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The minimal N=4 no-scale model from generalized dimensional reduction

We consider the generalized dimensional reduction of pure ungauged N=4, D=5 supergravity, where supersymmetry is spontaneously broken to N=2 or N=0 with identically vanishing scalar potential. We explicitly construct the resulting gauged D=4 theory coupled to a single vector multiplet, which provides the minimal N=4 realization of a no-scale model. We discuss its relation with the standard classification of N=4 gaugings, extensions to non-compact twists and to higher dimensions, the N=2 theories obtained via consistent Z_2 orbifold projections and prospects for further generalizations.


Introduction
Generalized dimensional reductions of higher-dimensional supergravity [1] and superstring theories [2] provide elegant and efficient mechanisms for supersymmetry breaking. In short, the periodicity conditions in the compact extra dimensions can be 'twisted' by an Rsymmetry of the higher-dimensional action. This induces four-dimensional gravitino mass terms and breaks the supersymmetries that do not commute with the twist, without generating a four-dimensional vacuum energy. Also M-theory and superstring compactifications with branes, orientifolds and fluxes (for a review and an extensive list of references, see e.g. [3]) can be related with twisted tori by suitable dualities [4,5]. The above theoretical constructions can play a rôle in solving some crucial problems of higher-dimensional supergravity and superstring theories: vacuum selection, supersymmetry breaking, moduli stabilization, generation of hierarchies. The corresponding D = 4 effective theories are gauged extended (N > 1) supergravities 1 , where a subgroup of the isometry group of the scalar manifold (which includes the R-symmetry group) is promoted to a local invariance.
It has been known for quite some time that generalized dimensional reductions of D = 11 [6] or minimal D = 10 [7] supergravity give rise to gauged N = 8 [8] or N = 4 [9] D = 4 supergravities, and that daughter theories with a lower number of supersymmetries can be obtained by suitable orbifold projections. Systematic investigations of the effective gauged D = 4 theories have recently been performed starting from N = 8 [10,11] or N = 2 [12] supergravities; many results have been obtained also for the N = 4 theories [13,14] that are the focus of the present paper. Indeed, even in the case of N = 1 orbifolds of superstrings or M-theory, the effective D = 4 superpotential for the light (bulk) states coming from the untwisted sector can be associated with the gauging of the underlying N = 4 theory [15].
The goal of the present paper is to consider generalized dimensional reductions of pure N = 4, D = 5 ungauged supergravity [16,17], and to derive explicitly Lagrangian and transformation laws for the resulting low-energy theory, a gauged N = 4, D = 4 supergravity coupled to a single vector multiplet. Such a theory can be legitimately called the minimal N = 4 no-scale model, in analogy with the minimal no-scale models with N = 1 [18] and N = 2 [19] constructed long ago. It is also the minimal N = 4 noscale model that allows for the partial breaking to N = 2, whereas partial breaking to N = 3 or N = 1 requires the presence of additional vector multiplets [20]: we will see that the minimal N = 2 no-scale model with partial breaking [21], which contains one vector multiplet and one hypermultiplet, corresponds to a consistent Z 2 orbifold projection of our minimal N = 4 no-scale model [22]. The choice of N = 4 for our detailed study is motivated not only by the need of filling a gap in the existing literature, but also by the fact that N = 4 is the minimal amount of supersymmetry shared by all stable superstring theories, and their corresponding supergravities, in D = 10. Within N = 4 theories, the choice of D = 5 as the starting point for the generalized dimensional reduction allows for minimality (when no D = 5 vector multiplets are included) as well as for maximum generality (when an arbitrary number of D = 5 vector or tensor multiplets are included), and prepares the ground for further discussions of the minimal Randall-Sundrum model [23] with extended supersymmetry [24], which is based on a pure gauged N = 4, D = 5 supergravity [17,25].
