive deformations of D=10 IIA supergravity. In Section 4 we review the case of D=10 IIB supergravity. This case does not allow massive deformations but will be needed for the discussion of the D=9 case. In Section 5 we present 7 massive deformations of the maximally supersymmetric D=9 supergravity theory. They all are gauged supergravities with gauge group SO(2), SO(1;1)+, R, R+ or CR1. In Section 6 we show, by combining the di(cid:11)erent gauged supergravities, that there exist (cid:12)ve di(cid:11)erent two{parameter massive supergravity theories. In Section 7 we make a systematic search for vacuum solutions of the supergravities we have obtained. Finally, in Section 8 we give our conclusions. In particular, we discuss which of the D=9 supergravities can be considered as candidate low-energy limits of (compacti(cid:12)ed) superstring theory. We give four Appendices. Appendix A contains our conventions. Appendix B discusses the Scherk{Schwarz reduction of a dilaton{gravity toy model. Appendix C contains the supersymmetry transformations of massless D=11, 10 and 9 supergravity plus the reduction Ansa¨tze to go from D=11 to D=10 to D=9. Finally, in Appendix D we discuss some manipulations with spinors and gamma-matrices in ten and nine dimensions. 2 D=11 Supergravity We (cid:12)rst consider eleven-dimensional supergravity. Its (cid:12)eld content is given by4 D=11: f^e^(cid:22)ˆˆaˆˆ;C^^(cid:22)ˆˆ(cid:23)ˆˆ(cid:26)ˆˆ; ^^(cid:22)ˆˆg: (2.1) The D=11 supersymmetry transformations are given in Appendix C, see eq. (C.1). These supersymmetry rules are covariant under an R+ symmetry with parameter (cid:11)^^ [26]. The weights of the D=11 (cid:12)elds under this R+ are given in Table 1. Note that the Lagrangian is not invariant but scales with weight w = 9. Therefore this R+ isa symmetry of the equations of motion only. R+ ^e^(cid:22)ˆˆaˆˆ C^(cid:22)ˆˆ(cid:23)ˆˆ(cid:26)ˆˆ ^^(cid:22)ˆˆ ^^(cid:15) L^^ ^ 1 1 (cid:11)^ 1 3 9 2 2 Table 1: The R+{weights of the D=11 supergravity (cid:12)elds, the supersymmetry parameters ^(cid:15)^ ^ and the Lagrangian L^. 4InordertodistinguishbetweenD=11,D=10andD=9weindicateD=11(cid:12)eldsandindiceswithadouble hat, D=10 (cid:12)elds and indices with a single hat and D=9 (cid:12)elds and indices with no hat. 3 No massive deformation of the eleven-dimensional supergravity theory is known. In particular, no cosmological constant can be added [27]. One problem with a D=11 super- symmetric cosmological constant is that its reduction gives rise to a D=10 cosmological constant with a dilaton coupling that di(cid:11)ers from Romans’ massive deformation. A gen- eral deformation of D=11 supergravity involving the use of extra Killing vectors has been considered in [28]. We will not consider this possibility in this paper. 