$dS_5$ vacua from matter-coupled 5D N=4 gauged supergravity

We study $dS_5$ vacua within matter-coupled $N=4$ gauged supergravity in five dimensions using the embedding tensor formalism. With a simple ansatz for solving the extremization and positivity of the scalar potential, we derive a set of conditions for the gauged supergravity to admit $dS_5$ as maximally symmetric background solutions. The results provide a new approach for finding $dS_5$ vacua in five-dimensional $N=4$ gauged supergravity and explain a number of notable features pointed out in previous works. These conditions also determine the form of the gauge groups to be $SO(1,1)\times G_{\textrm{nc}}$ with $G_{\textrm{nc}}$ being a non-abelian non-compact group. In general, $G_{\textrm{nc}}$ can be a product of $SO(1,2)$ and a smaller non-compact group $G'_{\textrm{nc}}$ together with (possibly) a compact group. The $SO(1,1)$ factor is gauged by one of the six graviphotons, that is singlet under $SO(5)\sim USp(4)$ R-symmetry. The compact parts of $SO(1,2)$ and $G'_{\textrm{nc}}$ are gauged by vector fields from the gravity and vector multiplets, respectively. In addition, we explicitly study $dS_5$ vacua for a number of gauge groups and compute scalar masses at the vacua. As in the four-dimensional $N=4$ gauged supergravity, all the $dS_5$ vacua identified here are unstable.


