All gaugings and stable de Sitter in D=7 half-maximal supergravity

We study the general formulation of gauged supergravity in seven dimensions with sixteen supercharges keeping duality covariance by means of the embedding tensor formalism. We first classify all inequivalent duality orbits of consistent deformations. Secondly, we analyse the complete set of critical points in a systematic way. Interestingly, we find the first examples of stable de Sitter solutions within a theory with such a large amount of supersymmetry.


Introduction
Supersymmetry preserving flux compactifications of string and M-theory are described by gauged supergravities. The introduction of background fluxes generates a scalar potential in the effective action, providing a successful arena for moduli stabilisation and supersymmetry breaking. The embedding tensor formalism [1,2] (see also [3] and references therein) provides an efficient formulation of such theories in which all possible flux-deformations are encoded into a single object, the embedding tensor. Much effort has been made in recent years to establish a correspondence between its components and the consistent flux-deformations that arise in string theory compactifications [4][5][6][7][8]. While some of these deformations (dubbed "geometric") can be rapidly identified with metric and p-form flux backgrounds, the higherdimensional origin of others is less clear (and were then named "non-geometric"). Interestingly, in this formulation the global symmetries inherited form the duality symmetries of the parent theories are manifest. These symmetries mix geometric and non-geometric deformations, allowing to identify the origin of non-geometric fluxes as the result of fluxcompactifications in dual backgrounds.
Regardless of their higher dimensional origin, it is clear that gauged supergravities offer a window to look into the vacuum structure of string theory. It is then of interest to explore and classify their possible gaugings and critical points. This is clearly a very ambitious programme, since the lower dimensional the gauged supergravity, the larger the space of possible deformations allowed by the embedding tensor. Still, it is worth the effort and here we give a small step in this direction. Before we comment on our main results, let us first offer a glimpse into previous results.
Among the three possible types of vacua: AdS, Mkw and dS, the latter is necessarily SUSY breaking and therefore the hardest to find in supersymmetric theories. Since we happen to live in such a vacuum, it is possibly the most interesting one. One finds instead a large landscape of AdS vacua which is mostly of interest in the context of the AdS/CFT correspondence, and there are many known examples of Mkw vacua of special interest in stringy constructions of standard-like models. Stable dS vacua, on the contrary, remain quite reluctant to discovery. We do have some knowledge of why this is the case, specially after some no-go theorems (see for example [9,10]) and after many studies that relate a positive cosmological constant with instabilities (among which we can mention [11][12][13]). Although in theories with small amount of supersymmetry there are very interesting examples of dS vacua [14][15][16], to our knowledge, there is no known fully stable dS vacuum in (half-)maximal gauged supergravity.
We then find it of interest to focus on particularly simple gauged supergravities in which an exhaustive analysis can be made outside the regions excluded by no-go theorems and previous surveys. Here we consider seven-dimensional half-maximal gauged supergravity [17], which is rich enough in structure to offer an intricate moduli space and at the same time is simple enough to make an exhaustive analysis. We begin by identifying the scalars of the theory (2.1)-(2.2), the possible gaugings (2.4) with their corresponding quadratic constraints (2.5), the scalar potential (2.6) and the shift matrices (2.8). The flux-deformations split into three types. On the one hand we have the universal half-maximal gaugings, consisting of the "three-form" f ABC and the unimodular deformations ξ A , and on the other there is a (non-gauging) massive deformation θ. We then classify all orbits of solutions to the quadratic constraints (tables 2 and 3), therefore finding all consistent seven-dimensional gauged supergravities with sixteen supercharges. The space of deformations splits into two branches, one with θ = 0 (branch 1) and another one with ξ A = 0 (branch 2). While the classification of orbits of branch 2 was exhaustively performed in [18], here we complete the classification for branch 1 by including the unimodular deformations ξ A .