3 Massive Deformations of D=10 IIA Supergravity A Kaluza-Klein reduction of the eleven-dimensional theory yields the IIA theory in ten dimensions5. The (cid:12)eld content of the D=10 IIA supergravity theory is given by D=10 IIA: fe^ aˆ;B^ ;(cid:30)^;A^ ;C^ ; ^ ;(cid:21)^g: (3.1) (cid:22)ˆ (cid:22)ˆ(cid:23)ˆ (cid:22)ˆ (cid:22)ˆ(cid:23)ˆ(cid:26)ˆ (cid:22)ˆ The supersymmetry transformations rules are given in eq. (C.5). For later purposes we indicate these (undeformed) supersymmetry transformations by (cid:14) . The transformation 0 rules have two R+{symmetries, one with parameter (cid:11)^ that scales the Lagrangian and one ^ with parameter (cid:12) that leaves the Lagrangian invariant. The (cid:12)rst symmetry follows via dimensional reduction from the D=11 R+{symmetry with parameter (cid:11)^^. The weights of these two R+{symmetries are given in Table 2. The gauge symmetry associated to the ^ Ramond-Ramond vector, with parameter (cid:21), reads A^ ! A^−d(cid:21)^; C^ ! C^ −d(cid:21)^B^: (3.2) R+ e^ aˆ B^ e(cid:30)ˆ A^ C^ ^ (cid:21)^ (cid:15)^ L^ Origin (cid:22)ˆ (cid:22)ˆ(cid:23)ˆ (cid:22)ˆ (cid:22)ˆ(cid:23)ˆ(cid:26)ˆ (cid:22)ˆ (cid:11)^ 9 3 3 0 3 9 − 9 9 9 (cid:11)^^ 8 2 16 16 16 (cid:12)^ 0 1 1 −3 −1 0 0 0 0 2 4 4 Table 2: The R+{weights of the D=10 IIA supergravity (cid:12)elds, the supersymmetry parameter (cid:15)^ and the Lagrangian L^. The D=10 IIA supergravity theory allows two massive deformations which we discuss one by one below. 3.1 Deformation m : D=10 massive supergravity R The (cid:12)rst massive deformation, with mass parameter m , is due to Romans [1]. In this case R (the same is true for all other cases) the supersymmetry transformations receive two types of massive deformations: explicit and implicit ones. The explicit deformations are terms, at most linear in m , that are added to the original supersymmetry rules. These explicit R 5We have used the reduction Ansa¨tze (C.4) with m11 =0. 4 ^ deformations are denoted by (cid:14) and are given, in terms of a superpotential W((cid:30)) and mR derivatives thereof, by ( (cid:14) ^ = −1WΓ^ (cid:15)^; with W = 1e5(cid:30)ˆ=4m ; m : mR (cid:22)ˆ 8 (cid:22)ˆ 4 R (3.3) R (cid:14) (cid:21)^ = 4(cid:14)W^(cid:15): mR (cid:14)(cid:30)ˆ There are further implicit massive deformations to the original supersymmetry rules (cid:14) , 0 which are given in eq. (C.5), due to the fact that in these rules one must replace all (cid:12)eld strengths by corresponding massive (cid:12)eld strengths which are given by F^ = dA^+m B^; H^ = dB^; G^ = dC^ +A^H^ + 1m B^B^: (3.4) R 2 R TheLagrangiancontainstermslinearandquadraticinm . Againthereareimplicitdeforma- R tions, via the massive (cid:12)eld strengths, and explicit deformations. The explicit deformations quadratic in the mass parameter de(cid:12)ne the scalar potential which can be written in terms ^ of the superpotential W((cid:30)) and derivatives thereof. Requiring closure of the supersymmetry algebra one (cid:12)nds the linear deformations of the fermionic (gravitino and dilatino) (cid:12)eld equations in Roman’s theory: ( X ( ^(cid:22)ˆ) (cid:17) m e5(cid:30)ˆ=4Γ^(cid:22)ˆ(cid:23)ˆ(1 ^ + 5 Γ^ (cid:21)^); m : mR R 4 (cid:23)ˆ 288 (cid:23)ˆ (3.5) R X ((cid:21)^) (cid:17) m e5(cid:30)ˆ=4Γ^(cid:23)ˆ(−5 ^ − 21 Γ^ (cid:21)^): mR R 4 (cid:23)ˆ 160 (cid:23)ˆ The undeformed equations, X ( ^(cid:22)ˆ) and X ((cid:21)^), are given in eqs. (C.7). 