\(dS_4\) vacua from mattercoupled 4D \(N=4\) gauged supergravity
Abstract
We study \(dS_4\) vacua within mattercoupled \(N=4\) gauged supergravity in the embedding tensor formalism. We derive a set of conditions for the existence of \(dS_4\) solutions by using a simple ansatz for solving the extremization and positivity of the scalar potential. We find two classes of gauge groups that lead to \(dS_4\) vacua. One of them consists of gauge groups of the form \(G_{\text {e}}\times G_{\text {m}}\times H\) with H being a compact group and \(G_{\text {e}}\times G_{\text {m}}\) a noncompact group with \(SO(3)\times SO(3)\) subgroup and dyonically gauged. These gauge groups are the same as those giving rise to maximally supersymmetric \(AdS_4\) vacua. The \(dS_4\) and \(AdS_4\) vacua arise from different coupling ratios between \(G_{\text {e}}\) and \(G_{\text {m}}\) factors. Another class of gauge groups is given by \(SO(2,1)_{\text {e}}\times SO(2,1)_{\text {m}}\times G_{\text {nc}}\times G'_{\text {nc}}\times H\) with SO(2, 1), \(G_{\text {nc}}\) and \(G'_{\text {nc}}\) dyonically gauged. We explicitly check that all known \(dS_4\) vacua in \(N=4\) gauged supergravity satisfy the aforementioned conditions, hence the two classes of gauge groups can accommodate all the previous results on \(dS_4\) vacua in a simple framework. Accordingly, the results provide a new approach for finding \(dS_4\) vacua. In addition, relations between the embedding tensors for gauge groups admitting \(dS_4\) and \(dS_5\) vacua are studied, and a new gauge group, \(SO(2,1)\times SO(4,1)\), with a \(dS_4\) vacuum is found by applying these relations to \(SO(1,1)\times SO(4,1)\) gauge group in five dimensions.
1 Introduction
De Sitter (dS) vacua are solutions of general relativity and gauged supergravity with positively constant curvature. Although these solutions are originally of mathematical interest, cosmological observations, see for example [1, 2, 3], suggest that the universe has a very small positive value of cosmological constant. Furthermore, these solutions have attracted much attention during the past twenty years due to the proposed dS/CFT correspondence [4], a holographic duality between a theory of gravity on dS background and a Euclidean conformal field theory along the line of the AdS/CFT correspondence [5].
Unlike the AdS counterpart found naturally in many gauged supergravities, dS vacua are very rare and (if they exist) the embedding in string/Mtheory is highly nontrivial. Various approaches have been devoted to search for these vacua with only a small number of solutions found, see [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] for an incomplete list. All these results even suggest that string/Mtheory does not admit de Sitter solutions, for a recent review see [24] and references therein. In addition, there are a number of previous works considering de Sitter solutions of gauged supergravities. In four dimensions, de Sitter vacua are extensively studied, see for example [25, 26, 27, 28, 29]. On the other hand, de Sitter vacua in higher dimensions are less known [30, 31, 32, 33, 34, 35, 36].
In this paper, we study \(dS_4\) vacua in fourdimensional \(N=4\) gauged supergravity coupled to vector multiplets constructed in [37] using the embedding tensor formalism, see [38, 39] for an earlier construction. We do not attempt to find new \(dS_4\) solutions but to introduce a new approach for finding \(dS_4\) vacua. We will extend a recent result initiated in the study of \(dS_5\) vacua in fivedimensional \(N=4\) gauged supergravity given in [36]. Unlike the previous results on \(dS_4\) solutions mostly obtained from using the old construction of [39], working in the embedding tensor formalism has the advantage that different deformations, gaugings and nontrivial SL(2) phases, are encoded in a single framework. Furthermore, an explicit form of the gauge group under consideration needs not be specified at the beginning. This allows to formulate a general setup and subsequently apply the results to a particular gauge group.
We now describe the procedure used in our analysis. In general, the scalar potential of a gauged supergravity can be written as a quadratic function of fermionshift matrices. In [36], the extremization and positivity of the scalar potential are solved by using a particular form of an ansatz such that the gravitinoshift matrix (usually denoted by the \(A_1\) tensor) vanishes. This guarantees the positivity of the potential and, with a suitable condition, the potential can be extremized. With the help of the embedding tensor formalism, a general form of gauge groups that lead to \(dS_4\) vacua can be determined from the conditions imposed on the fermionshift matrices.
It should be noted that the procedure and the resulting conditions are very similar to those arising from the existence of supersymmetric \(AdS_4\) vacua given in [40]. However, there is a crucial difference in the sense that the conditions for \(dS_4\) are derived from a particular ansatz, but those for \(AdS_4\) are obtained by requiring unbroken supersymmetry. While the latter guarantee that the results are vacuum solutions of the \(N=4\) gauged supergravity and, in particular, extremize the scalar potential, we need to explicitly impose the extremization of the potential in the former case. We will see that some of these extra conditions are already implied by the quadratic constraint. The remaining ones imply that the gauge groups must be dyonically embedded in the global symmetry group similar to the \(AdS_4\) case.
The paper is organized as follows. In Sect. 2, we review relevant formulae for computing the scalar potential of \(N=4\) gauged supergravity coupled to vector multiplets in the embedding tensor formalism. In Sect. 3, we derive the conditions for the scalar potential to admit \(dS_4\) vacua by solving the extremization and positivity of the potential. A general form of gauge groups is also determined by solving these conditions subject to the quadratic constraint. In Sect. 4, we explicitly verify that all the previously known \(dS_4\) vacua in \(N=4\) gauged supergravity are encoded in our results. Some relations between four and fivedimensional embedding tensors for gauge groups that admit de Sitter vacua are given in Sect. 5. Conclusions and comments on the results are given in Sect. 6. We also include an appendix collecting useful identities for SO(6) gamma matrices.