Introduction
Finding de Sitter (dS) vacua, with a positive cosmological constant, is one of the most interesting areas in gauged supergravities due to their importance in cosmology [1,2,3] and the dS/CFT correspondence [4]. Unlike the AdS vacua which are rather common in gauged supergravities, dS vacua are rare and occur only for a specific form of gauge groups. Moreover, embedding these vacua from lowerdimensional effective theories in string/M-theory is a highly non-trivial task, see for example [5,6,7,8], and a recent review [9].
In four dimensions, de Sitter vacua are extensively studied due to their direct relevance for cosmology, see for example [10,11,12,13,14]. In other dimensions, much less is known about this type of vacua. In this paper, we are interested in dS 5 vacua of N = 4 gauged supergravity coupled to vector multiplets constructed in [15,16]. There are no dS 5 vacua in pure N = 4 gauged supergravity of [17]. dS 5 vacua of N = 4 gauged supergravity coupled to two vector multiplets have already appeared in [18]. From a number of explicit examples, it has been pointed out that gauge groups admitting dS 5 vacua must contain non-compact abelian SO(1, 1) and non-compact non-abelian factors. To the best of our knowldege, these are the only known dS 5 vacua in the framework of matter-coupled N = 4 gauged supergravity. There are also a number of dS 5 vacua in the maximal N = 8 and the minimal N = 2 gauged supergravities [19,20,21,22,23]. However, most of these dS 5 vacua are unstable except for a few examples in [23].
We will use the embedding tensor formalism to analyze the extremization and positivity of the scalar potential in terms of fermion-shift matrices. With a simple ansatz, we can derive a number of conditions the embedding tensor must satisfy in order for the scalar potential to admit a dS 5 vacuum. In general, solving these conditions subject to the qudratic constraint will determine the form of possible gauge groups. The analysis and the resulting conditions are very similar to the conditions for the existence of maximally supersymmetric AdS 5 vacua studied in [24]. In a sense, our results can be regarded as a dS 5 analogue but are based on some assumption rather than supersymmetry. We will also show that our results explicitly lead to the fact that the gauge groups must be of the form SO(1, 1) × G nc with G nc being a non-abelian and non-compact group as pointed out in [18]. In general, for G nc being a semisimple group, G nc can be a product of SO(2, 1) and a smaller non-compact group G ′ nc . The paper is organized as follow. In section 2, we review relevant formulae for computing the scalar potential of N = 4 gauged supergravity in five dimensions coupled to vector multiplets in the embedding tensor formalism. In section 3, we derive the conditions that the embedding tensor needs to satisfy in order for the scalar potential to admit dS 5 vacua and determine a general form of gauge groups implied by these conditions. Some examples of these gauge groups are explicitly studied in section 4, and conclusions and comments are given in section 5. We also include an appendix containing useful identities for SO (5) gamma matrices and a collection of non-vanishing components of the embedding tensor for all gauge groups considered in this paper.
The vector multiplet contains a vector field A µ , four gaugini λ i and five scalars φ m . The n vector multiplets will be labeled by indices a, b = 1, . . . , n, and the component fields within these multiplets will be denoted by (A a µ , λ a i , φ ma ). From both gravity and vector multiplets, there are in total 6 + n vector fields which will be collectively denoted by . The 5n scalar fields from the vector multiplets parametrize the SO(5, n)/SO(5) × SO(n) coset. To describe this coset manifold, we introduce a coset representative V A M transforming under the global SO(5, n) and the local SO(5) × SO(n) by left and right multiplications, respectively. We use indices M, N, . . . = 1, 2, . . . , 5 + n for global SO(5, n) indices. The local SO(5) × SO(n) indices A, B, . . . will be split into A = (m, a). We can accordingly write the coset representative as The matrix V A M is an element of SO(5, n) and satisfies the relation with η M N = diag(−1, −1, −1, −1, −1, 1, . . . , 1) being the SO(5, n) invariant tensor. Gaugings are implemented by promoting a subgroup G 0 of the full global symmetry G = SO(1, 1) × SO(5, n) to be a local symmetry. The most general gaugings can be described by using the embedding tensor. N = 4 supersymmetry allows three components of the embedding tensor denoted by ξ M , [16]. The embedding tensor leads to minimal coupling of various fields via the covariant derivative in which ∇ µ is the usual space-time covariant derivative and t M N = t [M N ] and t 0 are generators of SO(5, n) and SO(1, 1), respectively. It should also be noted that ξ M , ξ M N and f M N P include the gauge coupling constants, and SO(5, n) indices M, N, . . . are lowered and raised by η M N and its inverse η M N , respectively. In term of the embedding tensor, gauge generators X MN P = (X M ) N P are given by To ensure that these generators form a closed subalgebra of G the embedding tensor must satisfy the following quadratic constraint Since we are only interested in maximally symmetric solutions, we will set all fields but the metric and scalars to zero. In this case, the bosonic Lagrangian of a general gauged N = 4 supergravity coupled to n vector multiplets can be written as where e is the vielbein determinant. The scalar potential is given by with M M N being the inverse of a symmetric matrix M M N defined by M M N P QRS is obtained from raising indices of M M N P QR defined by Equivalently, the scalar potential can also be written in terms of fermionshift matrices as or, after contraction of i and j indices, The fermion shift matrices are in turn defined by A aij where We have explicitly shown the symmetry of each tensor for later convenience. Note also that lowering and raising of USp(4) indices i, j, . . . with the symplectic form Ω ij and its inverse Ω ij correspond to complex conjugate for example A 1ij = Ω ik Ω jl A kl 1 = (A ij 1 ) * . V M ij is related to V M m by SO(5) gamma matrices and satisfies the relations 3 de Sitter vacua of N = 4 five-dimensional gauged supergravity We now consider gauge groups that lead to de Sitter vacua. Following [24], we first define V ij M in term of the coset representative V M m by where Γ ij m = Ω ik Γ mk j and Γ mi j are SO(5) gamma matrices. Similarly, the inverse element V ij M can be written as As in [24], it is useful to introduce "dressed" components of the embedding tensor Using the splitting of the index A = (m, a), we have the following components of the embedding tensor under the decomposition SO(5, n) → SO(5) × SO(n) In subsequent snalysis, we will determine the form of gauge groups that lead to dS 5 vacua. With some assumption, the analysis is very similar to that of the supersymmetric AdS 5 vacua studied in [24]. In order to have dS 5 vacua, we require that where, as in [24], means the quantity inside is evaluated at the vacuum. In terms of the fermion-shift matrices, these conditions read There are various ways to satisfy these relations. To make the analysis tractable, we will restrict ourselves to the following two possibilities: |µ| 2 denotes the cosmological constant, the value of the scalar potential at the vacuum V 0 .