0 0 Under supersymmetry the fermionic (cid:12)eld equations, X + X , transform into the de- 0 mR formed bosonicequations ofmotion. Since we will onlybe interested in(cid:12)nding solutions that are carried by the metric and the scalars it is convenient to truncate away all bosonic (cid:12)elds except the metric and the dilaton6. After this truncation we (cid:12)nd that under supersymmetry the fermionic (cid:12)eld equations transform into ((cid:14) +(cid:14) )(X +X )( ^(cid:22)ˆ) = 1Γ^(cid:23)ˆ(cid:15)^[R^(cid:22)ˆ − 1R^g^(cid:22)ˆ − 1(@(cid:22)ˆ(cid:30)^)(@ (cid:30)^)+ 1(@(cid:30)^)2g^(cid:22)ˆ + 1m2e5(cid:30)ˆ=2g^(cid:22)ˆ ]; 0 mR 0 mR 2 (cid:23)ˆ 2 (cid:23)ˆ 2 (cid:23)ˆ 4 (cid:23)ˆ 4 R (cid:23)ˆ ((cid:14) +(cid:14) )(X +X )((cid:21)^) = (cid:15)^[(cid:50)(cid:30)^− 5m2e5(cid:30)ˆ=2]: (3.6) 0 mR 0 mR 4 R At the right-hand side we thus (cid:12)nd the Romans’ bosonic (cid:12)eld equations for the metric and the dilaton, one solution of which is the D8-brane. Note that the bosonic (cid:12)eld equations contain terms quadratic in the mass parameter. Romans’ theory is not known to have a higher-dimensional supergravity origin. Neither ^ is it a gauged supergravity. A candidate symmetry of the Lagrangian to be gauged is the (cid:12) symmetry of Table 2. However, the candidate gauge (cid:12)eld A^ has a nontrivial weight under (cid:22)ˆ (cid:12)^. This means that the curl dA^ transforms with a non-covariant term proportional to d(cid:21)^A^. Such a term cannot be cancelled by adding an extra term, such as B^, to the de(cid:12)nition of the A^ curvature. In short, the (cid:12)^ symmetry cannot be gauged [3]. The same Table shows that on the other hand A^ has weight zero under the (cid:11)^{symmetry which is a symmetry of (cid:22)ˆ the equations of motion only. This (cid:11)^{symmetry can indeed be gauged at the level of the equations of motion. This gauging leads to the D=10 gauged supergravity discussed below. 6Note that a further truncation to (cid:30)=c is inconsistent. 5 3.2 Deformation m : D=10 gauged supergravity 11 The second massive deformation, with mass parameter m , has been considered in [8,9] 11 and is a gauged supergravity. It can be obtained by generalized Scherk-Schwarz reduction of D=11 supergravity using the R+ symmetry (cid:11)^^ of Table 1 [9]. The corresponding reduction Ans¨atze, with m 6= 0, are given in eq. (C.4). This reduction leads to the following explicit 11 massive deformations of the D=10 IIA supersymmetry rules: ( (cid:14) ^ = 9 m e−3(cid:30)ˆ=4Γ^ Γ (cid:15)^; m : m11 (cid:22)ˆ 16 11 (cid:22)ˆ 11 (3.7) 11 (cid:14) (cid:21)^ = 3m e−3(cid:30)ˆ=4Γ (cid:15)^: m11 2 11 11 The implicit massive deformations of the original supersymmetry rules (cid:14) are given by the 0 massive bosonic (cid:12)eld strengths ^ ^ 3 ^ ^ ^ ^ ^ ^ ^ ^ ^^ D(cid:30) = d(cid:30)+ m A; F = dA; H = dB +3m C; G = dC +AH; (3.8) 2 11 11 while the covariant derivative of the supersymmetry parameter is given by D ^(cid:15) = (@ +!^ + 9 m Γ^ A=^)(cid:15)^: (3.9) (cid:22)ˆ (cid:22)ˆ (cid:22)ˆ 16 11 (cid:22)ˆ The gauge vector in the de(cid:12)nition of the covariant derivative is required to make the deriva- tive of the supersymmetry parameter and the spin connection R+{covariant. The linear deformations of the fermionic (cid:12)eld equations read in this case ( X ( ^(cid:22)ˆ) (cid:17) m e−3(cid:30)ˆ=4Γ Γ^(cid:22)ˆ(cid:23)ˆ(−9 ^ + 17Γ^ (cid:21)^); m : m11 11 11 2 (cid:23)ˆ 48 (cid:23)ˆ (3.10) 11 X ((cid:21)^) (cid:17) m e−3(cid:30)ˆ=4Γ Γ^(cid:23)ˆ(3 ^ − 9 Γ^ (cid:21)^): m11 11 11 2 (cid:23)ˆ 16 (cid:23)ˆ We (cid:12)rst consider the truncation that all bosonic (cid:12)elds except the metric and the dilaton are set equal to zero. Under supersymmetry the fermionic (cid:12)eld equations transform into (cid:2) ((cid:14) +(cid:14) )(X +X )( ^(cid:22)ˆ) = 1Γ^(cid:23)ˆ^(cid:15) R^(cid:22)ˆ − 1R^g^(cid:22)ˆ − 1(@(cid:22)ˆ(cid:30)^)(@ (cid:30)^)+ 1(@(cid:30)^)2g^(cid:22)ˆ + 0 m11 0 m11 2 (cid:23)ˆ 2 (cid:23)ˆ 2(cid:3) (cid:23)ˆ 4 (cid:23)ˆ +36m2 e−3(cid:30)ˆ=2g^(cid:22)ˆ +Γ (cid:15)^[3m e−3(cid:30)ˆ=4@(cid:22)ˆ(cid:30)^]; 11 (cid:23)ˆ 11 11 ((cid:14) +(cid:14) )(X +X )((cid:21)^) =(cid:15)^[(cid:50)(cid:30)^]+Γ^(cid:23)ˆΓ ^(cid:15)[9m e−3(cid:30)ˆ=4@ (cid:30)^]: (3.11) 0 m11 0 m11 11 11 (cid:23)ˆ ThetermsinvolvingΓ arepartofthevector(cid:12)eldequation. Therefore,toobtainaconsistent 11 truncation, wemustfurthertruncatethedilatontozero. Oneisthenleftwithonlythemetric satisfying the Einstein equation with a positive cosmological constant, a solution of which is 10D de Sitter space [9]. The reduced theory is a gauged supergravity where the R+ symmetry (cid:11)^ of Table 2 has beengauged. Inparticular, thegaugeparameterandtransformationoftheRamond-Ramond potentials read as follows7: (cid:11)^ : (cid:3) = ewαˆm11(cid:21)ˆ with A^ ! A^−d(cid:21)^; C^ ! e3m11(cid:21)ˆ(C^ −d(cid:21)^B^); (3.12) 7It is understood that each (cid:12)eld with wαˆ 6=0 is multiplied by (cid:3). 6 where w are the weights under (cid:11)^. We note that one can take two di(cid:11)erent limits of the (cid:11)^ (cid:11)ˆ gauge transformations. First, the limit m ! 0 leads to the massless gauge transformations 11 (3.2). Note that C^ transforms trivially under this gauge symmetry in the sense that C^ can be made gauge-invariant after a simple (cid:12)eld-rede(cid:12)nition. Secondly, one can take the limit that (cid:11)^ is constant. This leads to the ungauged R+ (cid:11)^{symmetry of Table 2. A noteworthy feature of the D=10 gauged supergravity is that no Lagrangian can be de(cid:12)ned for it. In the search for supersymmetric domain wall solutions in D=5 dimensions many other examples of gauged supergravity theories without a Lagrangian have been given [29]. Note that one can write down a Lagrangian for the ungauged theory. The reason that one cannot write down a Lagrangianafter gauging is that the symmetry that is gauged is not a symmetry of the Lagrangian but only of the equations of motion. It would be instructive to construct the D=10 gauged supergravity from the ungauged theory by gauging the (cid:11)^{ symmetry. Apparently, it shows that one can gauge symmetries that leave a Lagrangian invariant up to a scale factor. 4 D=10 IIB Supergravity The other ten-dimensional supergravity theory is chiral IIB. Its (cid:12)eld content is D=10 IIB: fe^ aˆ;(cid:30)^;(cid:31)^;B^(1);B^(2);D^ ; ^ ;(cid:21)^g: (4.1) (cid:22)ˆ (cid:22)ˆ(cid:23)ˆ (cid:22)ˆ(cid:23)ˆ (cid:22)ˆ(cid:23)ˆ(cid:26)ˆ(cid:27)ˆ (cid:22)ˆ The supersymmetry variations are given in eq. (C.10). The IIB supersymmetry rules trans- form covariant under the SL(2;R) transformations (omitting indices): (cid:18) (cid:19) a(cid:28)^+b ~ ~ a b (cid:28)^ ! ; B^ ! ΩB^; D^ ! D^ ; with Ω = 