2 Fourdimensional \(N=4\) gauged supergravity coupled to vector multiplets
In this section, we breifly review \(N=4\) gauged supergravity coupled to an arbitrary number n of vector multiplets. We will mainly give relevant formulae for finding the scalar potential which is the most important part in our analysis. For more detail, interested readers are referred to [37].
For \(N=4\) supersymmetry, there are two types of supermultiplets, gravity and vector multiplets. The former contains the graviton \(e^{\hat{\mu }}_\mu \), four gravitini \(\psi _{\mu i}\), \(i=1,\ldots ,4\), six vectors \(A^m_\mu \), \(m=1,\ldots ,6\), four spin\(\frac{1}{2}\) fermions \(\lambda _i\), and a complex scalar \(\tau \). The field content of the latter is given by a vector field \(A_\mu \), four gaugini \(\lambda ^{i}\) and six scalars \(\phi ^{m}\). We use the following conventions on various types of indices. Spacetime and tangent space indices will be denoted by \(\mu , \nu , \ldots = 0,1,2,3\) and \(\hat{\mu }, \hat{\nu },\ldots = 0,1,2,3\), respectively. Indices \(m,n,\ldots =1,2,\ldots , 6\) label vector representation of \(SO(6)\sim SU(4)\) Rsymmetry while \(i,j,\ldots \) denote chiral spinor of SO(6) or fundamental representation of SU(4). The n vector multiplets are labeled by indices \(a,b,\ldots =1,\ldots ,n\). Accordingly, the field content of the vector multiplets can be collectively written as \((A^a_\mu ,\lambda ^{ai},\phi ^{am})\).
3 de Sitter vacua of \(N=4\) fourdimensional gauged supergravity
In this section, we will look for gauge groups that lead to de Sitter vacua. The analysis is similar to that given in [36] for \(dS_5\) vacua. Furthermore, the procedure is closely parallel to the case of maximally supersymmetric \(AdS_4\) vacua given in [40].
 1.
\(\langle A_1^{ij} \rangle =\langle {A_{2ai}}^j \rangle =0\) and \(\langle A^{ik}_2 A^*_{2jk} \rangle =\frac{9}{4}\mu ^2\delta ^i_j\) with \(A^*_{2ij}\delta A^{ij}_2+\delta A^*_{2ij} A^{ij}_2=0\).
 2.
\(\langle A_1^{ij} \rangle =\langle A^{ij}_2 \rangle =0\) and \(\langle {A_{2ai}}^k {A^*_{2ak}}^j \rangle =\frac{1}{2}\mu ^2\delta ^j_i\) with \(\delta {A_{2ai}}^j {A^*_{2aj}}^i+{A_{2ai}}^j\delta {A^*_{2aj}}^i=0\).
3.1 \(\langle A_1^{ij} \rangle =\langle {A_{2ai}}^j \rangle =0\) and \(\langle A^{ik}_2 A^*_{2jk} \rangle =\frac{9}{4}\mu ^2\delta ^i_j\)
At this point, all the remaining parts of the whole analysis are essentially the same as in [40]. In particular, the resulting gauge groups that can give rise to \(dS_4\) vacua must take the same form as in the \(AdS_4\) case. We will not repeat all the details here but simply summarize the structure of possible gauge groups. First of all, it should be noted that other components of the embedding tensors, \(\mathfrak {f}_{mab}\) and \(\mathfrak {f}_{abc}\), are not constrained by the existence of \(dS_4\) vacua.
As a final comment, we note that although the gauge groups giving rise to \(dS_4\) vacua are exactly the same as those leading to supersymmetric \(AdS_4\) solutions, the two types of vacua occur at different values of the ratio between the coupling constants of \(SO(3)_{\text {e}}\) (\(G_{\text {e}}\)) and \(SO(3)_{\text {m}}\) (\(G_{\text {m}}\)). More precisely, the two cases have the coupling ratios with opposite sign, recall the sign change in (39) as compared to the results of [40]. We will see this in explicit examples in the next section.
3.2 \(\langle A_1^{ij} \rangle =\langle A^{ij}_2 \rangle =0\) and \(\langle {A_{2ai}}^k {A^*_{2ak}}^j \rangle =\frac{1}{2}\mu ^2\delta ^j_i\)
Finally, using the result from (54), we find that the second condition in (57) is already implied by the (mnpq)component of the quadratic constraint. Therefore, we have a set of consistent conditions to be imposed on the embedding tensor. In the following, we will look for explicit solutions and possible forms of the corresponding gauge groups. The analysis will be closely parallel to that in the previous case.
We first note that components \(\mathfrak {f}_{mab}\) and \(\mathfrak {f}_{abc}\) are not constrained by the existence of \(dS_4\) vacua. These components can be anything without affecting the \(dS_4\) vacua. However, the structure of gauge groups will be different for different values of \(\mathfrak {f}_{mab}\) and \(\mathfrak {f}_{abc}\). We now look at various possibilities.