We then move to the analysis of critical points, following the going-to-the-origin approach [6]. While branch 1 contains only Mkw vacua, branch 2 allows for non-semisimple configurations that exhibit both AdS and Mkw, and semisimple configurations with a large variety of minima. Interestingly, we find an SO(1, 3) gauging configuration with an AdS-Mkw-dS transition vacuum containing a fully stable dS window. We believe this is the first example of stable dS vacua in half-maximal supergravity.
Let us briefly sketch the structure of the paper. In section 2 we introduce the theory and all the elements that are necessary to explore its moduli space. Section 3 is devoted to classify all the duality orbits of deformations. In section 4 we perform the analysis of critical points and conclude with a discussion in section 5. For completion we also include an appendix where we collected some technical material that may be relevant for our analysis.
2 The half-maximal D = 7 gauged supergravities Half-maximal (ungauged) supergravity in seven dimensions coupled to three vector multiplets can be obtained by reducing type I supergravity in ten dimensions on a T 3 . The theory possesses 16 supercharges which can be rearranged into a pair of symplectic-Majorana (SM) spinors transforming as a doublet of SU(2) R . The full Lagrangian enjoys a global symmetry given by . The (64 B + 64 F ) bosonic and fermionic propagating degrees of freedom (dof's) of the theory are then rearranged into irrep's of G 0 as described in table 1. We refer to the appendix for a summary of our notations for all different indices used throughout the paper.
where αβ αβ can be viewed as the invariant metric of SU(2) R × SU(2), which can be brought into the form of an 1 4 . Given a realisation of M mn , M AB can then be obtained as in terms of the 't Hooft symbols [G A ] mn introduced in appendix B.
The kinetic Lagrangian for the scalar sector reads where M AB and M mn denote the inverse of M AB and M mn , respectively.
As a consequence of the linear constraint (LC), the deformations of the theory described by a generalised embedding tensor need to transform in the following G 0 irrep's [18] Θ ∈ 1 (−4) θ ⊕ 10 (+1) , where θ can be viewed as a Stückelberg coupling defining as a so-called p = 3-type deformation [19], whereas all the other irreducible pieces correspond to gaugings. In particular, Q & Q can be used in order to gauge a subgroup of SL(4), whereas ξ necessarily gauges the R + Σ generator as well as a suitable subgroup of SL(4).
It is worth mentioning that the 't Hooft symbols given in appendix B may be used to map Q ⊕Q and ξ into a 3-form f ABC (self-dual (SD) and anti-self-dual (ASD) part) and a vector The generators of the gauge algebra can be written as in terms of the embedding tensor. Please note that the (X mn ) pq rs 's are in general not traceless. In particular their trace is proportional to ξ, thus implying that one needs an embedding tensor in the 6 in order to gauge the R + generator outside of SL(4).
The closure of the gauge algebra and more general bosonic consistency impose the following quadratic constraints (QC) on the various irreducible components of the embedding where ξ mn ≡ 1 2 mnpq ξ pq . The above QC contain irreducible pieces transforming in the . Gauge invariance and supersymmetry force the scalar potential of the theory to be of the where g denotes an arbitrary gauge coupling 3 . The above expression generalises the one given in ref. [20] to the case with ξ mn = 0.
The gauging procedure induces mass terms for the fermions proportional to the gauge coupling constant of the following form where the shift matrices components of the T-tensor. These can be written in terms of the embedding tensor as (2.8) 2 We hereby correct a typo in ref. [20] concerning the sign of the θQ term, which, though, does not affect any of the results obtained there. 3 In section 4, we will set g = 8 when analysing the set of critical points of the various gauged theories The conditions for preserving supersymmetry read for a pair of SM spinors q α transforming in the fundamental representation of SU(2) or, equivalently, In terms of the shift matrices defined in (2.8), the scalar potential in (2.6) can be rewritten , of SL(2) × SO(6, 6) by the LC [21].
By reducing a consistent gauged theory in 7D on a T 3 , one obtains a particular class of embedding tensors in 4D. Hence, in order to identify the parts of f a[MNP] & ξ aM which have a seven-dimensional origin, one needs to branch the N = 4 embedding tensor w.r.t. our global 7D symmetry G 0 . This is done through the following chain where R + Σ is a combination of the R + sitting inside SL(2) Σ and one of the Cartan generators of SO(3, 3)Â in the last step one is allowed to identify the fundamental representation of SO(3, 3) A with the two-form of SL(4) by using the mapping in appendix B.
The fundamental index M of SO(6, 6) splits as The ten scalars of the 7D theory are embedded as (2) SO (2) and SO(6,6) SO(6)×SO(6) coset representatives, respectively. The embedding tensor of the 7D theory is embedded into the objects f a