3.1
We begin with the first possibility. Since A aij 2 consists of three representations of USp(4) namely 10, 5 and 1 corresponding to ρ a(ij) , ζ a [ij] and Ω ij τ a , respectively, the condition A aij 2 = 0 then implies that these components must vanish separately The condition A ij 1 = 0 gives Using the definitions in (16), we have The conditions in (25) then imply that due to the non-vanishing Σ and Γ ij m and Γ ij mn being all linearly independent.
Using the identity given in (120) and the fact that In addition, the condition (26) gives We can easily see that, apart from some numerical differences, these conditions have a very similar structure to those for the existence of supersymmetric AdS 5 vacua. However, we still need to check whether these conditions extremize the potential. Note that for the AdS 5 case, the potential is automatically extremized since in this case we have where we have introduced the variations of the coset representative V M A with respect to scalars φ ma With A ij 2 = A aij 2 = 0, we immediately see that δV = −2 A 1ij δA ij 1 = 0. However, in the present case, with A ij 1 = A aij 2 = 0, we find where we have used the relation We see that δV = 0 implies ρ (ij) = A ij 2 = 0. The cosmological constant then vanishes, and there are no possible dS 5 vacua.
It is now useful to note that if all the above conditions extremized the potential, the structure of the resulting gauge groups would be the same as in the AdS 5 case namely U(1) × H with H containing an SU(2) subgroup. Therefore, the same gauge group would give two types of vacua, AdS 5 and dS 5 , with different ratios of the gauge coupling constants for the U(1) and H factors. According to the explicit form of the scalar potential studied in [25] and [26], this is not the case. There is no dS 5 vacuum for gauge groups of the form U(1) × H for any values of the coupling constants precisely in agreement with the above result.

3.2
Using (27) and the identity in (120) together with we find, after some manipulation, that the condition A aik After using (119), we arrive at We then move to the extremization of the potential, δV = 2 A a 2ij δA aij and With the relations the first condition gives From the definitions (16), we can easily derive the following relations After some manipulation together with (115), we find that the condition (45) gives where we have used the previous results ξ mn = 0 and f mnp = 0. We now need to find solutions to all of these conditions subject to the quadratic constraints (6). With ξ M = 0, the quadratic constraint simplifies considerably These also imply [X 0 , X M ] = 0, so X 0 generates an abelian factor. On the other hand, X M in general generate a non-abelian group with structure constants f M N P with the first quadratic constraint in (51) being the corresponding Jacobi's identity.
Using the relations ξ mn = f mnp = 0, the second relation in (51) gives The first three conditions in (52) are automatically satisfied by (42). Using components (mnpq) of the first quadratic constraint in (51), we find from (43) that We now study consequences of all the above conditions on the structure of gauge groups. First of all, using the relation (47) in (41), we find that This result implies that, for |µ| 2 = 0, we must have ξ am = 0 and f amn = 0. Therefore, the abelian generator X 0 corresponds to a non-compact group SO(1, 1), and the non-abelian part must be necessarily non-compact. This is in agreement with the result of [18] in which it has been pointed out, based on a few explicit examples, that only gauge groups of the form SO(1, 1) × G nc lead to dS 5 vacua. With a simple assumption, we are able to derive this result. From equation (55), we see that ξ am and f amn must be non-vanishing for different values of indices m and a. This also implies that gauge groups of the form SO(1, 1) × G nc are only possible for n ≥ 2 as shown by explicit computation in [18] that the existence of dS 5 vacua requires at least two vector multiplets. We now determine possible gauge groups allowed by the above conditions. By using SO(5, n) generators of the form (t M N ) P Q = δ Q [M η N ]P , we find that the gauge generators in the fundamental representation of SO(5, n) are given by (57) where we have set ξ M = 0. Note also that, for ξ M = 0 the gauge group SO(1, 1) × G nc is entirely embedded in SO(5, n).
We begin with the simplest possibility namely ξ ab = f mab = f abc = 0. These components are not constrained by the existence of dS 5 vacua, so we can have dS 5 vacua regardless of their values. Since in this case we have f abc = 0, the compact part of G nc must be an abelian SO(2) group. We find that f amn must correspond to SO(2, 1) group. The full gauge group then takes the form of a product SO(1, 1) × SO(2, 1).
For ξ ab = f abc = 0 but f mab = 0, the quadratic constraint implies that f mab and f amn cannot have common indices. Therefore, these components generate two separate non-compact groups with f amn corresponding to SO(2, 1) as in the previous case. Since the existence of dS 5 vacua requires f mnp = 0, the compact subgroup of the non-compact group corresponding to f mab must be SO (2), hence f mab generate another SO(2, 1) ′ factor. However, it should be noted that the compact parts of the two SO(2, 1)'s are embedded in the matter and Rsymmetry directions, respectively. The full gauge group in this case is then given by SO(1, 1) × SO(2, 1) × SO(2, 1) ′ .
For ξ ab = f mab = 0 but f abc = 0, we have the following non-vanishing components of the embedding tensors in which we have split indices as m = (m, m ′ ) and a = (ã, a ′ ). In this case, f a ′ b ′ c ′ together with f a ′ m ′ n ′ form a non-compact groupG nc whose compact subgroup H c is generated by f a ′ b ′ c ′ . The quadratic constraint gives the standard Jacobi's identity forH c . In general, there can be a subspace in which f a ′ b ′ c ′ do not have common indices with f a ′ m ′ n ′ . We will denote these components as f a ′′ b ′′ c ′′ with f a ′′ m ′ n ′ = 0. In this case, f a ′′ b ′′ c ′′ form a separate compact group H c while f a ′ m ′ n ′ and f a ′ b ′ c ′ still generate a smaller non-compact group G nc . The full gauge group can be written as SO(1, 1) × G nc × H c .
If in addition f m ′ a ′ b ′ = 0, we have an extra factor of SO(2, 1) ′ , hence a general gauge group takes the form of with the compact parts of SO(2, 1) and G nc are embedded along the R-symmetry and matter directions, respectively. The non-vanishing ξ ab on the other hand does not change the structure of the gauge groups.