4 \(dS_4\) vacua from different gauge groups
The six gauge groups giving rise to \(dS_4\) vacua as given in [28]. The embedding tensors for these gauge groups are obtained by imposing some relations between the coupling constants as shown in the last column
Gauge groups in [28]  Gauge groups in [29]  Conditions 

\(SO(3)^2_\times SO(3)^2_+\)  \(SO(4)_{\text {e}} \times SO(4)_{\text {m}}\)  \(g_1, \tilde{g}_1, g_2, \tilde{g}_2 \ne 0\) 
\(SO(3,1)_+\times SO(3,1)_+\)  \(SO(3,1)_{\text {e}}\times SO(3,1)_{\text {m}}\)  \(\begin{array}{l}\tilde{g}_1 = g_1, \tilde{g}_2 = g_2,\\ g_1, g_2 \ne 0\end{array}\) 
\(SO(3,1)_\times SO(3,1)_\)  \(SO(3,1)_{\text {e}}\times SO(3,1)_{\text {m}}\)  \(\begin{array}{l}\tilde{g}_1 = g_1, \tilde{g}_2 = g_2,\\ g_1, g_2 \ne 0 \end{array}\) 
\(SO(3,1)_\times SO(3)_\times SO(3)_+\)  \(\begin{array}{l}SO(3,1)_{\text {m}}\times SO(4)_{\text {e}}\\ SO(3,1)_{\text {e}}\times SO(4)_m\end{array}\)  \(\begin{array}{l}\tilde{g}_2 = g_2, \\ g_2, g_1, \tilde{g}_1 \ne 0\\ \tilde{g}_1 = g_1,\\ g_1, g_2, \tilde{g}_2 \ne 0\end{array}\) 
\(SO(2,1)^2_+\times SO(2,1)^2_\)  \(SO(2,2)_{\text {e}} \times SO(2,2)_{\text {m}}\)  \(g_1, \tilde{g}_1, g_2, \tilde{g}_2 \ne 0\) 
\(SO(3,1)_+\times SO(2,1)_+\times SO(2,1)_\)  \(\begin{array}{l}SO(3,1)_{\text {m}}\times SO(2,2)_{\text {e}}\\ \\ SO(3,1)_{\text {e}}\times SO(2,2)_{\text {m}}\end{array}\)  \(\begin{array}{l}\tilde{g}_2 = g_2,\\ g_2, g_1,\tilde{g}_1 \ne 0 \\ \tilde{g}_1 = g_1,\\ g_1, g_2, \tilde{g}_2 \ne 0\end{array}\) 
 For \(SO(3)_+\times SO(2,1)_+^3\), we rewrite it as \(SO(3)\times SO(2,1)\times SO(2,1)^2\) with the embedding tensor given bywith \(\alpha =\pm \) and \(\beta =\mp \) corresponding to the following electric and magnetic factors \(SO(3)_{\text {e(m)}}\times SO(2,1)_{\text {e(m)}}\times SO(2,1)_{\text {m(e)}}^2\)$$\begin{aligned} f_{\alpha \, 569} = g_1, \quad f_{\alpha \, 10,11,12} = g_2, \quad f_{\beta \, 127} = \tilde{g}_1, \quad f_{\beta \, 348} = \tilde{g}_2 \end{aligned}$$(83)
 For \(SU(2,1)_+\times SO(2,1)_+\), we choose the following gauge generators. The SO(2, 1) factor is generated by \(X_5, X_6\) and \(X_{11}\) while the SU(2, 1) is generated by \(X_{1},\ldots , X_4, X_7,\ldots X_{10}\) with the compact generators being \(X_7,\ldots X_{10}\). The associated embedding tensor is given bywith \(\alpha = \pm \) and \(\beta =\mp \) corresponding to \(SU(2,1)_{\text {e(m)}}\times SO(2,1)_{\text {m(e)}}\).$$\begin{aligned} f_{\alpha \, 129}= & {} f_{\alpha \,138}=f_{\alpha \,147} = f_{\alpha \,248} =g_1,\nonumber \\ f_{\alpha \,237}= & {} f_{\alpha \,349} = g_1,\nonumber \\ f_{\alpha \,789}= & {} 2g_1, \qquad f_{\alpha \,1,2,10} = f_{\alpha \,3,4,10}=\sqrt{3}g_1,\nonumber \\ f_{\beta \,5,6,11}= & {} g_2 \end{aligned}$$(84)
 For \(SL(3,\mathbb {R})_\times SO(3)_\), we choose the generators for SO(3) to be \(X_4, X_5\) and \(X_6\) while \(SL(3,\mathbb {R})\) is generated by the compact \(X_1, X_2, X_3\) and noncompact \(X_7,\ldots , X_{11}\) generators. Nonvanishing components of the embedding tensor are given bywith \(\alpha = \pm \) and \(\beta =\mp \) corresponding to \(SL(3,\mathbb {R})_{\text {e(m)}}\times SO(3)_{\text {m(e)}}\).$$\begin{aligned} f_{\alpha \,123}= & {} f_{\alpha \,1,9,10}=f_{\alpha \,279}=f_{\alpha \,2,8,10} =f_{\alpha \,3,7,10}\nonumber \\= & {} f_{\alpha \,3,8,9}=g_1,\nonumber \\ f_{\alpha \,178}= & {} 2g_1,\qquad f_{\alpha \,2,10,11}=f_{\alpha \,3,9,11} =\sqrt{3}g_1, \nonumber \\ f_{\beta \, 456}= & {} g_2 \end{aligned}$$(85)
4.1 \(SO(3)^2_+\times SO(3)_^2\)
 Setting \(\tilde{g}_2 g_2= g_1 \tilde{g}_1\) leads to an \(AdS_4\) critical point which is the trivial critical point of the same gauge group reported in [41] with \(V_0 = 6(g_1 \tilde{g}_1)^2\). As expected, this critical point satisfies the \(AdS_4\) conditions given in [40]$$\begin{aligned} \langle A_2^{ij}\rangle =\langle {A_{2ai}}^j\rangle =0, \qquad \langle A_1^{ij}A^*_{1kj}\rangle = \frac{4}{3} V_0 \delta ^i_k. \end{aligned}$$(93)