The paper is organized as follows. In section 2 we briefly recall the field content and the symmetries of pure, ungauged N = 4, D = 5 supergravity. In section 3 we derive the ungauged N = 4, D = 4 effective supergravity, coupled to a single vector multiplet, that originates from standard dimensional reduction. In particular, we identify the 'electric' subgroup of the D = 4 duality group that acts linearly on the gauge potentials and on the scalar fields. The gauged N = 4, D = 4 supergravity, originating from the pure ungauged D = 5 theory by generalized dimensional reduction, is derived in section 4. Twisting the periodicity conditions by an R-symmetry transformation, controlled by two independent parameters, does not generate a potential, and leads to N = 4 no-scale models with spontaneous breaking of half or all of the supersymmetries. As already stressed in similar contexts [20,4,10,11], the effective N = 4, D = 4 gauged supergravity does not fit the standard classification [26]: the reason is that formulations of the theory that are equivalent, in the ungauged case, via duality transformations, are no longer equivalent in the gauged case. For completeness, we also discuss the possibility of a non-compact twist, which generates a positive-definite four-dimensional potential without critical points [27]. In section 5, we explain how our results can be applied to generalized dimensional reductions of pure N = 4 supergravity from D to D − 1 dimensions, and we discuss the examples of the non-chiral (2,2) and the chiral (4,0) theories reduced from six to five dimensions. In the former case, we obtain new 'flat' gaugings of N = 4 in D = 5, not included in the classification of [28], which was limited to gaugings of semisimple and abelian groups. In the latter case, we generalize the flat gauging of ref. [25] to partial breaking. In section 6 we discuss consistent Z 2 orbifold truncations in the presence of the Scherk-Schwarz twist, and the main features of their N = 2, D = 4 effective theories: we find total or partial supersymmetry breaking with vanishing potential, in agreement with previous results [21,22,12]. In the concluding section, we summarize again our results and we comment on prospects for further work. Our conventions are explicitly spelled out in appendix A. The Lagrangian and the transformation laws of pure ungauged N = 4, D = 5 supergravity are recalled in appendix B. The details of the N = 4, D = 4 Lagrangian and transformation laws, relevant to both standard and generalized dimensional reduction, are collected in appendix C.

N = 4, D = 5 ungauged supergravity
The field content of pure N = 4, D = 5 supergravity, whose ungauged version was first constructed in [16,17], is just the gravitational multiplet: one graviton g M N , four gravitinos ψ M a , five plus one vectors V i M + v M , four spin-1/2 fermions χ a and one real scalar φ. The automorphism group of the N = 4, D = 5 supersymmetry algebra (R-symmetry) is U Sp(4). Fields with indices a = 1, . . . , 4 and i = 1, . . . , 5 transform in the 4 and 5 irreducible representations, respectively; all the other fields are singlets. In a schematic notation that will be convenient in the following, we summarize the content of the gravitational multiplet as: The numbers in brackets count the representations of different 'spin', with the latter decreasing from left to right in steps of 1/2. The subscript ' m=0 ' recalls that we are dealing with a massless multiplet. Further details on our conventions can be found in appendix A. The Lagrangian and the supersymmetry transformations are collected in appendix B. Before moving to the study of standard and generalized dimensional reductions to D = 4, it is useful to recall the local and global symmetries of the D = 5 theory. The local symmetries are general coordinate transformations, N = 4 supersymmetry and a U (1) 6 gauge invariance, associated with the six vector fields and with respect to which no fields are charged. The global symmetry of the theory, or 'U -duality' group, is in this case U Sp(4) × SO(1, 1). It is an electric subgroup of the D = 4 Sp(14, R) duality group acting on the space of the vector field strengths and their duals [29], in the sense that it acts linearly on the vector potentials. The action of SO(1, 1) on the fields is: while U Sp(4) ∼ SO(5) acts canonically on the a and i indices. If we couple N = 4, D = 5 pure supergravity to n vector multiplets, the U -duality group is enlarged to SO(5, n) × SO(1, 1), and the scalar manifold becomes: In this case, the combination of generators used for the twist and giving rise to a flat D = 4 gauging can be embedded in the maximal compact subgroup of SO(5) × SO(n). However, switching on the SO(n) sector does not increase the possibilities for supersymmetry breaking via generalized dimensional reduction, it just increases the dimension of the D = 4 gauge group 2 . Moreover, some of the resulting non-minimal N = 4 no-scale models have already been constructed as truncations of gauged N = 8 theories [13,14]. For these reasons, in the following we will focus on the minimal model (n = 0), corresponding to pure N = 4, D = 5 supergravity.

