Elsevier

Nuclear Physics B

Volume 655, Issues 1–2, 7 April 2003, Pages 93-126
Nuclear Physics B

On Lagrangians and gaugings of maximal supergravities

https://doi.org/10.1016/S0550-3213(03)00059-2Get rights and content

Abstract

A consistent gauging of maximal supergravity requires that the T-tensor transforms according to a specific representation of the duality group. The analysis of viable gaugings is thus amenable to group-theoretical analysis, which we explain and exploit for a large variety of gaugings. We discuss the subtleties in four spacetime dimensions, where the ungauged Lagrangians are not unique and encoded in an E7(7)Sp(56;R)/GL(28) matrix. Here we define the T-tensor and derive all relevant identities in full generality. We present a large number of examples in d=4,5 spacetime dimensions which include non-semisimple gaugings of the type arising in (multiple) Scherk–Schwarz reductions. We also present some general background material on the latter as well as some group-theoretical results which are necessary for using computer algebra.

Introduction

Maximal supergravity theories contain a number of vector gauge fields which have an optional coupling to themselves as well as to other supergravity fields. The corresponding gauge groups are non-Abelian. To preserve supersymmetry in the presence of these gauge couplings, the Lagrangian must contain masslike terms for the fermions and a potential depending on the scalar fields. In non-maximal supergravity these terms are often described by means of auxiliary fields and/or moment maps; in the maximally supersymmetric theories the effect of the masslike terms and the potential is encoded in the so-called T-tensor [1]. It is a subtle matter to determine which gauge groups and corresponding charge assignments are compatible with supersymmetry. Based on Kaluza–Klein compactifications of higher-dimensional maximal supergravities on spheres, one readily concludes that the gauge groups SO(8), SO(6) and SO(5) are possible options in d=4,5 and 7 dimensions, respectively, corresponding to the isometry groups of S7, S5 and S4 [1], [2], [3]. But also non-compact and non-semisimple groups turn out to be possible [2], [4], [5], which are non-compact versions and/or contractions of the orthogonal groups. More recent work revealed the so-called ‘flat’ gauge groups that one obtains upon Scherk–Schwarz reductions of higher-dimensional theories [6], as well as several other non-semisimple groups [7], [8], [9]. In d=3 dimensions there is no guidance from Kaluza–Klein compactifications and one must rely on a group-theoretical analysis [10]. In this paper we apply the same kind of analysis to gaugings in higher dimensions.

Apart from the choice of the gauge group, a number of other subtleties arise that depend on the number of spacetime dimensions. In d=3 dimensions supergravity does not contain any vector fields, because these can be dualized to scalar fields. Nevertheless a gauging can be performed by introducing vector fields via a Chern–Simons term (so that new dynamic degrees of freedom are avoided), which are subsequently coupled to some of the E8(8) invariances of the supergravity Lagrangian [11]. In that case there exists a large variety of gauge groups of rather high dimension. In d=4 dimensions there are 28 vector gauge fields, but the E7(7) invariance is not reflected in the Lagrangian but only in the combined field equations and Bianchi identities by means of electric-magnetic duality. This duality rotates magnetic and electric charges, but the gauge couplings must be of the electric type. Then, in d=5 dimensions, tensor and vector gauge fields are dual to one another in the absence of charges. The E6(6) invariance is only manifest when all the tensor fields have been converted to vector fields (transforming according to the 27-dimensional representation). In the presence of charges, however, the vector fields must either correspond to a non-Abelian gauge group or they must be neutral. Charged would-be vector fields that do not correspond to the non-Abelian gauge group, should be converted into antisymmetric tensor fields [12]. This implies that the field content of the d=5 theory depends on the gauge group.

The Lagrangian of ungauged maximal supergravity contains the standard Einstein–Hilbert, Rarita–Schwinger, and Dirac Lagrangians for the gravitons, the gravitini and the spinor fields. The kinetic terms of the gauge fields depend on the scalar fields and the kinetic term for the scalar fields takes the form of a non-linear sigma model based on a symmetric coset space G/H. Here H is the maximal compact subgroup of G; a list of these groups is given in Table 1. The Lagrangian (or the combined field equations and Bianchi identities) is invariant under the isometry group G which is referred to as the duality group. The standard treatment of gauged non-linear sigma models exploits a formulation in which the group H is realized as a local invariance which acts on the spinor fields and the scalars; the corresponding connections are composite fields. The gauging is based on a gauge group Gg⊂G whose connections are (some of the) elementary vector gauge fields of the supergravity theory. The matrix which encodes the embedding of the gauge group into the duality group is in fact linearly related to the T-tensor. The coupling constant associated with the gauge group Gg will be denoted by g. One can impose a gauge condition with respect to the local H invariance which amounts to fixing a coset representative for the coset space. In that case the G-symmetries will act non-linearly on the fields and these non-linearities make many calculations intractable or, at best, very cumbersome. Because it is much more convenient to work with symmetries that are realized linearly, the best strategy is therefore to postpone the gauge fixing till the end.