dS 5 vacua from different gauge groups
In this section, we will study N = 4 gauged supergravity with gauge groups that lead to dS 5 vacua identified in the previous section. We will restrict ourselves to a simple case of ξ ab = f mab = 0 given in (58). In general, the SO(1, 1) factor can be embedded as a diagonal subgroup SO(1, 1)

) factor is characterized by the values of indicesm andã.
In order to have non-vanishing f m ′ n ′ a ′ , we must have d ≤ 3 to allow for the antisymmetry in m ′ and n ′ indices. An explicit form of the gauge generators is given by Generators X m ′ and X a ′ correspond to non-compact and compact generators of G nc , respectively. Note also that we can have at most 5 − d non-compact generators for G nc . We now discuss possible gauge groups for different values of d.
For d = 1, G nc is a non-compact subgroup of SO(4, n − 1) with at most four non-compact generators. We can gauge the following groups For d = 2 and d = 3, we have the following admissible gauge groups We will explicitly study scalar potentials arising from these gauge groups and their dS 5 vacua. In most of the following analysis, we mainly work in the case of n = 5 vector multiplets for definiteness with an exception of SO(1, 1) × SO(4, 1) gauge group that requires n = 7. In addition, in many cases, we will consider only non-vanishing dilaton due to the complexity of including scalars from vector multiplets. In all of these gauge groups we have explicitly checked that the conditions A ij 1 = A ij 2 = 0 and A aij 2 A a 2kj = 1 4 V 0 δ i k are satisfied. We note here that SO(1, 1) × SO(2, 1) and SO(1, 1) (2) diag × SO(2, 1) gauge groups have already been studied in [18]. However, all the remaining gauge groups are new and arise from our analysis given in the previous section.
To compute the scalar potential, we need an explicit form of the coset representative for SO(5, n)/SO(5) × SO(n). It is convenient to define a basis of GL(5 + n, R) matrices (e M N ) P Q = δ M P δ N Q in terms of which SO(5, n) non-compact generators are given by Y ma = e m,a+5 + e a+5,m , m = 1, 2, . . . , 5, a = 1, 2, . . . , n .
To adopt the normalization used in [16] with we will use a slightly different definition of V M ij and V ij The explicit representation of the SO(5) gamma matrices is chosen to be where σ i , i = 1, 2, 3 are the usual Pauli matrices.