Standard dimensional reduction
We now describe the salient features of the standard dimensional reduction of the N = 4, D = 5 theory. For this purpose, we consider only the kinetic and Chern-Simons terms of the D = 5 Lagrangian eq. (B.1): The first step is to decompose the fields with five-dimensional indices into fields with fourdimensional indices: We can then assume that the zero modes do not depend on the fifth coordinate y and perform the following field redefinitions: 3) The first two lines allow us to 'ortho-normalize' the D = 4 fermionic kinetic terms, the third one avoids mixing terms of the form 'V µν A µ ∂ ν ϕ', and the last one makes the duality invariance of the scalar sector manifest. After moving from the field basis (e α µ , ψ µ , ψ y , χ, V i µ , v µ , A µ , V i 5 , v 5 , ρ, φ) to the field basis (e α µ , η µ , ψ ′ y , χ ′ , B i µ , b µ , A µ , ϕ i , τ, ϕ 0 , t), the part of the reduced Lagrangian coming from eq. (3.1) reads: where the dots stand for interaction terms and the primes in eq. (3.3) have been suppressed. The Lagrangian can be rewritten more compactly as: i ψ a y γ µ D µ ψ y a + . . . , (3.5) where the field strengths F I µν (I = 1, . . . , 7) are defined through and the symmetric field-dependent matrices g IJ and θ IJ are: Notice that the metric g IJ , which controls the vector kinetic terms, is a positive-definite and non-singular symmetric matrix for all the allowed scalar field configurations (ϕ 0 > 0, t > 0, ϕ i ∈ R and τ ∈ R). The reduced theory is a U Sp(4), N = 4, D = 4 supergravity coupled to one vector multiplet. In the schematic notation of eq. (2.1), we can write its spectrum as: The eight scalar fields (one coming from the metric, six coming from the vector fields and the D = 5 dilaton) parametrize the coset manifold: (3.10) The first factor is a Kähler manifold, parametrized by the complex scalar with a Kähler potential which implies (remembering our convention for the Einstein term) the kinetic term: The second factor is the scalar manifold of the real fields (ϕ 0 , ϕ i ).
The U -duality group of the reduced theory is SU (1, 1) × SO(6, 1). It is associated with a solvable Lie algebra, which can be decomposed as follows: (3.14) The U -duality group is not completely embedded in the electric subgroup of the full duality group Sp(14, R), acting on the seven field strengths and their duals. As a result, only part of the U -duality group can be gauged. In particular, the transformations associated with the generators 1 − and 5 − do not have a well-defined action on the elementary fields. As long as the theory remains ungauged, however, Sp(14, R) transformations can be used to connect different equivalent formulations of the same theory: besides the U Sp(4) of this paper, known examples are the SO(4) of [30] and the SU (4) of [31]. The non-trivial global symmetries of the theory with a well-defined action on the elementary fields are the 5 + +1 + translations, the two dilatations and the SO(5) ∼ U Sp(4) transformations.
The 5 + +1 + translations derive from the U(1) 6 gauge symmetry of the D = 5 theory. Their action on the 'axions' is then The vectors transform as where we used the definitions of (b µ , B i µ ) and the fact that (v µ , V i µ ) do not transform. The scalar sector is explicitly invariant. Since it is easily shown that also the other terms are invariant, up to total derivatives coming from the Chern-Simons term 3 . The two dilatations act on the scalar fields as follows: 18) and the invariance of the scalar sector is manifest. The invariance of the whole Lagrangian is obtained by requiring that the vector fields transform as Notice that the double SO(1, 1) is a symmetry of only the reduced theory, and it is not valid when the Kaluza-Klein modes are retained, unless , with a semidirect product structure, and under the local U (1) 7 group, with respect to which, however, no field is charged.