2 SL(2;R); c(cid:28)^+d c d (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) c(cid:28)^(cid:3) +d 1=4 c(cid:28)^(cid:3) +d 3=4 c(cid:28)^(cid:3) +d 1=4 ^ ! ^ ; (cid:21)^ ! (cid:21)^; (cid:15)^! (cid:15)^: (4.2) (cid:22)ˆ (cid:22)ˆ c(cid:28)^+d c(cid:28)^+d c(cid:28)^+d (cid:0) (cid:1) We have used here the vector notation B~^ = B^(1);B^(2) T. The group SL(2;R) contains a set of three one-parameter conjugacy classes de(cid:12)ning one compact and two non{compact subgroups. Since they are needed later we will describe them shortly. Each of the subgroups is generated by a SL(2;R) group element Ω with detΩ = 1. 1. One non{compact subgroup R is generated by (cid:18) (cid:19) ^ Ω = e12(cid:16)ˆ((cid:27)1+i(cid:27)2) = 1 (cid:16) : (4.3) p 0 1 Each element de(cid:12)nes a parabolic conjugacy class with TrΩ = 2. These parabolic transformations leave the combination (B^(2))2 invariant. Therefore the Cartan-Killing metric is given by diag(0,1). The action of the R (cid:16)^{symmetry on the (cid:12)elds can not be expressed by assigning weights to the standard basis of (cid:12)elds given in (4.1). 7 2. An SO(1;1)+ subgroup which is generated by elements (cid:18) (cid:19) eγˆ 0 Ω = eγˆ(cid:27)3 = : (4.4) h 0 e−γˆ Each element de(cid:12)nes a hyperbolic conjugacy class with TrΩ > 2. These hyperbolic transformations leave the combination B^(1)B^(2) invariant. After diagonalization this leads to a Cartan-Killing metric given by diag(1,{1). The weights corresponding to the SO(1;1)+ γ^{symmetry are given in Table 3. 3. There is a SO(2) subgroup which is generated by elements Ω of SL(2;R) with (cid:18) (cid:19) ^ ^ ˆ cos(cid:18) sin(cid:18) Ω = ei(cid:18)(cid:27)2 = : (4.5) e −sin(cid:18)^ cos(cid:18)^ Each element de(cid:12)nes an elliptic conjugacy class with TrΩ < 2. The elliptic transfor- mations leave (B^(1))2+(B^(2))2 invariant. After diagonalization this leads to a Cartan- ^ Killing metric given by diag(1,1). The action of the SO(2) (cid:18){symmetry on the (cid:12)elds can not be expressed by assigning weights to the standard real basis of (cid:12)elds given in (4.1). Table 3 contains the weights of the γ^{symmetry de(cid:12)ned above8 and of a new R+ symmetry (cid:14)^ which is not a subgroup of SL(2;R) and that does not leave the Lagrangian invariant. One could combine SL(2;R) with this new R+ into a GL(2;R) symmetry that leaves the IIB equations of motion invariant. Its action is the product of the two separate transformations: Ω~ = Ω(cid:3)ˆ. This exhausts all the symmetries of D=10 IIB supergravity. (cid:14) R+ e^ aˆ e(cid:30)ˆ (cid:31)^ B^(1) B^(2) D^ ^ (cid:21)^ (cid:15)^ L^ symmetry (cid:22)ˆ (cid:22)ˆ(cid:23)ˆ (cid:22)ˆ(cid:23)ˆ (cid:22)ˆ(cid:23)ˆ(cid:26)ˆ(cid:27)ˆ (cid:22)ˆ γ^ 0 −2 2 1 −1 0 0 0 0 0 SO(1;1)+ (cid:14)^ 1 0 0 2 2 4 1 −1 1 8 R+ 2 2 2 Table 3: The scaling weights of the D=10 IIB supergravity (cid:12)elds, the supersymmetry param- eter (cid:15)^ and the Lagrangian L^. The IIB supergravity theory is not known to have massive deformations. One of the reasons for this is that there is no candidate vector (cid:12)eld like in the IIA case. = 2 5 Massive deformations of D=9, N Supergravity The Kaluza-Klein reduction of either (massless) IIA or IIB supergravity gives the unique D = 9, N = 2 massless supergravity theory. Its (cid:12)eld content is given by D=9: fe a;(cid:30);’;(cid:31);A ;A(1);A(2);B(1);B(2);C ; ;(cid:21);(cid:21)~g: (5.1) (cid:22) (cid:22) (cid:22) (cid:22) (cid:22)(cid:23) (cid:22)(cid:23) (cid:22)(cid:23)(cid:26) (cid:22) 8The other two symmetries de(cid:12)ned above cannot be de(cid:12)ned in terms of weights of real (cid:12)elds only. 8 Thesupersymmetry rulesaregivenineq.(C.16). Themassless 9{dimensionaltheoryinherits several global symmetries from its parents: two R+ symmetries (cid:11);(cid:12) from IIA supergravity and one R+ symmetry (cid:14) plus a full SL(2;R) symmetry from IIB supergravity. The latter leads in particular to an SO(2) symmetry (cid:18), an SO(1;1)+ symmetry γ and an R{symmetry (cid:16). Theweights ofallthesesymmetries, except fortheSO(2)(cid:18){symmetry andR(cid:16){symmetry, and their higher-dimensional origin are given in Table 4 (see also [26]). R+ e a e(cid:30) e’ (cid:31) A A(1) A(2) B(1) B(2) C (cid:21) (cid:21)~ (cid:15) L Origin (cid:22) (cid:22) (cid:22) (cid:22) (cid:22)(cid:23) (cid:22)(cid:23) (cid:22)(cid:23)(cid:26) (cid:22) (cid:11) 9 0 p6 0 3 0 0 3 3 3 9 − 9 − 9 9 9 11D 7 7 14 14 14 14 p (cid:12) 0 3 7 -3 1 −3 0 −1 1 −1 0 0 0 0 0 IIA 4 4 4 2 4 4 2 4 γ 0 −2 0 2 0 1 −1 1 −1 0 0 0 0 0 0 IIB (cid:14) 8 0 −p4 0 0 2 2 2 2 4 4 −4 −4 4 8 IIB 7 7 7 7 7 7 Table 4: The scaling weights of the 9 dimensional supergravity (cid:12)elds, the supersymmetry parameter (cid:15) and the Lagrangian L. It turns out that only three out of the four scalings given in Table 4 are linearly inde- pendent. There is a relation 4(cid:11)− 8(cid:12) = γ + 1(cid:14): (5.2) 9 3 2 We observe the following pattern. Using (5.2) to eliminate one of the scaling{symmetries we are left with three independent scaling{symmetries. Each of the three gauge (cid:12)elds (1) (2) A ;A ;A has weight zero under two (linear combinations) of these three symmetries: (cid:22) (cid:22) (cid:22) one is a symmetry of the action, the other is a symmetry of the equations of motion only. The D=9 SL(2;R) symmetry acts in the following way: (cid:18) (cid:19) a(cid:28) +b a b (cid:28) ! ; A~ ! ΩA~; B~ ! ΩB~ ; with Ω = 2 SL(2;R); c(cid:28) +d c d (cid:18) (cid:19) (cid:18) (cid:19) c(cid:28)(cid:3) +d 1=4 c(cid:28)(cid:3) +d 3=4 ! ; (cid:21) ! (cid:21); (cid:22) (cid:22) c(cid:28) +d c(cid:28) +d (cid:18) (cid:19) (cid:18) (cid:19) c(cid:28)(cid:3) +d −1=4 c(cid:28)(cid:3) +d 1=4 (cid:21)~ ! (cid:21)~; (cid:15) ! (cid:15); (5.3) c(cid:28) +d c(cid:28) +d while ’ and C are invariant. We have used a vector notation for the two vectors and two antisymmetric tensors, like in D=10. Againone can combine SL(2;R) with anR+ symmetry to form GL(2;R) with parameter Ω~ = Ω(cid:3) . R+ In addition to the global symmetries there is a number of local symmetries. In particular, the gauge transformations of the vectors read A(1) ! A(1) −d(cid:21)(1); A(2) ! A(2) −d(cid:21)(2); A ! A−d(cid:21); B~ ! B~ −A~d(cid:21): (5.4) 9

via Scherk–Schwarz reduction and correspond to gauged supergravities. The massive supergravity of [1] is not a gauged supergravity and, at the field theory
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