De Sitter in extended supergravity

We show that known de Sitter solutions in extended gauged supergravity theories are interrelated via a web of supersymmetry-breaking truncations. In particular, all N=8 models reduce to a subset of the N=4 possibilities. Furthermore, a different subset of the N=4 models can be truncated to stable de Sitter vacua in N=2 theories. In addition to relations between the known cases, we also find new (un)stable models.


Introduction
In N = 1 supergravity theories, one has a fair amount of freedom in introducing a scalar potential. Starting from an ungauged or massless theory, one can introduce a holomorphic superpotential. This situation radically changes when considering extended, i.e. N 2, supergravities. The scalar potential of extended theories is fully determined by gaugings: one cannot introduce a scalar potential without turning on a gauging, and similarly one cannot turn on a gauging without inducing a scalar potential. One might think that in N = 2 theories without hypermultiplets there is the additional possibility of introducing Fayet-Iliopoulos terms, but in fact these can also be understood as a gauging of the SU(2) R-symmetry group [1]. Therefore a scalar potential can only be obtained by introducing a number of constants that specify the gauging, instead of using an arbitrary N = 1 holomorphic superpotential. For this reason the scalar potentials of extended gauged supergravity have a certain rigidity, and one may wonder to what extent these allow for cosmologically interesting solutions, e.g. stable de Sitter solutions or slow-roll inflationary models.
Starting from the theory with maximal supersymmetry, for particular gaugings the N = 8 scalar potential does indeed allow for stationary points with a positive scalar potential [2]. However, in all known cases these are saddlepoints rather than minima, and hence the corresponding de Sitter solution is unstable. In fact, the second slow roll parameter η, which is defined as the lowest eigenvalue of the scalar mass matrix divided by the value of the scalar potential in the extremum, * Corresponding author.  (1.1) where D i is the covariant derivative on the scalar manifold, always takes the value η = −2 [3]. Therefore these unstable de Sitter solutions are also unsuitable for slow-roll inflation, as this requires |η| 1.
The situation ameliorates somewhat when going down to N = 4 supersymmetries. A systematic classification of semi-simple gauge groups giving rise to de Sitter vacua was performed for N = 4 theories with six vectormultiplets in [4,5]. It turns out to be possible to raise the value of η above −2; however, it always remains negative. Again stable de Sitter and/or slow-roll inflation is impossible in all known cases.
Things become more interesting when considering N = 2 theories. The implications of this smaller amount of supersymmetries differ from the previous situations in a number of respects: one can introduce both vector-and hypermultiplets and the scalar manifolds are no longer uniquely determined by the matter content. Indeed it turns out to be possible to evade the no-go theorems of e.g. [6,7] and to construct models with (meta-)stable de Sitter vacua, i.e. with η vanishing or positive [8]. In these models, the gauge groups consist of two factors: a non-compact electric factor and a compact magnetic factor. Crucially, the compact factor should have a non-trivial action on the hypersector, or Fayet-Iliopoulos terms in the absence of hypermultiplets. The different gaugings for the various amounts of supersymmetry have been constructed independently and may seem unrelated. However, it is the purpose of this Letter to show that they are in fact interrelated via a web of supersymmetry truncations. Indeed, all known N = 8 gaugings with de Sitter solutions can be related to a subset of the N = 4 models. Moreover, one can construct all known N = 2 models with stable de Sitter solutions from a dif- ferent subset of the N = 4 models. This offers a unifying picture of all known gaugings of extended supergravity with (un)stable de Sitter solutions. In the process of showing this, we will also construct new unstable N = 4 and stable N = 2 models. This Letter is organised as follows. In Section 2 we will review and slightly generalise the N = 4 gaugings. Their relation to N = 8 gaugings is discussed in Section 3. Subsequently, we truncate to stable N = 2 gaugings in Section 4. Finally, Section 5 contains our conclusions and outlook.

The N = 4 models revisited
Our starting point will be the possible gaugings of N = 4 gauged supergravity. A thorough discussion of this theory can be found in e.g. [9], and we will only present the necessary details and formulae here. We will restrict to the case of six vectormultiplets, as this is most relevant for the discussion of de Sitter vacua in the literature. In this case the global symmetry is SL(2) × SO (6,6). The scalars parametrise the corresponding cosets of dimensions 2 and 36, while the 12 vectors transform in the fundamental representation of SO (6,6). Furthermore, the electric and magnetic parts of all vectors form a doublet under SL (2).