Generalized dimensional reduction
In the previous section we have identified the N = 4 supergravity, coupled to one vector multiplet, obtained from the pure ungauged D = 5 theory by standard dimensional reduction. In particular, we have observed that the ungauged theory does not have the form of the known SO(4) and SU (4) theories, but it is equivalent to them via a duality transformation. However, as argued in [20,4,10,11], inequivalent formulations of gauged supergravities can be obtained by considering different embeddings of the U -duality group in the full duality group Sp(14, R). We can then use the Scherk-Schwarz mechanism to obtain a new N = 4, D = 4 gauged supergravity. We are going to find, in analogy with [10], two remarkable properties. First, if the twist belongs to U Sp(4) the supergravity gauging corresponds to a flat but non semisimple group, which guarantees a vanishing scalar potential at the minimum. Moreover, a four-dimensional Chern-Simons term is automatically generated by the generalized reduction, thus preserving the consistency of the theory.
The Scherk-Schwarz reduction is performed by imposing generalized periodicity conditions on the fields: where U is a constant matrix corresponding to a symmetry of the five-dimensional theory.
For the moment we consider only the case U ∈ U Sp(4). The case of a non-compact twist U ∈ SO(1, 1) will be discussed separately at the end of this section: as we will see, in such a case a positive-definite scalar potential without critical points is generated, and the theory has no D = 4 maximally symmetric vacuum.
Since U Sp(4) has rank two, we can parametrize the twist U as where Y and T 3 are two representative generators in the Cartan subalgebra of U Sp (4), whose explicit representation is given in appendix A, and α 1,2 ∈ R. Following the standard procedure, we reparametrize the twisted fields in terms of periodic ones: (4.4) For simplicity, tildes will be dropped from now on, and fields will always be understood to be periodic.
With respect to the standard reduction, the generalized one produces extra terms in the D = 4 effective Lagrangian, proportional to (4.5) In our conventions, the explicit representations of the mass matrix M , acting respectively on the 4 and the 5 of U Sp(4), are: Some of the extra contributions to the D = 4 Lagrangian correspond to fermion mass terms. The remaining ones can be consistently organized to describe the 'gauging' of the N = 4 theory: the upgrade of a subgroup of the global U -duality group of the D = 4 theory to a local invariance. The gauge group, however, is not the direct product of simple and abelian factors. It can be identified with the semidirect product of the U (1) ⊂ U Sp(4) associated with the twist U , and four translations: U (1) T 4 . The associated Lie algebra is defined by the commutation relations: ], X 7 is the twist generator (with vector potential A µ ), and the X i are the generators of 5 + (with vector potentials B i µ ). Notice that only four independent X i enter non-trivially in the Lie algebra, as the fifth one commutes with the twist. Moreover, the generator X 6 of 1 + always remains ungauged, since it belongs to the SU (1, 1) sector, which does not carry charges with respect to U Sp(4). In the gauged theory, according to the algebra of eq. (4.8), covariant derivatives replace the ordinary derivatives of the ungauged theory, and covariant field strengths are modified accordingly: Consequently, the transformation laws under the gauged symmetry, with local parameters Ξ I (I = 1, . . . , 7), become: Under the gauged symmetry, the θ IJ matrix of eq. (3.8) transforms non linearly. This would require the addition of an extra Chern-Simons term in order to guarantee the gauge invariance of the D = 4 theory [32], namely: where d ı  k is a symmetric SO(5)-invariant tensor, normalized to: As argued in [10], and verified explicitly in the present case, this term is automatically produced by the generalized dimensional reduction. The detailed expression of the gauged Lagrangian can be found in appendix C. We display here only its bosonic part: For generic values of m 1 and m 2 , the gauging of the four translations allows us to shift away four spin-0 fields: they are absorbed by the corresponding vectors, which acquire a mass matrix proportional to M 5 . For |m 1 | = |m 2 |, only two spin-0 fields are absorbed by the vectors gauging the Euclidean group on the plane, U (1) T 2 . In the fermionic sector, besides the covariantization of derivatives and field strengths with respect to the gauged group, mass terms appear, proportional to M 4 , and a super-Higgs effect takes place, with the ψ y a playing the role of goldstinos. This can be deduced, for instance, by looking at the inhomogeneous terms in the supersymmetry transformations of eq. (C.4).
Depending on the specific choice of the mass parameters m 1 and m 2 appearing in the twist matrix, it is possible to break supersymmetry partially (N = 2) or completely (N = 0). We summarize the spectrum for the different cases in our short-hand notation: In the partially broken case, beside the gravitational multiplet and two vector multiplets of N = 2, we can recognize a massive N = 2 BPS short multiplet of spin 3/2 [33].
The spectrum of the theory can be easily extracted from the Lagrangian in eq. (C.1). For the bosonic sector, it is useful to observe that the kinetic terms for the vector fields are diagonal in the basis of the through covariantization with respect to the vector metric g IJ : The spectrum is twice degenerate and reads: If non-compact generators are used to perform the Scherk-Schwarz reduction, non-flat gaugings are generated [1], with a positive-definite four-dimensional potential without critical points. This is the case if we use the non-compact SO(1, 1) in the U -duality group for the twist, in analogy with [27].
Given the SO(1,1) field transformations in eq. (2.2), we obtain the following relations between periodic (with tildes) and non-periodic (without tildes) fields: After the SO(1, 1) generalized dimensional reduction, and removing the tildes as usual, we get the following D = 4 bosonic Lagrangian: where the coefficients C IJK are fully symmetrized. The non-vanishing independent ones are identified by The scalar covariant derivatives are 19) and the covariant field strengths are From eq. (4.17), we see that the SO(1, 1) twist produces a positive-definite, non-vanishing scalar potential without critical points, which does not admit maximally symmetric D = 4 vacua.
In the fermionic Lagrangian, all the field strengths and the scalar derivatives become covariant as described above. The fermions remain neutral with respect to these gauge interactions, but acquire a field-dependent mass term controlled by the scalar potential: The theory now has a non-abelian gauge group, which is the semidirect product of the six translations (B i µ , b µ ) and the dilatation (A µ ), namely SO(1, 1) T 6 . All the seven vectors acquire a field-dependent mass term proportional to the potential, absorbing the corresponding seven spin-0 fields. The surviving scalar, associated with the ungauged SO(1, 1), has a runaway behaviour described by the potential of eq. (4.17).

N = 4, D ≥ 6 reductions
The discussion given in section 4 can be extended to higher dimensions. Starting from a N = 4 pure ungauged supergravity in D dimensions, by generalized dimensional reduction we can obtain a minimal 4 no-scale model in D−1 dimensions. However, this strategy works only for D ≤ 8, where there is a non-trivial R-symmetry for the twist. As an example, in this section we will show that from the generalized reduction of an ungauged N = 4, D = 6 pure supergravity we can obtain some new N = 4, D = 5 gauged supergravities, spontaneously broken to N = 2 or N = 0. These theories were not considered in a previous classification [28] that concentrated on semisimple and abelian gaugings. Analogous results have recently been found [34] for the N = 8 and N = 2 cases, and the present discussion will complete that analysis. Since there exist two inequivalent N = 4, D = 6 supergravities, the non-chiral one, or (2,2) [35], and the chiral one, or (4,0) [36], we now discuss them in turn. We then conclude the section with a brief discussion of the remaining cases, D = 7 → 6 and D = 8 → 7, which complete the analysis of such gaugings for minimal N = 4 theories.

Reduction of D = 6 (2,2) supergravity
The R-symmetry of the D = 6 (2,2) non-chiral supergravity is U Sp(2) × U Sp (2). The scalar manifold of the generic theory with n vectors is: We thus get the following D = 5 theories: (5.4) where the N = 2 and N = 0 cases are obtained by appropriate choices of the twist parameters in the two U Sp (2). The masses of fermions and vectors have the same dependence on the twist parameters as in the D = 4 case.

Reduction of D = 6 (4,0) supergravity
Pure N = 4, D = 6 chiral supergravity is anomalous, and extra multiplets have to be added to obtain a consistent theory. The choice is unique [36] and consists in adding 21 antisymmetric tensor multiplets. This produces a theory with the following field content: The spin-1 entries 5 − and 1 + are (anti) self-dual tensors, transforming in the 5 and in the 1 of U Sp(4), respectively. The theory then has 105 scalars, parametrizing the manifold: (21) .