 Another possibility is to set \(\tilde{g}_2 g_2= (g_1  \tilde{g}_1)\) which leads to a \(dS_4\) critical point with \(V_0 = 2(g_1\tilde{g}_1)^2\) and satisfying the conditions in (89)$$\begin{aligned} \langle A_1^{ij}\rangle =\langle {A_{2ai}}^{j}\rangle =0, \qquad \langle A_2^{ij}A^*_{2kj}\rangle = V_0\delta ^i_k. \end{aligned}$$(94)
4.2 \(SO(3,1)_\pm \times SO(3,1)_\pm \)
 For \(SO(3,1)_+\times SO(3,1)_+\), we choose \(\tilde{g}_1 = g_1\) and \(\tilde{g}_2 = g_2\) which eliminate the following components of the embedding tensorAccordingly, the SO(3) subgroups of both SO(3, 1) factors are embedded along the mattermultiplet directions \(M =7,8,9\) and \(M=10,11,12\). Setting \(\tilde{g}_1 = g_1\) and \(\tilde{g}_2 = g_2\), we can rewrite the critical point (96) as$$\begin{aligned}&f_{+123} = f_{+783} =f_{+729} = f_{+189}=0, \nonumber \\&f_{456} = f_{10,11,12} = f_{10,5,12} = f_{4,11,12} =0. \end{aligned}$$(97)To bring this critical point to the values \(\chi = \phi =0\), we set \(g_2 = \pm g_1\), and both of these choices lead to the same \(dS_4\) critical point with \(V_0 =6g_2^2\) and satisfying (90)$$\begin{aligned} \chi = 0, \qquad \phi = \ln \left[ \pm \frac{g_2}{g_1}\right] \end{aligned}$$(98)This \(dS_4\) vacuum is then of the second type. There is no \(AdS_4\) vacuum in this case since the existence of \(AdS_4\) requires the embedding of \(SO(3)\times SO(3)\) along the Rsymmetry directions.$$\begin{aligned} \langle A_1^{ij}\rangle =\langle A_2^{ij}\rangle =0, \qquad \langle {A_{2ai}}^{k}{A^*_{2ak}}^j\rangle = 3g_2^2\delta ^j_i. \end{aligned}$$(99)
 For \(SO(3,1)_\times SO(3,1)_\), we set \(\tilde{g}_1 = g_1\) and \(\tilde{g}_2 = g_2\) which giveThe SO(3) subgroups of both SO(3, 1) factors are now embedded along the Rsymmetry directions \(M=1,2,3\) and \(M=4,5,6\). With this choice of the coupling constants, the critical point (96) becomes$$\begin{aligned} f_{+789}= & {} f_{+129} = f_{+183} = f_{+723}= 0,\\ f_{10,11,12}= & {} f_{4,5,12} = f_{4,11,6} = f_{10,5,6} =0. \end{aligned}$$In this case, however, setting \(g_2 = \pm g_1\) leads to two different critical points.$$\begin{aligned} \chi = 0, \qquad \phi = \ln \left[ \pm \frac{g_2}{g_1}\right] . \end{aligned}$$(100)In this case, the \(dS_4\) vacuum is of the first type. It should be noted that, as in the \(SO(3)^2_+\times SO(3)^2_\) gauge group, the \(SO(3,1)_\times SO(3,1)_\) gauge group gives two types of vacua with opposite ratios of the coupling constants.
 Setting \(g_2=g_1\) leads to a \(dS_4\) critical point satisfying (89)$$\begin{aligned} V_0 = 2g_2^2, \qquad \langle A_1^{ij}\rangle= & {} \langle {A_{2ai}}^{j}\rangle =0, \nonumber \\ \langle A_2^{ij}A^*_{2kj}\rangle= & {} \frac{9}{4}V_0\delta ^i_k. \end{aligned}$$(101)
 The choice \(g_2 = g_1\) gives an \(AdS_4\) critical point with \(V_0 = 6g_2^2\) and satisfying$$\begin{aligned} \langle A_2^{ij}\rangle =\langle {A_{2ai}}^{j}\rangle =0, \qquad \langle A_1^{ij}A^*_{1kj}\rangle = \frac{3}{4}V_0\delta ^i_k.\nonumber \\ \end{aligned}$$(102)

4.3 \(SO(2,1)^2\times SO(2,1)^2\)
In this case, the gauge group is given by \(SO(2,2)\times SO(2,2)\sim SO(2,1)^4\), and there are two possible gaugings to consider depending on the assignment of electric or magnetic gaugings to each SO(2, 1) factor. One gauging is described by \(SO(2,2)_{\text {e}}\times SO(2,2)_{\text {m}}\sim SO(2,1)_{\text {e}}\times SO(2,1)_{\text {e}}\times SO(2,1)_{\text {m}}\times SO(2,1)_{\text {m}}\) with the embedding tensor given in (81). The other one is \(SO(2,1)_{\text {e}}\times SO(2,1)_{\text {m}}\times SO(2,1)_{\text {m}}\times SO(2,1)_{\text {m}}\) with the embedding tensor given in (82) and its electricmagnetic dual \(SO(2,1)_{\text {e}}\times SO(2,1)_{\text {e}}\times SO(2,1)_{\text {e}}\times SO(2,1)_{\text {m}}\).