Orbit classification of deformations
Each solution to the QC in (2.5) identifies a consistent deformation of half-maximal 7D supergravity. The global symmetry group R + × SL(4) ≈ R + × SO(3, 3) of the theory can be interpreted as T-duality and, since the QC are manifestly covariant w.r.t. such global symmetry, the space of solutions is naturally split into duality orbits.
This section is the natural generalisation of the analysis carried out in ref. [18] in the case θ = ξ mn = 0. Due to the last condition in (2.5), the set of all consistent gaugings is naturally split into two independent branches: • branch 1:

Orbits of deformations in branch 1
When θ = 0, the corresponding deformation can be understood as a gauging in the traditional sense, i.e. it is obtained by promoting a suitable subgroup of G 0 to a local symmetry. Table 2: All the T-duality orbits of consistent gaugings in the 6 ⊕ 10 ⊕ 10 of half-maximal supergravity in D = 7. Any value of (α, ξ 0 ) parameterises inequivalent orbits; the range of The shorthand 2 denotes the 2D Levi-Civita symbol. Note that, whenever ξ 0 = 0, one Abelian gauge generator needs to coincide with R + Σ . For more details on algebras marked with *, see appendix A.
In particular, as we have already observed earlier, Q &Q purely gauge generators within the SL(4) factor, whereas ξ necessarily gauges the extra R + Σ as well as a subgroup of SL(4). The T-duality orbit classifications for gaugings in this branch is presented in table 2.
As far as the higher-dimensional origin of the orbits in this branch is concerned, they can all be regarded as generalised twisted reductions of heterotic supergravity and hence they should all be accessible by means of twisted reductions of Double Field Theory (DFT) [22]. However, as already noted 5 in [18], this will generically require a relaxation of the section condition as originally proposed in ref. [24].

Orbits of deformations in branch 2
When ξ mn = 0, the most general consistent deformation of half-maximal 7D supergravity is a combination of a massive deformation induced by θ and a gauging of an arbitrary (up to six-dimensional) subgroup of SL(4). The consistency conditions and therefore the resulting gauge algebras turn out to be identical to those in the θ = ξ mn = 0 case already analysed in detail in ref. [18]. We collect in table 3 the corresponding results suitably combined with a non-zero θ.
Please note that, whenever θ = 0, all the duality orbits belonging to this second branch of consistent deformations are related to orientifold compactifications of type II supergravities on a T 3 with dyonic generalised fluxes and hence go beyond those twisted reductions of DFT considered in ref. [18].

Systematic analysis of critical points
After having classified all the consistent deformations of half-maximal supergravity in D = 7, the aim of this section is that of studying the critical points of the potential (2.6). To this end we introduce the following explicit parametrisation for the SL(4) scalars containing three dilatons and six axions. Please note that the vielbein in (4.1) can be related to the object appearing in table 1 in the following way by using the Dirac matrices of SO(4).
In terms of the vielbein V m m , the coset representative M mn appearing in the scalar potential is given by  Table 3: All the T-duality orbits of consistent deformations in the 1 ⊕ 10 ⊕ 10 of halfmaximal supergravity in D = 7. Any value of α parameterises inequivalent orbits; the range of α is everywhere − π 2 < α < π 2 , except in orbits 1, 2, 3 and 11, where it is reduced to − π 4 < α ≤ π 4 due to the symmetry w.r.t. interchanges between Q &Q. Note that the value of κ, instead, can be restricted to being either 0 or 1 by using an R + Σ rescaling. For more details on algebras marked with *, see appendix A.
However, in the origin of the scalar manifold, they explicitly read The inverse of the above kinetic metric turns out to be needed in order to write down the correct physical normalised mass matrix for Φ I , which then reads We remind the reader that one needs to take the Breitenlohner-Freedman (BF) bound [25] into account when it comes to judging the stability of an AdS critical point. In D dimensions, this translates into the following lower bound for the normalised mass of the mode in question , (4.8) which equals − 3 5 in 7D. For the analysis of critical points of the scalar potential we will adopt the going to the origin (GTTO) approach [6], i.e. we will make use of a non-compact R + × SL(4) transformation in order to restrict the search of solutions to the origin of moduli space without loss of generality.
Furthermore, since in both branches (1 & 2) we retain a set of embedding tensor components which happens to be closed w.r.t. compact global symmetries as well, we are still allowed to use an SO(4) to further simplify the embedding tensor while keeping all the scalars in the origin. In our case, we will exploit this possibility in order to assume a diagonal form for the symmetric matrix Q mn .
No-go argument for Λ = 0 within branch 1 When θ = 0, all the non-vanishing embedding tensor irrep's happen to have the same R + Σ weight. As a consequence, the complete scalar potential within this class of deformations can be written as where V 0 is an arbitrary function of the SL(4) scalars but independent of Σ.
This immediately implies that Σ is generically a run-away direction. This statement is analogous to that in ref. [26] concerning the run-away behaviour of the SL(2) dilaton in every purely electric gauging within half-maximal supergravity in four dimensions.
The only way of solving the Σ field equation is having a vanishing Λ at the solution. As a consequence, one is only left with Minkowski solutions of the no-scale type as the only possibility. An example of such a solution with ξ mn =Q mn = 0 is with the following (non-)normalised mass spectrum Such a solution corresponds to a reduction of type I supergravity on an ISO(2) group manifold [27]. We have not explored this branch exhaustively but so far we have no evidence for the existence of Minkowski solutions with non-zero ξ mn .