As for gaugings, it was shown in [9]  to magnetic gaugings. Consistency of the gaugings imposes a number of quadratic constraints on these components, corresponding to Jacobi identities and orthogonality of charges. Together with the explicit form of the scalar potential these can be found in [9]. We will first restrict to a subset of all possible gaugings, namely those that correspond to a direct product of two six-dimensional gauge factors, G = G 1 × G 2 , embedded in an SO(3, 3) 2 subgroup of the global symmetry group. This implies that ξ αM = 0, as these components induce gaugings of (subgroups of) SL (2). Furthermore, only 40 of the 220 doublet components of f αMNP survive this truncation. By performing SL(2) transformations we can subsequently arrange G 1 to be an electric gauging, while G 2 is magnetic. 1 It will be useful to organise the remaining 40 components in four symmetric matrices with SL(4) indices i = 1, . . . , 4 Restricting to diagonal matrices this reduces to Similar expressions relate the structure constants of the second SO (3,3) factor to two matrices Q 2 andQ 2 . For this subset of gaugings, one can check that the general form of the scalar potential V as given in [9], restricted to the dilatons, can be written in terms of a superpotential W : 1 In particular, an SO(2) rotation can be used to bring e.g. G 1 to a purely electric gauging. Subsequently we can use the shift symmetry of the axion to make G 2 purely magnetic, provided it had a magnetic component to start with [10]. (2.4) where φ = (φ, φ 1 , . . . , φ 6 ). The first of these corresponds to the SL(2) factor, while the SO (6,6) dilatons are parametrised by two SL(4) factors of the form M ij 1 = diag e α 1 · φ , . . . , e α 4 · φ , (2.5) where the 4 vectors α i = {α iI } are weights of SL(4, R): 3. The definition for M 2 in terms of φ 4,5,6 is analogous.
The four matrices describe gaugings of CSO(p, q, 4 − p − q) in either of the two SO(3, 3) SL(4, R) factors. 2 To see this, let us look at the first SO (3,3) factor, spanned by (1, 2, 3,1,2,3), in detail. The formulae are completely analogous for the second factor. Restricting to (semi-)simple gaugings, 3 the Jacobi identities on the structure constants (2.2) imply that the matrices Q 1 andQ 1 are proportional. This leads to three physically distinct possibilities: where η ij is the invariant metric of the gauge group.
To make this more explicit, let us write out the structure constants in Cartesian coordinates with SO(6, 6) metric η MN = diag(−1, . . . , −1, 1, . . . , 1). For SO(4) gaugings we find (2.9) These gaugings have been previously considered in [5]. Concerning the two non-simple cases, one finds that both simple factors are non-vanishing for generic values of g 1 andg 1 . The singular cases arise when either of the two simple factors vanish, i.e. g 1 = ±g 1 .
The SO(3, 1) gauging is somewhat more intricate. For special values of g 1 andg 1 , it corresponds to either of the embeddings of [5]: for g 1 = ±g 1 it is the SO(3, 1) ± embedding. For these embeddings, the compact generators are given by either T 7,8,9 or T 1,2,3 , respectively. However, we find a one-parameter family of different embeddings, labelled by e.g.g 1 /g 1 . Another interesting special case isg 1 = 0, which we will refer to as the null embedding, denoted by SO(3, 1) 0 . For this embedding the compact generators are null linear combinations of the generators T 1,2,3 and T 7,8,9 .
To summarise the discussion so far, restricting to the (semi-)simple cases, there are three inequivalent gaugings in the first SO (3,3) factor, all specified by two parameters. In total there are six inequivalent gaugings, specified by four parameters: (2.10) In [5] it has been analysed which of these give rise to de Sitter vacua, with the restriction to the SO(3, 1) ± embeddings. All de Sitter vacua turned out to have an instability, either in the SL (2) or in the SO(6, 6) part. We will not analyse the remaining possibilities, with other SO(3, 1) embeddings, in detail at this point. However, we have looked at a few possibilities and found that the resulting de Sitter vacua are unstable as well. One interesting point is that the scalar mass matrix no longer splits up in an SL (2) and an SO(6, 6) part in general, as was the case for [5].