4.4 \(SO(3)_+\times SO(3)_\times SO(3,1)_\)
 The case of \(g_2 = g_1\) leads to a \(dS_4\) solution with \(V_0 = g_1^2\) and satisfies (89)$$\begin{aligned} \langle A_1^{ij}\rangle = \langle {A_{2ai}}^{j}\rangle =0, \qquad \langle A_2^{ij}A_{2kj}^*\rangle = \frac{9}{4}V_0\delta ^i_k. \end{aligned}$$(110)

For \(g_2 = g_1\), the critical point is an \(AdS_4\) vacuum with \(V_0 = 3g_1^2\).
4.5 \(SO(3,1)_+\times SO(2,1)_+\times SO(2,1)_\)
 The critical point of \(V_{\text {I}}\) is given bywhich can be brought to the origin by choosing$$\begin{aligned} \chi = 0, \qquad \phi = \ln \left[ \pm \frac{\sqrt{2(g_2^2+4g_2 \tilde{g}_2 + \tilde{g}_2^2)}}{(g_1+\tilde{g}_1)}\right] \end{aligned}$$(115)after setting \(g_2 = \tilde{g}_2\). Both choices lead to the same \(dS_4\) vacuum with \(V_0 =6g_2^2\) and satisfying (90).$$\begin{aligned} \tilde{g}_1 = g_1 \pm 2\sqrt{3} g_2 \end{aligned}$$(116)
 For \(V_{\text {II}}\), we find the following critical pointAfter setting \(\tilde{g}_1 = g_1\), this critical point can be brought to the origin by choosing$$\begin{aligned} \chi = 0, \qquad \phi = \ln \left[ \pm \frac{(g_2 + \tilde{g}_2)}{\sqrt{2(g_1^2 + 4 g_1 \tilde{g}_t + \tilde{g}_1^2)}}\right] . \end{aligned}$$(117)Both sign choices again give the same \(dS_4\) critical point which satisfies (90) and \(V_0 = 6g_1^2\).$$\begin{aligned} \tilde{g}_2 = g_2 \pm 2\sqrt{3}g_1. \end{aligned}$$(118)
4.6 \(SO(3)_+\times SO(2,1)_+^3\)
4.7 \(SU(2,1)_+\times SO(2,1)_+\)
4.8 \(SL(3,\mathbb {R})_\times SO(3)_\)
The embedding tensor for this gauge group is given in (85). In this gauge group, both \(AdS_4\) and \(dS_4\) solutions are possible, and, unlike the previous cases, the choice of which gauge group factor is electric or magnetic affects the resulting solutions. We will consider each choice separately.
5 Relations between gaugings with \(dS_4\) and \(dS_5\) vacua
In this section, we give some relations between gaugings of \(N=4\) gauged supergravities in four and five dimensions with de Sitter vacua. In general, a circle reduction of \(N=4\) fivedimensional theory gives rise to fourdimensional theory with the same number of supersymmetries. As pointed out in [37], the relations between the embedding tensors in four and five dimensions can be obtained from an analysis of group structures. We will follow this procedure in relating four and fivedimensional gaugings with de Sitter vacua.
A fivedimensional supergravity theory with \(\hat{n}\) vector multiplets gives, via a reduction on \(S^1\), a fourdimensional theory with \(n=\hat{n}+1\) vector multiplets. The global or duality symmetries in these two theories are given by \(\hat{G}=SO(1,1)\times SO(5,\hat{n})\) and \(G=SL(2)\times SO(6,\hat{n}+1)\), respectively. Accordingly, it is possible that gaugings in five dimensions can be encoded in those in four dimensions since \(\hat{G}\subset G\).
The identification between five and fourdimensional gaugings. Gauge groups with \(\#1,2,3\) in five dimensions admit only \(AdS_5\) vacua and are identified with fourdimensional gaugings with \(dS_4\) vacua satisfying the conditions (89). Gauge groups with \(\# 4,\ldots , 11\) give \(dS_5\) vacua and are identified with fourdimensional gaugings that lead to \(dS_4\) vacua satisfying the conditions (90)
\(\#_\text {5D}\)  5D gauge groups  4D gauge groups 

1  \(U(1)\times SU(2)\times SU(2)\)  \(SO(3)^2_+\times SO(3)^2_\) 
2  \(U(1)\times SO(3,1)\)  \(SO(3)_\times SO(3)_+\times SO(3,1)_\) 
3  \(U(1)\times SL(3,\mathbb {R})\)  \(SO(3)_\times SL(3,\mathbb {R})_\) 
4  \(SO(1,1)\times SU(2,1)\)  \(SO(2,1)_+\times SU(2,1)_+\) 
5  \(SO(1,1)\times SO(2,1)\)  \(SO(2,1)_+\times SO(2,1)_+\) 
6  \(SO(1,1)^{(2)}_\text {diag}\times SO(2,1)\)  \(SO(3,1)_+\times SO(2,1)_+\times SO(2,1)_\) 
7  \(SO(1,1)^{(3)}_\text {diag}\times SO(2,1)\)  \(SU(2,1)_+\times SO(2,1)_+\) 
8  \(SO(1,1)\times SO(2,1)^2\)  \(SO(2,1)^2_+\times SO(2,1)_^2\) 
9  \(SO(1,1)\times SO(3,1)\)  \(SO(3,1)_+\times SO(2,1)_+\times SO(2,1)_\) 
10  \(SO(1,1)^{(2)}_\text {diag}\times SO(3,1)\)  \(SO(3,1)_+\times SO(3,1)_+\) 
11  \(SO(1,1)\times SO(4,1)\)  \(SO(2,1)_+\times SO(4,1)_+\) 
5.1 Fourdimensional gaugings with \(AdS_4\) and \(dS_4\) vacua
In this case, the fourdimensional gaugings can give rise to both \(AdS_4\) and \(dS_4\) vacua with the \(dS_4\) solutions satisfying the conditions given in (89). The related fivedimensional gauge groups only admit supersymmetric \(AdS_5\) vacua. All the gauge groups considered here and the associated \(AdS_5\) vacua have already been studied in [42, 43].