Critical points in branch 2
In this second branch of consistent deformed theories θ offers us the only terms in the scalar potential having a different scaling behaviour w.r.t. Σ, thus allowing us to stabilise all the moduli at non-vanishing values of the cosmological constant. This case represents the 7D analog of introducing non-trivial de Roo-Wagemans phases.
In this branch, the QC (2.5) take the following simple form If one furthermore restricts, as argued earlier to a diagonal Q mn , the above QC imply a diagonal form forQ mn as well. This, in turn, guarantees that all the equations of motion for the axions will be automatically satisfied, thus simplifying our analysis enormously.   (× 1)  and hence all fall into the classification of ref. [28]. All critical points of this type are collected in table 6.
ID θ Q mn orbit mass spectrum Exploring semisimple gaugings in the 10 ⊕ 10 When considering more general semisimple gaugings with both Q &Q turned on, the space of solutions to the QC and field equations suddenly becomes much richer and a complete analytical treatment gets much harder to perform. However, we were able to exhaustively explore some particularly relevant subcases within this class.
By setting, e.g.  [29] in the context of the G 2 invariant sector of N = 8 supergravity in four dimensions.
In solution 5 instead, the gauge group evolves from SO(4) to SO(2, 2) via CSO(2, 0, 2) in correspondence of λ = 1. However in this case, the cosmological constant stays vanishing for any value of λ. Stable dS in the SO(3, 1)-gauged theory (orbit 3) The continuous branches of solutions (labelled by ±) read Q ± = diag(1, λ, λ, λ) ,Q ± = f ± (λ) diag(λ, 1, 1, 1) and , The above solutions exhibit a stable dS window respectively given by where µ ± represent the two real roots 6 of the following polynomial (µ + < µ − ) This situation is depicted in figure 1. Such stable dS windows lie in the vicinity of a stable Minkowski critical point in analogy to what has been observed in refs [14][15][16] in the context of N = 1 supergravity in 4D.
An explicit example of stable dS critical point within branch + (λ = −3) is given by This example of dS vacuum is highly non-geomeric. In the standard language of Neveu-Schwarz fluxes [30], it includes not only the three-form H-flux and the metric ω-flux, but also the non-geometric Q and R-fluxes. None of them can be removed through a duality transformation, and then the orbit to which they belong is genuinely non-geometric in the sense of [18]. While the universal half-maximal gaugings were uplifted to duality covariant higher-dimensional theories in [31,32], massive deformations have been considered in that context in [33]. We believe that combining these results can provide an uplift of this vacuum eventually.

Conclusions
In this paper we have studied various aspects of supergravities in D = 7 with sixteen supercharges coupled to three vector multiplets. The most general deformations include a combination of a "Romans-like" massive deformation and a traditional gauging of a subgroup of the global duality group.
By using the embedding tensor formalism, we were first able to classify the inequivalent duality orbits of consistent deformations. It is worth mentioning that all orbits with no massive deformation can be regarded as generalised twisted reductions of DFT, provided that one allows for a dependence on doubled coordinates generically violating the section condition. The above massive deformation happens to have a non-trivial S-duality phase. Hence, any attempt of uplifting those orbits where a gauging is combined with such a deformation would require going beyond DFT reductions.

Solvable algebras
This class includes the gaugings described in rows 6 -11 of table 2 and 6 -10 of table 3. In the former case, the central generator named z will be realised through R + Σ .
The g 0 algebra Note that this algebra is solvable and not nilpotent even though its Cartan-Killing metric is completely zero.

Nilpotent algebras
This family comprises the gaugings found in rows 12 and 11 in tables 2 and 3, respectively.
The CSO (1, 0, 3) algebra The details about this algebra can be again found in ref. [34]; briefly summarising, the 6 generators are given by {t m , z m } m=1,2,3 and they satisfy the following commutation relations [t m , t n ] = mnp z p , (A. 8) with all the other brackets being vanishing. The order of nilpotency of this algebra is 2.
The l algebra The 6 generators {t 1 , · · · , t 6 } satisfy the following commutation relations from which we can immediately conclude that its nilpotency order is 3.

B Mapping between SL(4) and SO(3, 3)
The We perform the following explicit choice for the chiral 2 × 2 blocks where {σ i } i = 1, 2, 3 are the usual Pauli matrices given by