SO(1, 1) × SO(2, 1) gauge group
We begin with the simplest possible gauge group SO(1, 1) × SO(2, 1). The embedding of SO(1, 1) factor is given by while the f M N P describing the embedding of SO(2, 1) has the following independent non-vanishing component This means that the compact part SO(2) ⊂ SO(2, 1) is generated by X 8 while the two noncompact generators are embedded in the R-symmetry directions M = 2, 3. Note also that this embedding tensor manifestly satisfies the quadratic constraint (51). At the vacuum, SO(1, 1) × SO(2, 1) gauge symmetry will be broken to its compact subgroup SO (2). In principle, we can study the scalar potential for scalars which are singlets under this residual SO(2) symmetry. However, the number of these singlet scalars is rather large, 15 singlets in this case. Therefore, we will mainly analyze the scalar potential only to second-order in fluctuations of scalar fields and give explicit form of the potential only for the dilaton Σ nonvanishing. With all scalars from vector multiplets set to zero, the scalar potential is given by This potential admits a unique critical point at The value of the dilaton Σ at the vacuum can be rescaled to Σ = 1 by setting g 2 = g 1 which gives the dS 5 vacuum in a simple form The linearized analysis of the full scalar potential for all 26 scalars shows that this is also the critical point of the full potential with scalar masses given by (73) The number in the subscripts of each mass value is the multiplicity of that mass value at the critical point, and the dS 5 radius is given by L 2 = 6/V 0 . This dS 5 critical point is unstable because of the negative mass value 2(1 − √ 33). The mass value 16 corresponds to that of Σ, and the nine massless scalars contain three Goldstone bosons of the symmetry breaking SO(1, 1) × SO(2, 1) → SO(2).
We now consider another embedding for the SO(1, 1) factor as a diagonal subgroup of SO(1, 1)×SO(1, 1) generated by the non-compact generators Y 41 and Y 52 of SO (5,5). The embedding tensor for SO(2, 1) remains the same as in the previous case. We will use the following non-vanishing components of the embedding tensor for the SO(1, 1) The scalar potential for this gauging is given by with a unique critical point at We can again shift this critical point to the value Σ = 1 by setting g 2 = √ 2g 1 and obtain the value of the cosmological constant Scalar masses at this critical point are given by Notice that all masses are quantized, and the dS 5 vacuum is unstable because of the negative mass values −8 and −6. The value m 2 L 2 = 16 corresponds to that of Σ. Three of the eight massless scalars correspond to the Goldstone bosons of the symmetry breaking SO(1, 1) diag × SO(2, 1) → SO(2) at the vacuum.

SO(1, 1)
In this case, the embedding tensor for SO(2, 1) remains the same as the previous two cases while the SO(1, 1) is a diagonal subgroup of SO(1, 1) × SO(1, 1) × SO(1, 1). Non-vanishing components of the full embedding tensor are now given by These give the following scalar potential for Σ which admits a critical point at After setting g 2 = √ 3g 1 , we find the cosmological constant Scalar masses are given by This dS 5 vacuum is unstable due to the negative mass value The embedding tensor for SO(1, 1) is still given by (68). Under the compact SO(2) × SO(2) symmetry, there are five singlet scalars corresponding to the noncompact generators Y 2a , a = 1, 2, . . . , 5. The coset representative can be written as The scalar potential is given by which admits only one critical point at φ i = 0, i = 1, 2, . . . , 5 and Σ = g 2 2 + g 2 The scalar mass spectrum is The dS 5 vacuum is unstable because of the negative mass value 2(3 − √ 17). The mass value 16 corresponds to that of the dilaton while the five scalars with zero mass correspond to the Goldstone bossons of the symmetry breaking SO(1, 1) × SO(2, 1) × SO(2, 1) → SO(2) × SO(2).
This potential admits a critical point at As in other cases, we can bring this critical point to the value Σ = 1 by setting . We have not found other critical points from the above potential. The scalar mass spectrum at the dS 5 critical point is given by which implies that the dS 5 is unstable because of the negative mass values −20/3 and 2 3 (7− √ 145). There are four Goldstone bosons corresponding to the symmetry breaking SO(1, 1) × SO(3, 1) → SO (3).
We now move to a similar gauge group of the form SO(1, 1) Apart from a dS 5 critical point with all scalars from vector multiplets vahishing, we have not found any other critical point. Therefore, to simplify the expression, we will give only the potential with only the dilaton Σ non-vanishing Choosing g 2 = 2 3 g 1 , the dS 5 critical point is given by with the scalar mass spectrum This dS 5 vacuum is again unstable because of negative mass values −2/3 and 4 3 (5 − √ 73). The mass value m 2 L 2 = 16 corresponds to that of Σ.