5.1.1 \(U(1)\times SU(2)\times SU(2)\) 5D gauge group
5.1.2 \(U(1)\times SO(3,1)\) 5D gauge group
5.1.3 \(U(1)\times SL(3,\mathbb {R})\) 5D gauge group
5.2 Fourdimensional gaugings with only \(dS_4\) vacua
From the results of the previous sections and in [36], we know that when a nonabelian compact subgroup of the Rsymmetry is not gauged, \(\hat{f}_{\hat{m}\hat{n}\hat{p}}=0\) and \(f_{\alpha mnp}=0\), the gauged supergravities admit de Sitter vacua. Both in four and five dimensions, these gauge groups take the form of a product of noncompact groups. We will give some relations between this type of gaugings in four and five dimensions.
5.2.1 \(SO(1,1)\times SU(2,1)\) 5D gauge group
5.2.2 \(SO(1,1)\times SO(2,1)\) 5D gauge group
5.2.3 \(SO(1,1)^{(2)}_{\text {diag}}\times SO(2,1)\) 5D gauge group
5.2.4 \(SO(1,1)^{(3)}_{\text {diag}}\times SO(2,1)\) 5D gauge group
5.2.5 \(SO(1,1)\times SO(2,2)\) 5D gauge group
5.2.6 \(SO(1,1)\times SO(3,1)\) 5D gauge group
5.2.7 \(SO(1,1)^{(2)}_{\text {diag}}\times SO(3,1)\) 5D gauge group
5.2.8 \(SO(1,1)\times SO(4,1)\) 5D gauge group
6 Conclusions
In this paper, we have studied \(dS_4\) vacua of fourdimensional \(N=4\) gauged supergravity coupled to vector multiplets. By requiring that the scalar potential is extremized and positive, we have derived a set of conditions for determining a general form of gauge groups admitting \(dS_4\) vacua by adopting a simple ansatz. This extends the previous result in fivedimensional \(N=4\) gauged supergravity and provides a useful approach for finding \(dS_4\) vacua in \(N=4\) gauged supergravity. We have also given some relations between the embedding tensors of four and fivedimensional gauge groups that could be related by a simple circle reduction. From this analysis, we have given a new example of fourdimensional gauge group, \(SO(2,1)\times SO(4,1)\), that gives a \(dS_4\) vacuum. This has not previously been studied since the gauging requires the coupling to at least seven vector multiplets.
Unlike in five dimensions, we find two large classes of gauge groups that give \(dS_4\) vacua as maximally symmetric backgrounds of the mattercoupled \(N=4\) gauged supergravity. For the first class, the gauge groups take a general form of \(G_{\text {e}}\times G_{\text {m}}\times G_0^v\) in which \(G_{\text {e(m)}}\) is electrically (magnetically) gauged and contains an SO(3) subgroup. \(G_0^v\) is a compact group gauged by vector fields in the vector multiplets. These gauge groups are precisely the ones that lead to supersymmetric \(AdS_4\) vacua studied in [40]. Two different types of vacua, \(AdS_4\) and \(dS_4\), arise from different coupling ratios between \(G_{\text {e}}\) and \(G_{\text {m}}\) factors. This result is obtained from imposing the conditions that \(\langle A^{ij}_1\rangle =\langle {A_{2ai}}^j\rangle =0\). These conditions take a very similar form to those for the existence of supersymmetric \(AdS_4\) vacua. We have explicitly verified that the potential is extremized by these conditions.
For the second class, we have imposed another set of conditions, \(\langle A^{ij}_1\rangle =\langle A^{ij}_2\rangle =0\), and found that the gauge groups generally take the form \(SO(2,1)\times SO(2,1)' \times G_{\text {nc}}\times G'_{\text {nc}}\times H_{\text {c}}\). \(G_{\text {nc}}\) and \(G'_{\text {nc}}\) are noncompact groups with the compact parts embedded in the matter directions while the compact \(SO(2)\times SO(2)'\subset SO(2,1)\times SO(2,1)'\) is embedded along the Rsymmetry directions. As in the previous case, \(SO(2,1)\times G_{\text {nc}}\) (\(SO(2,1)'\times G'_{\text {nc}}\)) is electrically (magnetically) gauged, and \(H_{\text {c}}\) is a compact group. It should be emphasized that only \(G_{\text {nc}}\) and \(G'_{\text {nc}}\) are necessary for the \(dS_4\) vacua to exist.
Given that our ansatz is rather simple, it is remarkable that the above results encode all semisimple gauge groups that are previously known to give \(dS_4\) vacua of \(N=4\) gauged supergravity. Two different sets of these gauge groups have also been noted in [28], and these correspond to the two sets of conditions given in this paper. The results given here are hopefully useful for finding \(dS_4\) vacua and could be interesting in the dS/CFT correspondence and cosmology.