A number of generalisations is possible at this point [5]. First of all, in the case of non-simple gauge groups in (2.10), one can choose a different SL (2)  In the following sections we will show how all known N = 8 and N = 2 gauged supergravity models with de Sitter vacua are related to this simple set of N = 4 models. The scalar potential V is given in terms of a superpotential W : where the scalars in the SL(8, R)/SO(8) part of the E 7(7) /SU (8) coset are parametrised by a scalar matrix M ij . The dilatons are the diagonal part of this scalar matrix, 4 There are other formulations or duality frames of N = 8 supergravity, where a different subgroup of E 7 (7) is manifest, i.e. realised on the electric vectors. An example is SL(3) × SL (6). It would be of interest to see if this formulation allows for gaugings that might reduce to the more exceptional gaugings of N = 4 supergravity with a 9 + 3 split. Table 1 Truncations of N = 8 gaugings with unstable de Sitter solutions to N = 4 gaugings.
The resulting theories always have two factors with orthogonal angles and null embeddings, i.e.g 1 =g 2 = 0. All de Sitter solutions remain unstable in N = 4.
where M 1,2 have been defined in the previous section.
These gaugings give rise to Anti-de Sitter vacua for the SO(8) gauging, and de Sitter vacua for SO (5,3) and SO(4, 4) (see Table 1) [2]. For the intermediate cases SO (7,1) and SO (6,2) there is no maximally symmetric vacuum. It can be checked that the dilatons of the two de Sitter vacua always contain one tachyonic direction. In particular, the second derivatives of the scalar potential have the following eigenvalues in the vacuum:  In this section, we will discuss truncations of the N = 4 gauged supergravities with de Sitter vacua, as identified in [4,5], that break supersymmetry down to N = 2. Our goal is to obtain N = 2 theories where the de Sitter vacuum is stable. Since models with tachyonic SL(2) scalars do not lead to stable N = 2 de Sitter vacua, 5 The truncation of the SO(8) gauged theory to SO(4) 2 in N = 4 was already pointed out in [12].
for reasons that will become clear later on, we will discard these from now on. This leaves us with five possible gauge groups, that can be found in e.g. Table 2.
In general, truncations to N = 2 are achieved by modding out with respect to a Z 2 element of SO(6) × SO (6) ⊂ SO(6, 6) of the with n V + n H = 7. The spectrum of fields that are even under this Z 2 truncation, is that of an N = 2 supergravity, with n V vectormultiplets and n H hypermultiplets. The scalar fields then span the following symmetric scalar manifold 7 : The first two factors span a special Kähler manifold, while the third factor corresponds to a quaternionic-Kähler space. The SO (6,6) part of the N = 4 scalar manifold is truncated into two SO(p, q) parts, while the SL(2) part remains intact. This is the reason for discarding the N = 4 models with unstable SL(2) scalars: upon truncation, the instability is inherited by the N = 2 theory.
In order for the truncation (4.1) to be consistent in the presence of a gauging, the structure constants of the gauge group have to be even under the Z 2 truncation. The N = 4 gaugings in general then lead to N = 2 gaugings and possible Fayet-Iliopoulos terms. Which N = 2 gauging one is left with and whether or not the tachyonic scalars are truncated, has to be checked in all different cases.
For reasons of clarity, we will first discuss an example of such a truncation in more detail, before tackling the general case. Consider the gauge group SO(3, 1) × SO(3, 1). We will embed the adjoint of the first SO(3, 1) factor along the indices 1, 2, 3, 7, 8, 9, and the adjoint of the second factor along the directions 4, 5, 6, 10, 11, 12. Furthermore, the rotation subgroups of the two SO(3, 1) subgroups lie along the indices 7, . . . , 12, the boosts along the indices 1 . . . 6. The structure constants corresponding to the first gauge factor can be read off from (2.8) with g 1 =g 1 , and similar for the second gauge factor. Consider then the Z 2 trunca- The structure constants (2.8) are even under this Z 2 element and the resulting N = 2 supergravity has n V = 5 vectormultiplets and n H = 2 hypermultiplets. The resulting N = 2 gauge group is given by SO(2, 1) × SO (3). The first factor is spanned by the gauge vectors in the (1, 2, 9)-directions, while the second factor is spanned by the (10, 11, 12) gauge vectors. Crucially, both gauge factors act in both the special Kähler SO (2,4) and in the quaternionic-Kähler SO(4, 2) part, as can be seen from the adjoint representation of these generators. This truncation corresponds to the third model of [8] with r 0 = 1. This identification is confirmed by looking at the value of the scalar potential and its second derivatives. From [5] one finds that the value of the scalar potential in the de Sitter extremum is 6 The form of this Z 2 -truncation is defined up to permutations of the diagonal elements. 7 Amusingly, it has been argued in [8] that these particular N = 2 scalar manifolds are the only ones that can allow for stable de Sitter solutions.