SO(1, 1) × SU (2, 1)
We now consider the last gauge group that can be embedded in SO (5,5). Nonvanishing components of the embedding tensor for the non-abelian part SU (2,1) are given by Gauge generators (X 7 , X 8 , X 9 ) and X 10 correspond to the compact SU(2) and U(1), respectively. Among the 25 scalars in SO(5, 5)/SO(5) × SO (5), there are two SU(2) singlets corresponding to the following non-compact generators Y 51 and Y 55 . The coset representative can be written as and, with the embedding tensor of SO(1, 1) given by (68), the scalar potential reads There is only one critical point at The scalar mass spectrum for all the 26 scalars is identical to that of SO(1, 1) × SO(2, 2) gauge group with the five massless scalars being the Goldstone bosons of the symmetry breaking SO(1, 1) × SU(2, 1) → SU(2) × U(1).
The embedding of SO(1, 1) is again given in (68). Under SO(4) ⊂ SO(4, 1) symmetry, there are three singlet scalars from SO(5, 7)/SO(5) × SO(7) coset corresponding to non-compact generators Y 51 , Y 61 and Y 62 − Y 71 . Using the coset representative we find the scalar potential for this gauge group which admits only one critical point at Scalar masses at this vacuum are given by which implies the instability of the dS 5 critical point because of the negative mass value 3 − √ 17. The five massless scalars are Goldstone bosons of the symmetry breaking SO(1, 1) × SO(4, 1) → SO(4).

Conclusions
In this paper, we have studied dS 5 vacua of five-dimensional N = 4 gauged supergravity coupled to vector multiplets. By using a simple ansatz for solving the extremization and positivity of the scalar potential, we have derived a set of general conditions for determining the form of gauge groups admitting dS 5 vacua as maximally symmetric backgrounds. In particular, these gauge groups must be of the form SO(1, 1) × SO(2, 1) × G nc × H c with SO(1, 1) gauged by one of the graviphoton that are SO(5) R singlet. G nc is a non-compact group whose compact subgroup is gauged by vector fields in vector multiplets. On the other hand, the compact and non-compact parts of SO(2, 1) are embedded along the R-symmetry and matter directions, respectively. H c ⊂ SO(n) is a compact group gauged by vector fields from the vector multiplets. The results provide a new approach for finding dS 5 vacua which could be of particular interest in various contexts such as in the dS/CFT correspondence and cosmology. We have also explicitly computed scalar potentials and studied dS 5 vacua for a number of gauge groups. However, as in the N = 4 gauged supergravity in four dimensions, all of these dS 5 vacua are unstable.
It would be interesting to carry out a similar analysis in gauged supergravities in other dimensions in particular in four-dimensional N = 4 gauged supergravity and compare with the results given in [11] and [13]. Another direction would be to embed the known dS 5 vacuum of N = 8 gauged supergravity with SO(3, 3) gauge group in N = 4 gauged supergravity considered here. On the other hand, embedding the dS 5 vacua identified here in N = 2 gauged supergravity and truncating out some scalar fields might give stable dS 5 vacua within N = 2 gauged supergravity as in a similar study in four-dimensional gauged supergravity [14]. Finally, relations between dS 4 and dS 5 vacua in N = 4 gauged supergravity similar to the N = 2 case studied in [22] also deserve further study. We hope to come back to these issues in future works.

A Useful formulae
In this section, we collect some useful identities involving SO(5) gamma matrices which are useful in the analysis of the constraints on the embedding tensor. For convenience, we also give a summary of non-vanishing components of the embedding tensor for gauge groups considered in the main text.

A.1 SO(5) gamma matrices
The SO(5) gamma matrices Γ mi j satisfy the Clifford algebra Γ mi k Γ nk j + Γ ni k Γ mk j = δ j i δ mn .
The charge conjugation matrix C ij and its inverse C ij are related to the USp(4) symplectic form as follow C ij = −Ω ij and C ij = Ω ij .
We can now define the gamma matrices with all indices up or down In particular, we have the following relations Other useful identities are Γ ij m Γ nij = 4δ mn , (114) Tr(Γ mn Γ pq ) = 4(δ mq δ np − δ mp δ nq ), with ǫ 12345 = +1. The sign choices are related to the definition Γ 5 = ±Γ 1 Γ 2 Γ 3 Γ 4 . In our calculations, we choose the convention with the upper sign.

A.2 Embedding tensor for possible gauge groups with dS 5 vacua
Examples of gauge groups that lead to dS 5 vacua are listed in table 1 along with non-vanishing components of the corresponding embedding tensor.