In this paper, we have looked at only semisimple gauge groups. It is also interesting to consider nonsemisimple gauge groups listed in [44] and those arising from flux compactification studied in [45] and [46]. In deriving all the conditions for the existence of \(dS_4\) vacua, we have not restricted the gauge groups to be semisimple. Therefore, our conditions are also valid for nonsemisimple gauge groups. In particular, it can be verified that, for the \(dS_4\) vacuum from \(ISO(3)\times ISO(3)\) gauge group considered in [46], we have \(\langle A^{ij}_1\rangle =\langle {A_{2ai}}^j\rangle =0\). This \(dS_4\) solution is accordingly of the first type described by the criteria given in (89). A systematic classification of nonsemisimple groups leading to \(dS_4\) vacua in \(N=4\) gauged supergravity is worth considering.
Given the success in \(N=4\) gauged supergravities in both four and five dimensions, it is natural to extend this approach to other gauged supergravities with different numbers of supersymmetries in various dimensions. The success of this approach also suggests that the conditions we have derived might have deeper meaning although they are originally obtained from a simple assumption. It would be of particular interest to have a definite conclusion whether there is some explanation for these conditions within gauged supergravity and string/Mtheory or these conditions are just a tool for finding de Sitter solutions.
Notes
Acknowledgements
P. K. is supported by The Thailand Research Fund (TRF) under grant RSA6280022.
References
 1.C.L. Bennet et al., First year Wilkinson microwave anisotropy probe (WMAP) observations: preliminary maps and basic results. Astrophys. J. Suppl. 148, 1 (2003). arXiv:astroph/0302207 ADSCrossRefGoogle Scholar
 2.A.G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998). arXiv:astroph/9805201 ADSCrossRefGoogle Scholar
 3.S. Perlmutter et al., Measurement of omega and lambda from 42 highredshift supernovae. Astrophys. J. 517, 565 (1999). arXiv:astroph/9812133 ADSzbMATHGoogle Scholar
 4.A. Strominger, The dS/CFT correspondence. JHEP 10, 034 (2001). arXiv:hepth/0106113 ADSMathSciNetCrossRefGoogle Scholar
 5.J.M. Maldacena, The large \(N\) limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998). arXiv:hepth/9711200 Google Scholar
 6.L. Randall, R. Sundrum, An alternative to compactification. Phys. Rev. Lett. 83, 4690–4693 (1999). arXiv:hepth/9906064 ADSMathSciNetCrossRefGoogle Scholar
 7.G.W. Gibbons, C.M. Hull, De Sitter space from warped supergravity solutions. arXiv:hepth/0111072
 8.I.P. Neupane, De Sitter braneworld, localization of gravity, and the cosmological constant. Phys. Rev. D 83, 086004 (2011). arXiv:1011.6357 ADSCrossRefGoogle Scholar
 9.M. Minamitsuji, K. Uzawa, Warped de Sitter compactifications. JHEP 01, 142 (2012). arXiv:1103.5326 ADSMathSciNetCrossRefGoogle Scholar
 10.M. Dodelson, X. Dong, E. Silverstein, G. Torroba, New solutions with accelerated expansion in string theory. JHEP 12, 050 (2014). arXiv:1310.5297 ADSMathSciNetCrossRefGoogle Scholar
 11.B. de Carlos, A. Guarino, J.M. Moreno, Flux moduli stabilisation. Supergravity algebras and nogo theorems. JHEP 01, 012 (2010). arXiv:0907.5580 MathSciNetCrossRefGoogle Scholar
 12.B. de Carlos, A. Guarino, J.M. Moreno, Complete classification of Minkowski vacua in generalised flux models. JHEP 02, 076 (2010). arXiv:0911.2876 MathSciNetCrossRefGoogle Scholar
 13.J. Blaback, U. Danielsson, G. Dibitetto, Fully stable dS vacua from generalised fluxes. JHEP 08, 054 (2013). arXiv:1301.7073 ADSMathSciNetCrossRefGoogle Scholar
 14.C. Damian, L.R. DiazBarron, O. LoaizaBrito, M. Sabido, Slowroll inflation in nongeometric flux compactification. JHEP 06, 109 (2013). arXiv:1302.0529 ADSMathSciNetCrossRefGoogle Scholar
 15.C. Damian, O. LoaizaBrito, More stable de Sitter vacua from Sdual nongeometric fluxes. Phys. Rev. D 88, 046008 (2013). arXiv:1304.0792 ADSCrossRefGoogle Scholar
 16.R. Blumenhagen, C. Damian, A. Font, D. Herschmann, R. Sun, The fluxscaling scenario: De Sitter uplift and axion inflation. Fortsch. Phys. 64, 536–550 (2016). arXiv:1510.01522 ADSMathSciNetCrossRefGoogle Scholar
 17.K. Dasgupta, G. Rajesh, S. Sethi, M theory, orientifolds and Gflux. JHEP 08, 023 (1999). arXiv:hepth/9908088 ADSMathSciNetCrossRefGoogle Scholar
 18.G.W. Gibbons, Aspects of supergravity theories, in Supersymmetry, Supergravity and Related Topics, ed. by F. Del Aguila, J.A. de Azcarraga, L.E. Ibañez (World Scientific, Singapore, 1985)Google Scholar