We could perform a Scherk-Schwarz twist with a generator in the maximal compact subgroup of the U -duality group, namely U Sp(4) × SO (21). However, we consider only a twist in the U Sp(4) factor, since a twist in SO (21) does not break supersymmetry. Notice also that no vectors are present in this D = 6 theory, and that the only spin-1 fields charged under U Sp(4) are anti-self-dual tensors. This means that in the present example there are no shift symmetries to be gauged. Indeed, the only gauged symmetry can be a U (1) subgroup of U Sp(4), via the vector potential e 6 M . The generalized reduction can then be performed as in the D = 5 case, by choosing the twist of eq. (4.4). In the D = 5 reduced theory, the fermions in the 4 of U Sp(4) acquire mass as in the D = 4 case. Four of the 5 − tensors transform into two complex D = 5 tensors (2 c ) with masses m 1 ± m 2 . The other 22 tensors remain massless and neutral, so that they can be dualized to D = 5 vectors. Finally, the scalars acquire a potential, which vanishes at its minimum. The potential has 21 flat directions, corresponding to 21 massless scalars. The other 42+42 scalars acquire instead a mass m 1 ± m 2 .
In terms of D = 5 multiplets, the spectrum can be summarized as follows: where the massive fields are also charged with respect to the local U (1) ⊂ U Sp(4).
We notice finally that, starting from the (anomalous) pure D = 6 (4,0) supergravity, or performing a consistent truncation of the reduced theory, it is possible to recover the flat pure D = 5 gauged theory of [25]. This happens because the reduction of the pure (4,0) theory does not produce extra matter in D = 5. In particular, we can obtain the flat case of [25] by setting m 1 = m 2 . However, we can also choose m 1 = m 2 , which leads to two massive complex anti-self-dual tensors. In particular, the case m 2 = 0 gives partial breaking of supersymmetry. All this generalizes the minimal N = 4, D = 5 no-scale model given in [25].

Reductions from D = 7 and D = 8
We complete the higher-dimensional case with a brief discussion of D = 7 and D = 8. In pure N = 4, D = 7 supergravity, we have the global U Sp(2) × SO(1, 1) group, and the gravitational multiplet is: After a twist in U Sp(2), we end up with: the fermions getting a mass m proportional to the twist, and two vectors getting a mass 2m, via the gauging of the semidirect product U (1) T 2 . Finally, the D = 8 theory is globally invariant only under SO(2) × SO(1, 1). With respect to SO(2), the gravitational multiplet can be decomposed as where (1 ± ) means a complex Weyl spinor charged under U (1) ∼ SO (2). As in the previous case, the reduced theory is coupled to one vector multiplet: The scalar manifold is SO(3, 1) U Sp(2) × SO(1, 1) , (5.15) and the algebra of the duality group decomposes into Again, a U (1) twist leads to the gauging of the subgroup U (1) T 2 , with matter content fermions of mass m and vectors of mass 2m.

Z 2 orbifold
We have seen that generalized dimensional reduction can give either partial or complete breaking of supersymmetry, but the number of residual supersymmetries is always even, as the mechanism cannot generate chirality. However, with the additional help of an orbifold projection, it is also possible to obtain a reduced theory with an odd number of unbroken supersymmetries. It is then interesting to study how the results obtained in section 4 can be modified by orbifold projections. With only one internal dimension, it is not restrictive to consider the Z 2 orbifold associated with the parity y → −y, which leaves the classical D = 5 action invariant. The corresponding action on the fields can be written as: where for consistency Z 2 must square to 1. In the absence of a twist, a consistent assignment of the Z 2 parities to the fields is the following: In our conventions, the above assignment corresponds to the following Z 2 representation: where Y is the U (1) ⊂ U Sp(4) generator given in appendix A. All other parity assignments follow from eq. We are now ready to discuss what happens if the Z 2 orbifold projection and generalized dimensional reduction are combined. For a consistent space-time interpretation, the orbifold symmetry Z 2 and the twist U must obey the relation: If we express the twist as U = exp(i T ), we can identify two ways of satisfying eq. (6.5): either [T, Z 2 ] = 0 or {T, Z 2 } = 0. In the first case, eq. (6.5) reduces to U 2 = 1, which discretizes the possible values of the twist T . In the second case, instead, eq. (6.5) is satisfied for every value of the twist. In both cases the gauging identified in section 4 is nullified by the orbifold, since the projection removes the zero mode of A µ , which gauges U Sp(4) and is crucial for the nonabelian character of the algebra in eq. (4.8). However, there is still room for a smaller abelian group to be gauged. We therefore analyse the two cases in more detail.