This paper aims at exploiting the group-theoretical constraints on the T-tensor, which are essential in order to have a consistent, supersymmetric gauging. It is well-known that the T-tensor must be restricted to a certain representation of the duality group. For instance, in four dimensions, this is the 912 representation of E7(7), and in five dimensions it is the 351 representation of E6(6). We derive these allowed representations for dimensions d=3,…,7. Possible gaugings can then be explored by investigating which gauge groups lead to T-tensors that belong to the required representation. This proves to be sufficiently powerful to completely identify the possible gauge groups within a given subgroup of G, and to determine which gauge fields and generators of G are involved in each of the gaugings. Part of the analysis is done with help of the computer. To demonstrate the method and its potential, we analyze a number of gaugings in d=4,5 dimensions, including the known cases. Applications with hitherto unknown gauge groups are relegated to a forthcoming publication [13].

In four dimensions the Lagrangian is not unique in the absence of charges, because of electric/magnetic duality. Once the charges are switched on, the possibility of obtaining alternative Lagrangians is restricted, because electric charges cannot be converted to magnetic ones. Without introducing the gauging, there exist different Lagrangians (i.e., not related by local field redefinitions) with different symmetry groups, whose field equations and Bianchi identities are equivalent and share the same invariance group. This feature makes the four-dimensional case more subtle to analyze and therefore considerable attention is given to this case. In particular, we show that the different Lagrangians of the ungauged theory are encoded in a matrix E belonging to E7(7)Sp(56;R)/GL(28).

Some of the gaugings can be interpreted as originating from a Scherk–Schwarz truncation of a higher-dimensional theory [6]. In order to identify such gaugings we have included some material on these reductions and we exhibit examples in four and five dimensions. Both of them are single reductions, originating from a theory with one extra dimension. However, also multiple reductions are possible from theories with more than one extra dimensions, which lead to more complicated gauge groups, as we shall discuss in more detail in [13].

This paper is organized as follows. In Section 2 we review the structure of the non-linear sigma models that appear in maximal supergravity and the symmetries of the Lagrangians. In Section 3 we focus on the definition of the T-tensor in four dimensions and discuss a large number of relevant features. In Section 4, we derive the group theoretical constraint on the T-tensor (and equivalently on the embedding matrix of the gauge group) for dimensions d=3,…,7. This constraint provides an efficient criterion for identifying consistent gaugings. In Section 5 we review characteristic features of Scherk–Schwarz reductions, which correspond to some of the gaugings. Finally, in 6 The known gaugings in, 7 Some gaugings in, we demonstrate how this framework naturally comprises the known gaugings in d=4 and 5 dimensions. In addition, we exhibit a new gauging in d=5 dimensions which can be interpreted as a Scherk–Schwarz reduction. Appendix A is included with some group-theoretical results.

Section snippets

Coset-space geometry and duality

In this section we review the coset-space structure of the maximally supersymmetric supergravity theories. Because of the subtleties of supergravity in four spacetime dimensions special attention is devoted to this theory. In particular, we discuss its inequivalent Lagrangians, encoded in a matrix E. The existence of different Lagrangians makes the analysis of the various gaugings more complicated, as they are associated with different classes of gaugings.

The scalar fields in maximal

The T-tensor

The gauging of supergravity is effected by switching on the gauge coupling constant, after assigning the various fields to representations of the gauge group embedded in G. Again we mainly focus on the four-dimensional theory, where G=E7(7), but we will occasionally comment on other spacetime dimensions. In d=4 dimensions only the gauge fields themselves and the spinless fields transform under the gauge group. In other dimensions, the ungauged supergravity theory has tensor gauge fields

Group-theoretic analysis

The three SU(8) covariant tensors, A1, A2 and A3, which depend only on the spinless fields, must be linearly related to the T-tensor, because they were introduced for the purpose of cancelling the variations proportional to the T-tensors. To see how this can be the case, let us analyze the SU(8) content of the T-tensor. As we mentioned already, the T-tensor is cubic in the 56-bein, and as such it constitutes a tensor that transforms under E7(7). The transformation properties were given in (3.10)

Scherk–Schwarz reductions

In this section we summarize a number of features related to the modified dimensional reduction scheme proposed by Scherk and Schwarz [6]. This scheme applies to a theory in higher dimensions with a rigid internal symmetry group G. It consists of an ordinary dimensional reduction on a hypertorus, where an extra dependence on the torus coordinates is introduced by applying a finite, uniform G-transformation that depends non-trivially on the these coordinates. Subsequently one retains only the

The known gaugings in d=4 dimensions

In this section we demonstrate how the group-theoretical approach of this paper allows us to straightforwardly establish the viability of the known gaugings of maximal supergravity in 4 dimensions. The strategy is to make an assumption about the gauge group Gg, or about the electric subgroup Ge which contains Gg as a subgroup, and then analyze the constraint (4.11). We do this by comparing the branching of the 133×56 representation under a given electric subgroup Ge, to the representations that

Some gaugings in d=5 dimensions

As yet another application, we analyze some of the gaugings in d=5 maximal supergravity. In this case there are no subtleties related to electric–magnetic dualities and the search for viable gauge groups should be based on arbitrary subgroups of E6(6) without the need for referring to a specific basis. We consider several classes. First we assume that the gauge group is a subgroup of the SL(2,RSL(6,R) maximal subgroup of E6(6). Then we consider the case where the gauge group is embedded in

Acknowledgements

We thank H. Nicolai and S. Ferrara for useful discussions. This work is partly supported by EU contracts HPRN-CT-2000-00122 and HPRN-CT-2000-00131. M.T. is supported by a European Community Marie Curie Fellowship under contract HPMF-CT-2001-01276.

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