 19.B. de Wit, D.J. Smit, N.D. Hari Dass, Residual supersymmetry of compactified D = 10 supergravity. Nucl. Phys. B 283, 165 (1987)ADSCrossRefGoogle Scholar
 20.J. Maldacena, C. Nunez, Supergravity description of field theories on curved manifolds and a no go theorem. Int. J. Mod. Phys. A 16, 822 (2001). arXiv:hepth/0007018 ADSMathSciNetCrossRefGoogle Scholar
 21.M.P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark, Inflationary constraints on type IIA string theory. JHEP 0712, 095 (2007). arXiv:0711.2512 ADSMathSciNetCrossRefGoogle Scholar
 22.P.K. Townsend, Cosmic acceleration and Mtheory. arXiv:hepth/0308149
 23.U. Danielsson, G. Dibitetto, On the distribution of stable de Sitter vacua. JHEP 03, 018 (2013). arXiv:1212.4984 ADSCrossRefGoogle Scholar
 24.U. Danielsson, T. van Riet, What if string theory has no de Sitter vacua? Int. J. Mod. Phys. D 27, 12, 1830007 (2018). arXiv:1804.01120 MathSciNetCrossRefGoogle Scholar
 25.C.M. Hull, Noncompact gaugings of \(N=8\) supergravity. Phys. Lett. B 142, 39 (1984)Google Scholar
 26.M. de Roo, D.B. Westra, S. Panda, De Sitter solutions in \(N = 4\) matter coupled supergravity. JHEP 02, 003 (2003). arXiv:hepth/0212216 Google Scholar
 27.P. Fre, M. Trigiante, A. Van Proeyen, Stable de Sitter vacua from \(N = 2\) supergravity. Class. Quantum Gravity 19, 4167–4194 (2002). arXiv:hepth/0205119 Google Scholar
 28.M. de Roo, D.B. Westra, S. Panda, M. Trigiante, Potential and massmatrix in gauged N = 4 supergravity. JHEP 11, 022 (2003). arXiv:hepth/0310187 MathSciNetCrossRefGoogle Scholar
 29.D. Roest, J. Rosseel, deSitter in extended gauged supergravity. Phys. Lett. B 685, 201–207 (2010). arXiv:hepth/0912.4440 ADSMathSciNetCrossRefGoogle Scholar
 30.G. Smet, PhD thesis, de Sitter space and supergravity in various dimensions (2006)Google Scholar
 31.M. Gunaydin, M. Zagermann, The vacua of 5d, \(N = 2\) gauged Yang–Mills/Einstein/tensor supergravity: Abelian case. Phys. Rev. D 62, 044028 (2000). arXiv:hepth/0002228 Google Scholar
 32.O. Ogetbil, A general study of ground states in gauged \(N = 2\) supergravity theories with symmetric scalar manifolds in 5 dimensions. arXiv:hepth/0612145
 33.O. Ogetbil, Stable de Sitter Vacua in 4 dimensional supergravity originating from 5 dimensions. Phys. Rev. D 78, 105001 (2008). arXiv:0809.0544
 34.B. Cosemans, G. Smet, Stable de Sitter vacua in \(N = 2\), \(D = 5\) supergravity. Class. Quantum Gravity 22, 2359–2380 (2005). arXiv:hepth/0502202 Google Scholar
 35.G. Dibitetto, J.J. FernandezMelgarejo, D. Marques, All gaugings and stable de Sitter in \(D = 7\) halfmaximal supergravity. JHEP 11, 037 (2015). arXiv:1506.01294 Google Scholar
 36.H.L. Dao, P. Karndumri, \(dS_5\) vacua from mattercoupled \(5D\) \(N=4\) gauged supergravity. arXiv:1906.09776
 37.J. Schon, M. Weidner, Gauged \(N=4\) supergravities. JHEP 05, 034 (2006). arXiv:hepth/0602024 Google Scholar
 38.E. Bergshoeff, I.G. Koh, E. Sezgin, Coupling of Yang–Mills to \(N=4\), \(d=4\) supergravity. Phys. Lett. B 155, 71–75 (1985)Google Scholar
 39.M. de Roo, P. Wagemans, Gauged matter coupling in \(N=4\) supergravity. Nucl. Phys. B 262, 644–660 (1985)Google Scholar
 40.J. Louis, H. Triendl, Maximally supersymmetric \(AdS_4\) vacua in \(N=4\) supergravity. JHEP 10, 007 (2014). arXiv:1406.3363 Google Scholar
 41.P. Karndumri, K. Upathambhakul, Holographic RG flows in N = 4 SCFTs from halfmaximal gauged supergravity. Eur. Phys. J. C 78, 626 (2018). arXiv:hepth/1806.01819 ADSCrossRefGoogle Scholar
 42.H.L. Dao, P. Karndumri, Holographic RG flows and \(AdS_5\) black strings from 5D halfmaximal gauged supergravity. Eur. Phys. J. C 79, 137 (2019). arXiv:1811.01608 Google Scholar
 43.H.L. Dao, P. Karndumri, Supersymmetric \(AdS_5\) black holes and strings from 5D \(N=4\) gauged supergravity. Eur. Phys. J. C 79, 247 (2019). arXiv:1812.10122 Google Scholar
 44.M. de Roo, D.B. Westra, S. Panda, Gauging CSO groups in \(N=4\) supergravity. JHEP 09, 011 (2006). arXiv:hepth/0606282 Google Scholar
 45.G. Dibitetto, R. Linares, D. Roest, Flux compactifications, gauge algebras and De Sitter. Phys. Lett. B 688, 96–100 (2010). arXiv:1001.3982 ADSMathSciNetCrossRefGoogle Scholar
 46.G. Dibitetto, A. Guarino, D. Roest, Charting the landscape of \(N=4\) flux compactifications. JHEP 03, 137 (2011). arXiv:1102.0239 Google Scholar
 47.A. Borghese, D. Roest, Metastable supersymmetry breaking in extended supergravity. JHEP 05, 102 (2011). arXiv:1012.3736 ADSMathSciNetCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}