S t a b l e ? which again coincides with Table 2 of [8].
The previous example clarifies and corroborates the truncation procedure. One can fix the Z 2 truncation, according to the number of vector-and hypermultiplets one wants to end up with. One then writes a form of the structure constants of the N = 4 gauge group that is Z 2 invariant. In the following we will discuss truncations to N = 2 theories with either n V = 3, n H = 4, or n V = 5, n H = 2.
Exhaustive lists of such gaugings that stem from truncation of an N = 4 supergravity with an SO(6, 6) unstable de Sitter vacuum are given in Tables 2 and 3. We use the following notation: • In some cases one has to remove N = 4 gauge factors in order to be able to truncate to N = 2, leading to less gauge factors Table 3 Truncations of N = 4 gaugings with de Sitter solutions having SO (6,6) instabilities to N = 2 gaugings with n V = 5, n H = 2. N = 4 gauging → N = 2 gauging with n V = 5, n H = 2 S t a b l e ?
on the right side of the table. Of course one should make sure that such singular limits of the gauge group do not affect the de Sitter solution. This is the case for all possibilities listed in the tables.
Subsequently, a careful analysis of the potential and its second derivatives can be carried out in order to determine whether the tachyonic scalars are truncated out. Using this procedure, we have been able to identify five stable de Sitter vacua in N = 2, that we will now list. For each specific truncation, we will indicate the extremum value of the potential, as well as the mass eigenvalues of the scalar fields (normalized by the potential) and their multiplicities (we will not list the mass eigenvalues of the SL(2) scalars as these are positive in all cases). Whenever possible, we will explicitly indicate whether these masses are associated to vectormultiplet or hypermultiplet scalars. Note that this is only possible when the mass matrix splits up in two blocks, corresponding to vector-or hyperscalars respectively. As in [4,5], we use the notation a ij = g i g j sin(α i − α j ). The indices i, j indicate the specific gauge factor of the N = 4 gauge group, in the order as they are written here. The g i , α i then denote the coupling constant, resp. SL(2) angle of the ith gauge factor.
Stable de Sitter vacua with n V = 3, n H = 4: One has to put the coupling constants of the two SO(2, 1) − factors equal to zero for consistency (i.e. g 3 = g 4 = 0). The potential then reaches the value V 0 = a 12 at the extremum. The masses of the vector-and hypermultiplet scalars are given by: (4.7) Stable de Sitter vacua with n V = 5, n H = 2: In this case the value of the potential at the vacuum is given by V 0 = 3a 12 . The mass matrix is not block diagonal. Its eigenvalues are explicitly given by: 2 3 (2×), 4 3 (6×) . (4.8) As indicated in the example given above, this model was discussed in [8] and we find perfect agreement with their results.
In order for this truncation to be consistent, one has to put the coupling constant g 3 of SO(2, 1) − equal to zero. The value of the potential at the extremum is then given by V 0 = √ 3a 12 . The mass eigenvalues for vector and hypermultiplet scalars are given by (6×) . (4.9) This truncation corresponds to the third model discussed in [8]. 8 Again, the value of the potential and the mass eigenvalues are in agreement with their results.