To satisfy [T, Z 2 ] = 0, we must search for generators of U Sp(4) that commute with Y , but the most general twist with these properties is the one already analysed in section 4. To satisfy also U 2 = 1, we must choose α 1 ± α 2 = 0 (mod. 2π), or, equivalently, m 1,2 = (0, ±1)/(2r). If we do so, also the twist acts like a parity, in particular U = Z 2 · Z ′ 2 , where Z 2 is the reflection with respect to y = 0 and Z ′ 2 the one with respect to y = π. This is equivalent to imposing a Z 2 × Z ′ 2 projection on a circle of radius 2r. Choosing for instance m 1 = 0, m 2 = 1/(2r), the (Z 2 , Z ′ 2 ) parity assignments will be: This means that the reduced theory is an N = 1, D = 4 unbroken supergravity with one gravitational multiplet [e α µ , ψ µ 1 ] and two chiral multiplets [(ψ y 2 , χ 2 ), (ρ, φ, v 5 , V 5 5 )]: We can finally move to the more interesting case {T, Z 2 } = 0. Six generators of U Sp(4) have this property, namely those in the coset U Sp(4)/[SU (2) × U (1)]. We can easily find two of these generators that commute with each other (and obviously anticommute with Y ). Without loss of generality we choose: (6.8) Our twist then reads: The action of the twist gives the same results as those in section 4, but with different matrices M 4,5 . In particular, we have now: where (σ i ) ij means a σ i matrix taken in the subspace ( ij ). The Z 2 orbifold projection removes some fields from the reduced theory, in particular the field A µ , responsible for the gauging described in section 4. However, only half of the gauged translations are removed, while the others survive. Observe that the orbifold preserves only the vectors V 1,2 µ and the scalars V 3,4,5 5 . However, now the twist connects 1 → 3 and 2 → 4, still allowing the surviving scalars to be gauged. This can be explicitly checked in the covariant derivative for the scalars, eq. (4.9): The mass matrices in eq. (6.10) are non-diagonal, but their squares are: The Z 2 orbifold projection removes the second and the fourth entry in M 4 , as well as the last three entries in M 5 . The resulting N = 2 theory is spontaneously broken either to N = 1 or to N = 0, with two independent mass parameters, and inherits the N = 4 mass sum rule Str M 2 = 0. The spectrum of the N = 4, D = 5 theory, after the orbifold and the partial (m 1 = 0, m 2 = 0) or total (m 1 m 2 = 0) Scherk-Schwarz breaking, reads: Notice that in the partially broken theory the massive multiplet is a long spin-3/2 multiplet, because this is the only possibility with N = 1 (the short one, being BPS-charged, requires an even number of supersymmetries).
This mechanism, already considered in [22], allows us to break an N = 2 supergravity, via generalized reduction, with two independent parameters, starting from the Z 2 orbifold of an N = 4, D = 5 theory. The corresponding D = 4 effective theory is the minimal N = 2 no-scale model with partial breaking of [21].
If we start from N = 8, D = 5 supergravity, we can get a N = 4, D = 4 spontaneously broken supergravity with four independent mass parameters, recovering the results of [14] for a Type IIB supergravity compactified on a T 6 /Z 2 orientifold with fluxes.

Conclusions and outlook
In this paper we constructed the minimal N = 4 no-scale model in four dimensions, by generalized dimensional reduction of pure N = 4, D = 5 supergravity. We explicitly derived Lagrangian and transformation laws of the D = 4 effective theory, a N = 4 supergravity where a flat, non-semisimple group is gauged, and half or all of the supersymmetries are spontaneously broken. We found that the Scherk-Schwarz reduction automatically generates the extra Chern-Simons term that must be added to the D = 4 Lagrangian to ensure gauge invariance. We also studied how this procedure extends to non-compact twists and to higher dimensions, and found new N = 4 gauged supergravities not included in earlier classifications. Finally, we discussed the consistent orbifold projections of the theory, recovering the minimal partially broken N = 2 no-scale model of refs. [21,22].
