Elsevier

Nuclear Physics B

Volume 651, Issues 1–2, 17 February 2003, Pages 263-290
Nuclear Physics B

Gauged/massive supergravities in diverse dimensions

https://doi.org/10.1016/S0550-3213(02)01125-2Get rights and content

Abstract

We show how massive/gauged maximal supergravities in 11−n dimensions with SO(nl,l) gauge groups (and other non-semisimple subgroups of Sl(n,R)) can be systematically obtained by dimensional reduction of “massive 11-dimensional supergravity”. This series of massive/gauged supergravities includes, for instance, Romans' massive N=2A, d=10 supergravity for n=1, N=2,d=9 SO(2) and SO(1,1) gauged supergravities for n=2, and N=8,d=5 SO(6−l,l) gauged supergravity. In all cases, higher p-form fields get masses through the Stückelberg mechanism which is an alternative to self-duality in odd dimensions.

Introduction

Massive/gauged supergravities are very interesting theories which have become fashionable1 due to the relation between the presence in their Lagrangians of mass/gauge coupling parameters with the existence of domain-wall-type solutions and the holographic relation between (in general non-conformal) the gauge field theories that live on the domain wall and the superstring/supergravity theories that live in the bulk [2], [3] (for a review, see Ref. [4]).

These supergravity theories appear in the literature in essentially 3 different ways in the compactification of ungauged (massless) supergravities:

  • (1)

    In compactifications in non-trivial internal manifolds (particularly Freund–Rubin-type [5] spontaneous compactifications on spheres). Some notable examples are the S7 [6] compactification of 11-dimensional supergravity that is supposed to give the SO(8)-gauged N=8,d=4 supergravity, the S4 compactification of 11-dimensional supergravity [7] that gives [8] the SO(5)-gauged N=4,d=7 theory, and the S5 compactification of N=2B,d=10 supergravity [9], [10], [11] that gives the SO(6)-gauged N=8,d=5 supergravity theory;

  • (2)

    In Scherk–Schwarz generalized dimensional reductions [12], in which the global symmetry is geometrical or non-geometrical.

    Examples in which a global geometrical SU(2) symmetry has been used to obtain a gauged/massive supergravity are Salam and Sezgin's compactification of 11-dimensional supergravity to obtain SU(2)-gauged N=4,d=8 supergravity [13], and Chamseddine and Volkov's obtention of SU(2)×SU(2)-gauged N=4,d=4 supergravity [14], [15] and Chamseddine and Sabra's obtention of and SU(2)-gauged N=2,d=7 supergravity [16] from N=1,d=10 supergravity in both cases.

    An example in which a global symmetry of non-geometrical origin is used to obtain a gauged/massive supergravity by generalized dimensional reduction is the obtention of massive N=2,d=9 supergravity from N=2B,d=10 supergravity exploiting the axion's shift symmetry [17]. If one exploits the full global Sl(2,R) symmetry of the N=2B,d=10 theory one obtains a 3-parameter family of supergravity theories [18] (see also [19]) some of which are gauged supergravities [20]. From the string theory point of view, the three parameters take discrete values which must be considered equivalent when they are related by an Sl(2,Z) duality transformation (i.e., when they belong to the same conjugacy class) and they describe the low-energy limit of the same string theory [21]. There is, actually, an infinite number of Sl(2,Z) conjugacy classes and for each of them one gets a massive/gauged supergravity with either SO(2), SO(1,1) or no gauge group [22].2 We will take a closer look later to this Sl(2,Z) family of theories;

  • (3)

    In compactifications with non-trivial p-form fluxes (see, e.g., [25]).

These three instances are not totally unrelated. To start with, compactifications with fluxes can be understood as non-geometrical Scherk–Schwarz reductions in which the global symmetry exploited is the one generated by p-form “gauge” transformations with constant parameters. The axion shift symmetry can be understood as the limit case p=−1 and can be used in compactifications on circles. Higher p-form fluxes can only be exploited in higher-dimensional internal spaces that can support them. On the other hand, the geometrical Scherk–Schwarz compactifications used by Salam and Sezgin, and Chamseddine, Volkov and Sabra could be understood as compactification on the SU(2) group manifold S3 although one would expect a gauge group SO(4)∼SU(2)×SU(2) since this is the isometry group of the S3 metric used.

Finally, the Freund–Rubin spontaneous sphere compactifications are compactifications on a brane background (more precisely, in a brane's near-horizon geometry) and there is a net flux of the form associated to the brane, while the non-geometrical Scherk–Schwarz compactifications can also be seen as compactifications on a (d−3)-brane background [18], in which the brane couples to the (d−2)-form potential dual to the scalar.

Historically, almost all the gauged/massive theories we just discussed had been constructed by gauging or mass-deforming known ungauged/massless theories3 the only exception being the N=2,d=9 theories that, in principle, could have been constructed in that way as well.

A crucial ingredient in the gauging of some of the higher-dimensional supergravities with p-form fields transforming under the global symmetry being gauged is that these fields must be given a mass whose value is related by supersymmetry to the gauge coupling parameter: if the p-form fields remained massless, they should transform simultaneously under their own massless p-form gauge transformations (to decouple negative-norm states) and under the new gauge transformations, which is impossible. A mass term eliminates the requirement of massless p-form gauge invariance but introduces another problem, because the number of degrees of freedom of the theory should remain invariant. In the cases of the SO(5)-gauged N=4,d=7 and the SO(6)-gauged N=8,d=5 theories this was achieved by using the “self-duality in odd dimensions” mechanism [34], [35] which we will explain later on.

The need to introduce mass parameters together with the gauge coupling constant is one of the reasons why we call these theories gauged/massive supergravities. In some cases no mass parameters will be needed in the gauging and in some others no gauge symmetry will be present when the mass paramaters are present but they can nevertheless be seen as members of the same class of theories. Another reason is that in many cases the gauge parameter has simultaneously the interpretation of gauge coupling constant and mass of a domain-wall solution of the theory4 that can correspond to the near-horizon limit of some higher-dimensional brane solution apart from that of the mass of a given field in the Lagrangian.

The gauging and mass-deformation procedures are very effective tools to produce gauged theories in a convenient form but hide completely their possible higher-dimensional or string/M-theorical origin. In fact, there are many gauged/massive supergravity theories whose string- or M-theoretical origin is still unknown, which, in supergravity language means that we do not know how to obtain them by some compactification procedure from some higher-dimensional (ungauged/massless) theory. In some cases, it is known how to obtain it from N=2B,d=10 supergravity but not from the N=2A,d=10 or 11-dimensional supergravity. A notorious example is Romans' massive N=2A,d=10 supergravity [33] that cannot be obtained from standard 11-dimensional supergravity (a theory that cannot be deformed to accommodate a mass parameter preserving 11-dimensional Lorentz invariance [36], [37], [38]) by any sort of generalized dimensional reduction, but there are many more. Let us review some other examples:

  • (1)

    The Sl(2,Z) family of N=2,d=9 gauged/massive supergravities are obtained by Scherk–Schwarz reduction of the N=2B,d=10 theory, but it is not known how to obtain them from standard 11-dimensional or N=2A,d=10 supergravity.

    In these theories, the Sl(2,R) doublet of 2-form potentials gets masses through the Stückelberg mechanism;

  • (2)

    The massless N=4,d=8 supergravity contains two SU(2) triplets of vector fields. The two triplets are related by Sl(2,R) S-duality transformations. It should be possible to gauge SU(2) using as SU(2) gauge fields any of the two triplets. If we gauged the triplet of Kaluza–Klein vectors, we would get the theory that Salam and Sezgin obtained by Scherk–Schwarz reduction of 11-dimensional supergravity. It is not known how to derive from standard 11-dimensional supergravity the “S-dual” theory that one would get gauging the other triplet, that comes from the 11-dimensional 3-form.

    In these two theories, the SU(2) triplet of 2-form potentials gets masses through the Stückelberg mechanism, eating the 3 vectors that are not SU(2)-gauged;

  • (3)

    The SO(5)-gauged N=4,d=7 supergravity theory is also just a particular member of the family of SO(5−l,l) gauged N=4,d=7 supergravities constructed in Ref. [39]. The 11-dimensional origin of the SO(5) theory is well understood, but not that of the theories with non-compact gauge group;

  • (4)

    The SO(6)-gauged N=8,d=5 supergravity theory is also just a particular member of the family of SO(6−l,l) gauged N=8,d=5 supergravities constructed in Refs. [29], [30]. Again, while the N=2B,d=10 origin of the SO(6) theory is well understood, its 11-dimensional origin and the higher-dimensional origin of the theories with non-compact gauge groups is unknown;

  • (5)

    Essentially the same can be said about the SO(8)-gauged N=8,d=4 supergravity theory since it is possible to generate from it by analytical continuation theories with non-compact groups SO(8−l,l) [40] whose higher-dimensional origin is also unknown.

In the search for an 11-dimensional origin of Romans' massive N=2A,d=10 theory a “massive 11-dimensional supergravity” was proposed in Ref. [41]. This theory is a deformation of the standard 11-dimensional supergravity that contains a mass parameter and, to evade the no-go theorem of Refs. [36], [37], [38], a Killing vector in the Lagrangian that effectively breaks the 11-dimensional Lorentz symmetry to the 10-dimensional one even if the theory is formally 11-dimensional covariant. Standard dimensional reduction in the direction of the Killing vector gives the Lagrangian of Romans' theory.

This theory was little more than the straightforward uplift of Romans' but it could be generalized to one with n Killing vectors and a symmetric5 n×n mass matrix Qmn [18]. The reduction of the n=2 theory in the direction of the two Killing vectors turns out to give all the SO(2−l,l)-gauged N=2,d=9 supergravities obtained by Scherk–Schwarz reduction from N=2B,d=10 supergravity [18], [19]: each of these theories is determined by a traceless 2×2 matrix mmn of the sl(2,R) Lie algebra which is related to the symmetric mass matrix Qmn by Qmnmpmpn,ηmn01−10mn=−ηmn.

The reduction of the n=3 theory gives the “S-dual” SU(2)-gauged N=4,d=8 theory mentioned above when we make the choice Qmn=gδmn [42] but can also give the theories with non-compact gauge group SO(2,1) if we choose Q=gdiag(++−). Singular Qs give rise other 3-dimensional non-semisimple gauge groups and massive/ungauged supergravities, as we are going to show.

In this paper we are going to study these and other gauged/massive theories obtained by dimensional reduction of “massive 11-dimensional supergravity” with n Killing vectors [18], [41]. Generically, the theories obtained in this way are (11−n)-dimensional supergravity theories with 32 supercharges determined by a mass matrix Qmn. They are covariant under global Sl(n,R) duality transformations that in general transform Qmn into the mass matrix of another theory6 of the same family.

The subgroup of Sl(n,R) that preserves the mass matrix is a symmetry of the theory and at the end it will be the gauge group. If we use Sl(n,R) transformations and rescalings to diagonalize the mass matrix so it has only +1,−1,0 in the diagonal, it is clear that SO(n,nl) will be amongst the possible gauge groups and corresponds to a non-singular mass matrix. These theories with non-singular mass matrices have n(n−1)/2 vector fields coming from the Cμmn components of the 11-dimensional 3-form and transforming as SO(nl,l) l=0,…,n gauge vector fields plus n 2-forms with the same origin and n Kaluza–Klein vectors coming from the 11-dimensional metric that transform as SO(nl,l) n-plets. The n vectors act as Stückelberg fields for the 2-forms which become massive. In this way the theory is consistent with the SO(nl,l) gauge symmetry.

Finally, all these theories have a scalar potential that contains a universal term of the form V=−12eαϕTr(QM)2−2Tr(QM)2, where M is a (symmetric) Sl(n,R)/SO(n) scalar matrix, plus, possibly, other terms form the scalars that come from the 3-form. That scalar potentials of this form appears in several gauged supergravities was already noticed in Refs. [13], [44]. The d=5 case is special because α=0. This is related to the invariance of the Lagrangian under the N=2B,d=10 Sl(2,R) symmetry.

Some of these theories are known, albeit in a very different form. The case n=6 is particularly interesting: we get SO(6−l,l)-gauged N=8,d=5 supergravities which were constructed by explicit gauging in Refs. [29], [30], with 15 gauge vectors that originate in the 3-form, 6 Kaluza–Klein vector fields that originate in the metric and give mass by the Stückelberg mechanism to 6 2-forms that come from the 3-form. That is: the field content (but not the couplings nor the spectrum) is the same as that of the ungauged theory that one would obtain by straightforward toroidal dimensional reduction. In fact, the ungauged theory can be recovered by taking the limit Q→0 which is non-singular. In Refs. [29], [30] the gauged theories were constructed by dualizing first the 6 vectors into 2-forms that, together with the other 6 2-forms, satisfy self-duality equations [34] and describe also the degrees of freedom of 6 massive 2-forms. In this theory the massles limit is singular and can only be taken after the elimination of the 6 unphysical 2-forms [35].

Thus, we have, presumably, two different versions of the same theory in which the 6 massive 2-forms are described using the Stückelberg formalism or the self-duality formalism. We will try to show the full equivalence between both formulations at the classical level.

Something similar happens in d=7, although we get SO(4−l,l)-gauged theories and in the literature only SO(5−l,l)-gauged theories have been constructed [28], [39].

This paper is organized as follows: in Section 2 we describe the “massive 11-dimensional supergravity”, its Lagrangian and symmetries. In Section 3 we briefly review how for n=1 we recover Romans' massive N=2A,d=10 supergravity. In Section 4 we revise how for n=2 we get the Sl(2,Z) family of gauged/massive N=2,d=9 supergravities and how they are classified by their mass matrix. In Section 5 we study the case n=3 and the gauged/massive N=2,d=8 supergravities that arise which allows us to describe the general situation for arbitrary n. In Section 6 we study the n=6 case and try to argue that we have obtained an alternative but fully equivalent form of the SO(6−l,l)-gauged N=8,d=5.

Section snippets

Massive 11-dimensional supergravity

Massive 11-dimensional supergravity can be understood as a deformation of standard 11-dimensional supergravity [46] that breaks 11-dimensional Lorentz invariance. The bosonic fields of the standard N=1,d=11 supergravity are the Elfbein and a 3-form potential7 êμ̂â,Cμ̂ν̂ρ̂. The field strength of the 3-form is G=4∂C, and is obviously invariant under the

Romans' massive N=2A,d=10 supergravity from d=11

The reduction of the n=1 case in the direction of the unique Killing vector present with the same Kaluza–Klein ansatz as in the massless case gives Romans' massive N=2A,d=10 supergravity [41] with the field content (in stringy notation) gμν,φ,Bμν,C(3)μνρ,C(1)μμ and with a mass parameter9 m equal to minus the mass matrix m=−Q. Thus, setting for the bosonic fields êμ̂â=e13φeμae23φC(1)μ0e23φ,êâμ̂=e13φeaμ−e13φC(1)a0e23φ,Cμνρ=C(3)μνρ,Cμνz=Bμν,

Massive N=2,d=9 supergravities from d=11

The reduction of the n=2 case in the direction of the two Killing vectors present with the same Kaluza–Klein ansatz as in the massless case gives gauged/massive N=2,d=9 supergravities characterized by the mass matrices Qmn [18], [20], [21], [22], [23], [24]. The field content of these theories is gμν,ϕ,Lmi,Cμνρ,Bmμν,Vμ,Amμiμi. The Lmi parametrize an Sl(2,R)/SO(2) coset. The field Vμ comes from the 11-dimensional 3-form components Cμmn and will be a gauge field. Its presence is the main new

Massive N=4,d=8 supergravities from d=11

The reduction of next case n=3 in the direction of the three Killing vectors gives 8-dimensional gauged theories [42]. Only the SO(3) case was studied in Ref. [42] but we are going to show that more general non-compact and non-semisimple gaugings naturally arise as in the previous case. We are going to use the general formalism and field definitions that will be valid in any dimension to show that in the general case n one can get SO(nl,l)-gauged (11−n)-dimensional supergravities.

The field

Massive N=8,d=5 supergravities from d=11

From the discussions and examples in the previous sections it should be clear that in the n=4 case we will obtain SO(4−l,l)-gauged 7-dimensional supergravities etc. A particularly interesting case is the n=6 one, in which we can obtain SO(6−l,l)-gauged N=8,d=5 supergravities which were constructed in Refs. [29], [30]. This offers us the possibility to check our construction and show that, as we have claimed, it systematically gives gauged/massive supergravities.

The derivation of the

Acknowledgements

N.A.-A. would like to thank P. Meessen and P. Resco for interesting conversations, and specially E. Lozano-Tellechea. T.O. would like to thank E. Bergshoeff for interesting conversations and specially P.K. Townsend for pointing us to Ref. [35], the Newton Institute for Mathematical Sciences and the Institute for Theoretical Physics of the University of Groningen for its hospitality and financial support and M.M. Fernández for her continuous support. This work has been partially supported by the

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