The value of the potential at the extremum is now given by    In the previous we have listed all eigenvalues of the scalar mass matrix. However, for every non-compact generator in the gauge group, there is always a flat direction in the scalar potential corresponding to a Goldstone boson [8]. The associated scalar is being eaten up by the gauge vector in order to render it massive via the BEH effect. Due to the SO(2, 1) + factors there are therefore always two non-physical vanishing eigenvalues in the vectorsector. Furthermore, in the fourth example there are two non-physical zero eigenvalues in the hypersector. Both the fourth and the fifth example are therefore fully stable, with all physical scalars having strictly positive mass eigenvalues. These are the first such examples in the presence of hypermultiplets. 9 A subsequent question could be whether these models allow for a truncation of the hypersector. In terms of SO(6) × SO (6), such a truncation would correspond to modding out with a Z 2 element whose first SO (6) factor is identical to that of the element to go to N = 2, while the second SO (6) factor is the identity I 6 .
It can be seen that such a subsequent truncation is only possible in the absence of any gaugings of non-compact isometries of the quaternionic-Kähler manifold, i.e. in the absence of any SO (1,1) or SO(2, 1) H factors. In this way the hypermultiplet truncation of (4.7) leads to the first model of [8]. Similarly, truncating the hypersector of (4.9) leads to the second model of [8]. Hence also Fayet-Iliopoulos parameters can be generated in this way in models without a hypersector. We have also checked that none of the unstable models of Tables 2 and 3 become stable after a truncation of the hypersector.

Discussion
In this Letter we have shown that all known extended supergravity models with de Sitter solutions are related via supersymmetry truncations. In particular, we have discussed relations between the N = 8 and N = 4 models, and between the N = 4 and N = 2 models. A natural question concerns the relation between N = 8 and N = 2. As follows from the previous discussion, only one of the five models of Section 4 can be obtained in this way. This is the fourth model with eigenvalues listed in (4.10). When descending in supersymmetry, the gauge groups of this model are SO (4,4) → SO(2, 2) 2 ± → SO(2, 1) + × SO(1, 1) 2 + × SO(2) + . (5.1) As the N = 4 model has the restrictiong 1 =g 2 = 0, we should impose a 14 = a 24 = a 23 = a 13 and a 12 = a 34 = 0 on the N = 2 side.
Let us discuss the crucial ingredients of the N = 2 models with stable de Sitter vacua. It was already pointed out in [8] that their three models have the following features in common. First of 9 We thank Mario Trigiante for pointing this out to us.
all, the gauge group is a direct product of a compact and a noncompact gauge factor, which have different SL (2) angles. Moreover, the compact factor needs to act non-trivially on the hypersector. From a rather large survey of candidate models with stable de Sitter vacua (listed in Tables 2 and 3), we have found only two additional possibilities. These satisfy the requirements identified by [8]. The new n V = 5 models are generalisations of [8] by hav- ing Abelian in addition to non-Abelian gauge factors. These are the first examples of fully stable de Sitter vacua in N = 2 theories with hypermultiplets. From the survey one can also extract the effect of a non-trivial action of the non-compact factor on the hypersector. As mentioned before, for the compact factor this was absolutely crucial. In contrast, it turns out that the opposite conclusion can be drawn for the non-compact factor. Indeed, having a non-compact gauge factor that acts on the hypermultiplet sector has a "destabilising" effect. This can e.g. be seen from Table 2, where the unique stable model becomes unstable when one replaces SO(2, 1) + by SO(2, 1) H + in the gauge group. This holds for both the second and the last line, which differ in the representation in which the SO(2, 1) + acts on the hypersector.
Points that merit further investigation include the following. First of all, we have not considered the most general possibility to obtain stable de Sitter vacua by truncations of N = 4 theories. One could in principal also obtain gaugings of N = 2 theories with a different number of vector-and hypermultiplets than considered here. One could also start from different N = 4 theories (e.g. with more vectors) and explore whether they allow for de Sitter vacua that become stable upon truncation.
A final question regards the possible higher-dimensional origin of the stable vacua. In this respect it is useful to note that most of the stable de Sitter vacua we found cannot be directly obtained by truncation of an N = 8 theory. It was shown in [13] that the non-compact N = 8 gaugings can be associated to solutions of eleven-dimensional supergravity with non-compact internal spaces. Our analysis however suggests that one cannot directly use this mechanism to interpret most of the stable N = 2 vacua from a higher-dimensional viewpoint. 10 As an intermediate step towards a better understanding of this, one might consider stable N = 2 vacua in five dimensions [14,15] and their relation to the four-dimensional ones [16].