An interesting extension of the present work would consist in performing a similar investigation with pure but gauged N = 4, D = 5 supergravity [17,25] as the starting point. This may lead to a better understanding of spontaneous supersymmetry breaking and boundary actions for warped compactifications, in the context of the Randall-Sundrum model [23] with an underlying N = 4 supersymmetry [24], explicitly broken to N = 2 by the Z 2 orbifold projection.
where Ω ≡ Ω ab is for us a real, antisymmetric symplectic metric with upper indices: The matrix U acts from the left on column, four-component vectors with We can then define symplectic vectors with upper indices by: and write the symplectic product as: As a short-hand notation for the symplectic product, we adopt the NW-SE convention on the contraction of symplectic indices: Our convention on how to lower the indices is set by defining the inverse symplectic metric with lower indices: The generators a ≡ a b a of the U Sp(4) Lie algebra, defined by U = exp(a), satisfy: Exploiting the isomorphism Spin(5) ∼ U Sp(4), the ten U Sp(4) generators a can be identified with those of Spin(5), whose elements (Γ i ) b a satisfy the Euclidean Clifford algebra where i, j = 1, . . . , 5 are the indices 6 of Spin(5) and a, b = 1, . . . , 4 are those of U Sp (4). Notice that we can define antisymmetric Γ matrices with upper U Sp(4) indices by: Then, the antisymmetric two-index representation of Spin(5) of dimension 10, with gener- satisfies the algebra of eq. (A.8) automatically. This means that the generic tensor S a 1 ...an transforms under U Sp(4) as where the α ij are real coefficients. The generic U Sp(4) algebra element with two indices can now be decomposed as (A.14) The last two relations link the 5 and the 10 of U Sp(4) to those of SO (5).
An explicit representation of the (Γ i ) b a appearing in (A.9), useful for discussing the gaugings and the Scherk-Schwarz twists in N = 4, D = 5 supergravity is: (4) is: The fields of N = 4, D = 5 supergravity fall only in the 1, 4 and 5 irreducible representations of U Sp(4): we denote them here by the generic symbols φ, χ a and A ab , with A ab antisymmetric and Ω-traceless. If we parametrize the generic transformation by the non-trivial field transformations read Using the decomposition in eq. (A.13) and the Clifford algebra in eq. (A.9), the infinitesimal transformation for A ab in the 5 of U Sp(4) reads and is just the transformation law for the fundamental representation of SO(5). We can thus exploit this further isomorphism to rewrite the finite transformation of eq. (A.21): where O hk are the 10 generators of SO (5). Hence the U (1)×SU (2) generators in eqs. (A. 16) and (A.17) correspond to the SO(2) × SO(3) subgroup of this SO(5). A generic field A i transforming in the 5 of U Sp(4) can then be decomposed as where X c denotes a X of SU (2) with U (1) charge equal to c, and we have introduced another short-hand notation to identify the composition of a charged field. Considering only two commuting generators of U (1) × SU (2) ⊂ U Sp(4), Y and T 3 , we can write the explicit transformation laws of the relevant fields as

A.2 Space-time and spinors
We choose a 'mostly plus' metric: The total antisymmetric tensor ε M N P QR is defined in such a way that: Connections, covariant derivatives and curvature tensors are defined as follows: In our conventions, and neglecting four-fermion terms that are not relevant to the present work, the ungauged N = 4, D = 5 Lagrangian for pure supergravity [16,17] reads: where: Neglecting three-fermion terms, the supersymmetry transformation laws are: C. The N = 4, D = 4 reduced supergravity We give here Lagrangian and transformation laws for the fully broken D = 4 effective theory, obtained via generalized reduction from the N = 4, D = 5 ungauged theory. The unbroken theory corresponding to the standard reduction can be easily extracted by setting the Scherk-Schwarz twist M = 0, while the partially broken theory can be obtained by choosing |m 1 | = |m 2 | = 0 in M . In the broken cases one can move to the unitary gauge, removing the goldstinos according to the standard procedure. Symplectic indices are suppressed according to the NW-SE convention. Neglecting as before four-fermion terms, the Lagrangian reads: