Cosmological solutions from 4D N=4 matter-coupled supergravity
aa r X i v : . [ h e p - t h ] F e b Cosmological solutions from 4D N = 4 matter-coupled supergravity H. L. Dao
Department of Physics,National University of Singapore,3 Science Drive 2, Singapore 117551 [email protected]
Abstract
From four-dimensional N = 4 matter-coupled gauged supergravity, we studysmooth time-dependent cosmological solutions interpolating between a dS × Σ spacetime, with Σ = S and H , in the infinite past and a dS spacetime in theinfinite future. The solutions were obtained by solving the second-order equationsof motion from all the ten gauged theories known to admit dS solutions, of whichthere are two types. Type I dS gauged theories can admit both dS solutions as wellas supersymmetric AdS solutions while type II dS gauged theories only admit dS solutions. We also study the extent to which the first-order equations that solve theaforementioned second-order field equations fail to admit the dS vacua and theirassociated cosmological solutions. ontents N = 4 matter-coupled supergravity 43 dS solutions from 4D N = 4 supergravity 74 dS × Σ solutions from 4D N = 4 supergravity 125 Type I dS gauged theories 15 SO (3) × SO (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Other gauge groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 dS gauged theories 17 SO (2 , × SO (2 ,
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 SO (2 , × SO (2 ,
2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3 SO (3 , × SO (3 ,
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 SO (2 , × SO (3 ,
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.5 SO (2 , × SO (4 ,
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.6 SO (2 , × SU (2 ,
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.7 Summary of all solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
AdS case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 dS case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Cosmological solutions connecting a dS spacetime with a very large Hubble constantin the past to another dS spacetime with a very small Hubble constant (both of theright order of magnitude) in the future can provide a good description of our universe.In the context of the dS/CFT holography, as formulated in [1], [2], these and the moregeneral cosmological solutions connecting dS spacetimes of different dimensions, such as dS D − d × Σ d and dS D , are fundamental building blocks on the gravity side. These are dualto Euclidean conformal field theories with different dimensions which are connected bytime evolution since the holographic dimension is time itself. Despite much effort, a meta-stable dS vacuum remains elusive from string theory [9]. Furthermore, there exist manyoutstanding issues regarding the quantum nature of dS space itself [6]. Because of thesedifficulties, dS/CFT holography based on the correspondence between a dS D +1 solution of( D +1)-dimensional supergravity and a Euclidean CFT in D dimensions, analogous to the2ow routinely-applied AdS/CFT correspondence between an AdS ( D +1) spacetime and aCFT D theory, remains very poorly understood. To date, arguably the most concrete formof dS/CFT is found in higher spin theories [3], [4], [5] instead of ordinary supergravity.A radically different way to obtain dS solutions is from unconventional gauged dS supergravities that arise from the compactification of the exotic 11D M* and 10D typeIIB* theories [14], which are themselves obtained from the timelike T-dualities of theoriginal M/string theories [11], [12], [13]. This framework, originally proposed in [10],naturally gives dS space the priviledged position occupied by AdS space in the conven-tional framework of M/string theories. Consequently, dS/CFT holography, in the senseproposed by [10], naturally arises in the context of these *-theories in the same mannerthat AdS/CFT holography does in conventional M/string theories. It is then unsurprisingthat many cosmological solutions can be found in this framework, see for examples [16],[18].On the other hand, only a few examples involving cosmological solutions in conven-tional gauged supergravities are known. These include [15] and [17]. The cosmologicalsolution in [15] interpolates between a dS spacetime and a singularity, while the cosmo-logical solution in [17], interpolating between a dS × S D − spacetime in the past and a dS D spacetime in the future, is an analytic continuation of the domain wall solution inter-polating between an AdS × S D − spacetime and an AdS D spacetime. In [7], cosmologicalsolutions in 5D N = 4 supergravity interpolating between a dS − d × H d (with d = 2 , dS spacetime in the infinite future were presentedin all the gauged theories capable of admitting dS solutions. Since the dS vacua of 5D N = 4 supergravity are unstable, these and their associated cosmological vacua studiedin [7] are not relevant in the context of dS holography. Nonetheless, independently ofholography, it was instructive to derive and study the cosmological solutions in 5D N = 4supergravity in their own right. The main motivation for the study in [7] was straightfor-ward - we would like to know whether there exist fixed point solutions dS − d × Σ d , withΣ d = S d , H d where d = 2 ,
3, and interpolating solutions connecting these in the infinitepast to the dS solutions in the infinite future. It is our goal in this work to carry outthe same analysis in the four-dimensional, N = 4 supergravity that is structurally similarto, but more complex than the five-dimensional theory studied in [7]. More specifically,our primary goal will be to study cosmological solutions interpolating between a dS × Σ spacetime, with Σ = S , H , at early times and a dS spacetime at late times from 4D N = 4 matter-coupled gauged supergravities. The solutions will be derived by solving thesecond-order field equations from the ten gauged theories known to admit dS solutions.These gauged theories can be classified into two types, depending on whether the gaugegroup under consideration allows for an AdS solution in addition to a dS solution, or just dS solutions [30]. Secondly, similar to the 5D case in [7], our secondary goal will be tostudy the possibility of obtaining these dS and cosmological solutions from some systemof first-order equations that solve the second-order field equations. Although the fact thatall dS vacua of 4D N = 4 supergravity are unstable precludes the existence of pseudosu-persymmetry which guarantees the existence of first-order pseudo-BPS equations capableof giving rise to these solutions, we will nevertheless perform our analysis, analogous to3he 5D case, to pinpoint the exact failure characterizing the lack of first-order systems.This rest of this paper is organized as follows. In section 2, we review the theoryof 4D N = 4 gauged supergravity coupled to vector multiplets in the embedding tensorformalism. In section 3, we review the dS solutions derived within the framework of theembedding tensor formalism and their classifications into type I and type II. In section 4,the ansatze for the dS × Σ solutions are specified and the associated equations of motionare derived. In section 5 and 6, we present the cosmological solutions from the type I andtype II dS gauged theories, respectively. In section 7, we investigate how the cosmologicalsolutions found in sections 5,6 fail to arise from the first-order equations that solve thesecond-order field equations. Section 8 concludes the paper. N = 4 matter-coupled supergravity To make this paper self-contained, we highlight some relevant details of four-dimensional N = 4 matter-coupled gauged supergravity. The detailed construction of theory can befound in [24] and the references therein.The field content of the theory includes an N = 4 supergravity multiplet and anarbitrary number n of vector multiplets. The supergravity multiplet (cid:0) e ˆ µµ , ψ µ i , A mµ , λ i , τ (cid:1) contains the graviton e ˆ µµ , four gravitini ψ µi , i = 1 , ...,
4, six vectors A mµ , m = 1 , ...,
6, fourspin- fermions λ i , and a complex scalar τ . The vector multiplet (cid:0) A µ , λ i , φ m (cid:1) contains a vector field A µ , four gaugini λ i and six scalars φ m . Spacetime and tangentspace indices will be denoted by µ, ν, . . . = 0 , , , µ, ˆ ν, . . . = 0 , , ,
3, respectively.Indices m, n, . . . = 1 , , . . . , SO (6) ∼ SU (4) R-symmetrywhile i, j, . . . denote chiral spinor of SO (6) or fundamental representation of SU (4). The n vector multiplets are labeled by indices a, b, . . . = 1 , ..., n . Collectively, the fied contentof the n vector multiplets can be written as( A aµ , λ ai , φ am ) . Altogether, there are (6 n + 2) scalars from both the gravity and vector multiplets.These scalars span the coset manifold M = SL (2 , R ) SO (2) × SO (6 , n ) SO (6) × SO ( n ) . (1)The first factor in (1) is parameterized by a complex scalar τ consisting of the dilaton φ and the axion χ from the gravity multiplet, where τ = χ + i e φ . (2)4he second factor in (1) is parameterized by the 6 n scalars from the vector multiplets.Both factors of (1) will be described in terms of coset representatives. For SL (2) /SO (2),this coset representative is V α = 1 √ Im τ (cid:18) τ (cid:19) (3)with an index α = (+ , − ) denoting the SL (2) fundamental representation. In the nextsection, we will also be using the notation α = ( e, m ). V α satisfies the relation M αβ = Re( V α V ∗ β ) and ǫ αβ = Im( V α V ∗ β ) (4)in which M αβ is a symmetric matrix with unit determinant, while ǫ αβ is anti-symmetricwith ǫ + − = ǫ + − = 1.The coset manifold SO (6 , n ) /SO (6) × SO ( n ) will be described by a coset representative V M A transforming under the global SO (6 , n ) and local SO (6) × SO ( n ) symmetry by leftand right multiplications, respectively. The local index A can be split as A = ( m, a ) with m = 1 , , . . . , a = 1 , , . . . , n denoting vector representations of SO (6) × SO ( n ).Accordingly, V M A can be written as V M A = ( V M m , V M a ) . (5)With V M A being an SO (6 , n ) matrix, the following relation is satisfied η MN = −V M m V N m + V M a V N a (6)where η MN = diag ( − , − , − , − , − , − , , . . . ,
1) is the SO (6 , n ) invariant tensor. SO (6 , n ) indices M, N, . . . are lowered and raised using η MN and its inverse η MN . Equiv-alently, the symmetric matrix M MN = V M m V N m + V M a V N a , (7)which is manifestly SO (6) × SO ( n ) invariant, is used to describe the SO (6 , n ) /SO (6) × SO ( n ) coset.Gaugings of the matter-coupled N = 4 supergravity are performed using the embed-ding tensor formalism [24] that provides an embedding of a gauge group G in the globalsymmetry group SL (2 , R ) × SO (6 , n ). N = 4 supersymmetry restricts the embeddingtensor to include only the components ξ αM and f αMNP = f α [ MNP ] . A closed subalgebraof SL (2 , R ) × SO (6 , n ) satisfying the required commutation relations is ensured by thefollowing quadratic constraints on the components of the embedding tensor0 = ξ αM ξ βM , ǫ αβ ( ξ αP f β MNP + ξ αM ξ βN ) , ξ ( αP f β ) MNP , f αR [ MN f βP Q ] R + 2 ξ ( α [ M f β ) NP Q ] , ǫ αβ (cid:0) f αMNR f βP QR − ξ αR f βR [ M [ P η Q ] N − ξ α [ M f βN ] P Q + ξ α [ P f βQ ] MN (cid:1) . (8)5he gauge group generators are constructed from these components of the embeddingtensor as follows. X αM, βN γP = − δ γβ f αMN P + 12 (cid:2) δ PM δ γβ ξ αN − δ PN δ γα ξ βN − δ γβ η MN ξ Pα + ǫ αβ δ PN ξ κM ǫ κγ (cid:3) (9)In this work, we are only interested in solutions involving the metric, scalars and someAbelian gauge fields. The bosonic Lagrangian reads e − L = 12 R + 116 D µ M MN D µ M MN + 18 D µ M αβ D µ M αβ − V −
14 Im τ M MN H M + µν H N + µν + 18 Re( τ ) η MN ǫ µνρλ H M + µν H N + ρλ + e − L top (10)where e is the vielbein determinant, and L top is the topological term whose explicit formwill not be of relevance to our purpose because this term will always vanish for thesolutions considered here. Note that in the Lagrangian above, only covariant electric gaugefield strengths H M + µν are present while magnetic gauge fields H M − µν enter the equations ofmotion. The covariant derivatives acting on the scalars read D µ M MN = ∂ µ M MN + 2 A µP α Θ αP ( M Q M N ) Q D µ M αβ = ∂ µ M αβ + A Mγµ ξ ( αM M β ) γ − A Mδµ ξ κM ǫ δ ( α ǫ κγ M β ) γ . (12)where Θ αMNP = f αMNP − ξ α [ N η P ] M . (13)In general, the covariant field strengths H Mαµν contain the auxiliary two-form fields B [ MN ] µν and B ( αβ ) µν that transform in the antisymmetric representation of SO (6 , n ) and inthe symmetric representation of SL (2 , R ), respectively. These two-forms are needed topreserve the covariance of the gauge field strengths H Mαµν . At this point we refrain fromgiving the full form of H Mαµν that contain B MNµν and B αβµν due to their complexity and thefact that these two-forms can be consistently truncated out for the specific embeddingtensor components corresponding to all the gauge groups studied in this work. With B [ MN ] µν = B αβµν = 0 and the fact that only Abelian gauge fields are turned on, the gaugefield strengths read H Mαµν = 2 ∂ [ µ A Mαν ] . (14)The scalar potential V reads V = g (cid:20) f αMNP f βQRS M αβ (cid:20) M MQ M NR M P S + (cid:18) η MQ − M MQ (cid:19) η NR η P S (cid:21) − f αMNP f βQRS ǫ αβ M MNP QRS + 3 ξ αM ξ βN M αβ M MN (cid:21) (15) Note that there are two equivalent ways to write the kinetic term for the supergravity scalar τD µ M αβ D µ M αβ = − τ ) D µ τ D µ τ ∗ . (11) M MN and M αβ are the inverse matrices of M MN and M αβ , respectively. M MNP QRS is obtained by raising indices of M MNP QRS defined by M MNP QRS = ǫ mnpqrs V mM V nN V pP V qQ V rR V sS . (16)Alternatively, the potential V can be written in terms of the fermion-shift matrices A ij , A ij and A aij that appear in the fermion mass-like terms and supersymmetry trans-formations of fermions as follows. V = 12 A aij A ∗ aj i + 19 A ij A ∗ ij − A ij A ∗ ij . (17)In terms of the scalar coset representatives, the fermion shift-matrices are given by A ij = ǫ αβ ( V α ) ∗ V klM V N ik V P jl f βM NP , (18) A ij = ǫ αβ V α V klM V N ik V P jl f βM NP + 32 ǫ αβ V α V M ij ξ βM , (19) A aij = ǫ αβ V α V aM V ikN V P jk f βMN P − δ ji ǫ αβ V α V aM ξ βM . (20) V M ij is obtained from V M m by converting the SO (6) vector index to an anti-symmetricpair of SU (4) fundamental indices using the chiral SO (6) gamma matrices. The explicitgamma matrices used are given in the Appendix of [30]. dS solutions from 4D N = 4 supergravity In this section we summarize the known results on dS solutions from [30]. dS solutionsof 4D N = 4 gauged supergravity coupled to n vector multiplets were originally derivedin [20], [21], and rederived within the framework of the embedding tensor formalism in[30] using the following two ansatze for the fermionic shift matrices (18, 19, 20) h A ij i = h A aij i = 0 , h A ik A ∗ jk i = 94 V δ ij (21) h A ij i = h A ij i = 0 , h A aik A ∗ akj i = 12 V δ ij (22)where V is the extremized value of the potential, and h i denotes the enclosed quantitiesbeing evaluated in the considered background. We recall that all solutions from [20]and [21] were recovered in [30] and in addition, a new solution with the gauge group SO (2 , × SO (4 ,
1) was found in [30]. While the ansatze (21, 22) ensure the positivity ofthe scalar potential, they must also be subject to the quadratic constraints (8), and theextremization condition on the scalar potential. For more details, see [30]. Solutions toeither (21) or (22) consist of the set of embedding tensor components ( ξ α M , f α MNP ) fromwhich the gauge group generators can be constructed using (9). As shown in [30], thereexist solutions to both (21) and (22) which form the two types of gaugings that give rise7o two different types of dS solutions in gauged N = 4 four dimensional supergravity.The common features of these two types are as follows. Firstly, all gaugings with dS solutions must have ξ αM = 0 which simplifies the formula for the gauge generators (9) X αM N P = − f α MN P . (23)Secondly, the gauge groups G must be dyonic, comprising a product of at least oneelectric G e and one magnetic G m gauge factor under the SL (2 , R ) duality group . G = G e × G m . (24)The differences between these two types of gaugings are as follows. • Type I dS solutions arise from gaugings of the first type, which can admit both dS and AdS solutions. The compulsory components of the 4D embedding tensor inthis case are f α mnp = 0 . (25)Other allowed (but not required) components of the embedding tensors are f α mab = 0 , f α abc = 0 . (26)The gauge group is of the form G = G e, − × G m, − × . . . (27)where G e, − , G m, − , . . . are in general non-compact. The compact parts of (27) arealways SO (3) e × SO (3) m , generated by f α mnp , and lie entirely in the R-symmetrygroup SO (6) ⊂ SO (6 , n ) of (1). The non-compact parts of (27), generated by f α mab ,lie completely in the matter-symmetry directions SO ( n ) ⊂ SO (6 , n ) of (1). Accord-ingly, the − subscript in the G e ( m ) factors in (27) is used to denote the embedding ofthe compact parts of (27) in the R-symmetry directions. Additionally, f α abc generatea purely compact gauge factor, embedded in the matter symmetry directions SO ( n ),that is optional and does not contribute anything to the scalar potential. When f α mab vanish, we can have the completely compact gauging SO (3) e, − × SO (3) m, − which can exist in the pure supergravity theory without any coupled matter. • Type II dS solutions arise from gaugings of the second type, which can only admit dS solutions with no AdS solutions possible. The compulsory components of the4D embedding tensor in this case are f α amn = 0 . (28) Here we use the notation α = ( e, m ) instead of α = (+ , − ) as used in section 2 to avoid the potentialconfusion arising from the use of subscript (+ , − ) reserved for other meanings. For
AdS vacua, this compulsory SU (2) e × SU (2) m factor can be identified with the R-symmetrygroup of 3d N = 4 SCFTs according to the AdS/CFT correspondence. f α abc = 0 . (29)The gauge group is always noncompact and is of the product form G = G e, + × G m, + × . . . (30)The non-compact parts of (30) are generated by the components f α amn and lieentirely along the R-symmetry directions SO (6) ⊂ SO (6 , n ) of (1). The compactparts of (30) are generated by f α abc and lie fully in the matter symmetry directions SO ( n ) ⊂ SO (6 , n ) of (1). Accordingly, the + subscript in the G e ( m ) factors of (30) isused to denote the embedding of the compact parts of (30) in the matter symmetrydirections. This is in direct constrast to those gaugings of the first type above.When f α abc = 0, the only gauging possible is SO (2 , e, + × SO (2 , m, + . Just likein 5D [31], gaugings of this second type cannot exist in the pure theory, but only inthe matter-coupled theories.We emphasize that the + / − subscripts in (27) and (30) are not to be mistaken with theearlier notation of α = (+ , − ) used in section 2. Here the α subscript is explicitly labeledas α = ( e, m ), while the + or − subcripts are used to denote whether the compact parts ofthe gauge group are embedded in the matter or R-symmetry directions. The embeddingtensor components for all 4D gaugings with dS solutions are listed in table 1 where wehave dropped the e ( m ) subscripts, but retained the (+ / − ) subscripts in the gauge factors.Because of the SL (2 , R ) duality, either G + or G − can be electric or magnetic and viceversa - this is reflected from the embedding tensor components f αMNP where α can assumeeither the value of e or m .While the full form of all gauge groups is given as in Table 1, we must point outthat not all gauge factors in these gauge groups contribute to the scalar potentials, asmentioned above. Only the compulsory components (25), (28) are needed for each type ofgaugings. Consequently, to simplify the analysis, we can reduce some of the gauge groupsto the bare forms needed to admit dS solutions without any loss of generality. For laterconvenience, we summarize the explicit scalar potentials corresponding to the simplifiedversions of those gauge groups listed in Table 1 in Table 2. These scalar potentials areconstructed from the explicit components of the embedding tensor given in Table 1 using(15) or (17). For concreteness, g and g are used to denote the electric factor G e andmagnetic factor G m , respectively, of the gauge group G . From here on, to lighten thenotations, the explicit + / − subscripts in the gauge groups will also be dropped sincetheir distinction is made clear from their assignment to either type I or type II gaugings.It is important to note that while the scalar potentials for the type II gaugings are alldifferent, there is only a single scalar potential resulting from the type I gaugings.9auge group G Embedding tensor f αMNP Type I:
AdS & dS possible SO (3) × SO (3) − f α = g , f α = ˜ g ,f β = g , f β , , = ˜ g SO (3 , − × SO (3 , − f α = − f α = f α = f α = g f β = − f β , , = g f β , , = f β , , = g SO (3) + × SO (3) − × SO (3 , − f α = f α = g f β = − f β , , = g f β , , = f β , , = g SO (3) − × SL (3 , R ) − f α = f α = f α = − g f α, , , = − f α, , , = f α, , , = − g f α , , = f α , , = √ g f α = 2 g , f β = g Type II:Only dS possible SO (2 , × SO (2 , − f α = g , f α = ˜ g f β , , = g , f β , , = ˜ g SO (2 , + × SO (2 , + × SO (3) + f α = g , f α, , , = g f β = g , f β = g SO (2 , + × SO (3 , + × SO (2 , − f α = − f α = f α = f α = g f β , , = g , f β , , = ˜ g SO (3 , + × SO (3 , + f α = − f α = f α = f α = g f β , , = − f β , , = g f β , , = f β , , = g SO (2 , + × SO (4 , + f α = f α = f α , , = f α = g f α , , = f α , , = g f α = f α , , = f α , , = − g f α , , = − g , f β , , = g SO (2 , + × SU (2 , + f α = f α = f α = f α = − g f α = f α = g f α , , = f α , , = −√ g f α = 2 g , f β , , = g Table 1: Embedding tensor components and gauge groups for the two types of gaugingsthat yield dS vacua as given in [30]. Some of the coupling constants have been rescaledcompared to the original ones used in [30]. 10auge group Scalar potential V ( φ, χ )and g /g scalingfor dS vacuum at φ = χ = 0Type I SO (3) × SO (3) SO (3 , × SO (3 , SO (3) × SO (3 , SO (3) × SL (3 , R ) 2 g g − e − φ g − e φ ( g + g χ ) g = + g ( dS ) g = − g ( AdS )Type II SO (2 , × SO (2 ,
1) 12 e − φ (cid:2) g + e φ ( g + g χ ) (cid:3) g = ± g SO (2 , × SO (2 ,
2) 12 e − φ (cid:2) g + e φ { g + g χ } (cid:3) g = ± √ g SO (2 , × SO (3 ,
1) 12 e − φ (cid:2) g + e φ (3 g + g χ ) (cid:3) g = ± √ g SO (3 , × SO (3 ,
1) 32 e − φ (cid:0) g + e φ [ g + g χ ] (cid:1) g = ± g SO (2 , × SO (4 ,
1) 12 e − φ (cid:2) g + e φ (6 g + g χ ) (cid:3) g = ± √ g SO (2 , × SU (2 ,
1) 12 e − φ (cid:2) g + e φ (12 g + g χ ) (cid:3) g = ± √ g Table 2: Scalar potentials constructed from the embedding tensor f αMNP given in Table1 using (15) or (17) for all gauge groups. For concreteness, we use g for magnetic gaugefactor G m and g for electric factor G e . Note that the magnetic gauge coupling alwaysappears with χ . 11 dS × Σ solutions from 4D N = 4 supergravity To obtain cosmological solutions interpolating between a dS × Σ solution and a dS solution in each of the gauge groups listed in Table 2 above, we need to turn on anAbelian U (1) gauge field, together with the metric and supergravity scalars φ, χ . Allother fields are truncated out. Specifically, φ ma = 0 , M MN = M MN = n (31)Accordingly, the full Lagrangian (10) reduces to the following general form e − L = 12 R −
14 (Im τ ) ∂ µ τ ∂ µ τ ∗ −
14 Im τ H M + µν H M + µν + 18 Re( τ ) η MN ǫ µνρλ H M + µν H N + ρλ − V . (32)However, this Lagrangian (32) will not be the final one that we will work with. Inparticular, for the dS × Σ solutions that are of interest to us, we will turn on the gaugefield U (1) diag ⊂ U (1) e × U (1) m ⊂ G e × G m . Since all our gaugings are dyonic, we willdualize the magnetic gauge factor of each gauging into an electric one using the procedureoutlined in [19], so that the SL (2 , R ) frame under consideration is purely electric. Thisis necessary for us in order to use the Lagrangian (32) in which only SL (2 , R ) electricfield strengths are present. Effectively, this means that instead of G e × G m we will have G e × ˜ G e where ˜ G e is the dualized G m . The dualization in this case is simply the following SL (2 , R ) transformation acting on τ as τ → τ ′ = − τ , (33)so that the part of the Lagrangian (32), involving the two gauge groups G e × ˜ G e , takesthe following specific form e − L gauge = − e − φ F Mµν F Mµν + 18 χ η MN ǫ µνρλ F Mµν F Nρλ − (cid:18) e φ χ e φ (cid:19) ˜ F Mµν ˜ F Mµν − (cid:18) χe φ χ e φ (cid:19) η MN ǫ µνρλ ˜ F Mµν ˜ F Nρλ , (34)with F M , ˜ F M being the field strengths corresponding to U (1) ⊂ G e and (1) ⊂ ˜ G e , respec-tively F M = 2 ∂ [ µ A Mν ] , ˜ F M = 2 ∂ [ µ ˜ A Mν ] . (35) A general SL (2 , R ) transformation acting on τ has the form τ → τ ′ = aτ + bcτ + d , ad − bc = 1so in the case of (33) a = d = 0 , b = 1 , c = − e − L = 12 R −
14 (Im τ ) ∂ µ τ ∂ µ τ ∗ + e − L gauge − V . (36)Next, we will truncate the axion χ in (36). This axion truncation is consistent as long asthe following terms in (34) which source χ vanish η MN ǫ µνρλ F Mµν F Nρλ = 0 , η MN ǫ µνρλ ˜ F Mµν ˜ F Nρλ = 0 . (37)This is the case because of the purely magnetic gauge ansatz that we will use, so χ canbe safely truncated out.Consequently, setting χ = 0 in (36) gives us the following Lagrangian that we willwork with e − L = 12 R − ∂ µ φ ∂ µ φ − e − φ F Mµν F M µν − e φ ˜ F Mµν ˜ F Mµν − V ( φ ) . (38)The explicit scalar potentials V ( φ ) for all gauge groups are given by those specified inTable 2 with χ = 0.Having established the Lagrangian (38), we now move on to specify the various ansatzefor the dS × Σ solutions. For the metric, the ansatz is ds = − dt + e f ( t ) dr + e g ( t ) d Ω (39)where d Ω is the line element for S or H , d Ω = ( dθ + sin θ dφ , Σ = S dθ + sinh dφ , Σ = H . (40)The ansatz for the Abelian U (1) diag ⊂ U (1) e × U (1) m gauge fields is A Mφ = ˜ A Mφ = ( a cos θ, Σ = S a cosh θ, Σ = H . (41)The exact gauge field ansatz with the specified values for M will be given in subsequentsections for each gauge group. For type I gaugings, we need to turn on the gauge fieldscorresponding to the U (1) × U (1) subgroup of the SO (3) × SO (3) compact subgroupsthat lie entirely along R-symmetry directions, so M = m . For type II gaugings, the This is the same as the Lagrangian given in [8] where the case SU (2) × SU (2) gauge group wasconsidered. M assumes different values for A and ˜ A . At this stage, although we only have SL (2 , R ) electric field strengths in the Lagrangian, we willcontinue to refer to the gauge ansatz as being either electric or magnetic when specifying M later in eachgauged theory. (1) × U (1) have to be the subgroup of the compact parts that are embedded in thematter symmetry directions, so M = a . The corresponding gauge field strengths to (41)read F Mθφ = ˜ F Mθφ = ( a sin θ, Σ = S a sinh θ, Σ = H . (42)The equations of motion for dS × Σ solutions resulting from using the ansatze (39,41) in the Lagrangian (38) are0 = λ e − g − a e − g (cid:0) e − φ + e φ (cid:1) + 2¨ g + 3 ˙ g − V ( φ ) + 14 ˙ φ , a e − g (cid:0) e − φ + e φ (cid:1) + ¨ f + ˙ f ˙ g + ˙ f + ¨ g + ˙ g − V ( φ ) + 14 ˙ φ , a e − g (cid:0) e − φ − e φ (cid:1) − ˙ f ˙ φ − g ˙ φ − V ′ ( φ ) − ¨ φ (43)with λ = ( +1 , Σ = S − , Σ = H . (44)A dS × Σ fixed-point solution of the equations (43) is given by φ ( t ) = φ , g ( t ) = g , f ( t ) = f t. (45)The full solution described by (39) and φ = φ ( t ) of (43) is a cosmological solution inter-polating between the above dS × Σ fixed point at early times t → −∞ and a dS fixedpoint at late times t → + ∞ .Before moving on, we note that real solutions can only be obtained if the product a g of the gauge flux a and gauge coupling g is negative. In particular, we will impose thefollowing constraint a g = ± i . (46)This situation resembles the case of cosmological solutions in those dS supergravities,arising from dimensionally reducing the exotic ⋆ -theories, with the wrong sign for thegauge field strengths [16], and is similar to the result obtained from the 5D analyses in5D N = 4 supergravity done in [7]. 14 Type I dS gauged theories There are four gauge groups in the first type of dS gaugings, namely SO (3) × SO (3), SO (3 , × SO (3 , SO (4) × SO (3), and SO (3) × SL (3 , R ). These four gauged theoriescan give rise to both dS and AdS solutions. The SU (2) × SU (2) common subgroupof the four gaugings are generated by X , X , X and X , X , X , as can be seen fromthe embedding tensor components given in Table 1. When the field content is truncatedto only the metric and the scalars φ, χ from the supergravity multiplet, all four gaugedtheories produce the same scalar potential V = 2 g g − e − φ g − e φ (cid:0) g + g χ (cid:1) . (47)that admits the following vacua at φ = χ = 0 AdS : g = − g V = − g dS : g = g V = g . (48) SO (3) × SO (3) This gauge group can be embedded entirely in the R-symmetry group SO (6) without theneed for any coupled matter. It is the simplest and only fully compact gauging for both dS and AdS solutions. To get dS solution, we will set g = g in (47) and also χ = 0 sothat the scalar potential of the type I dS gauged theory is V = − g e − φ (cid:0) − e φ + e φ (cid:1) (49)with the following dS solution φ = 0 , f = g √ . (50)As mentioned above, the SO (3) × SO (3) group is generated by X , X , X and X , X , X .Therefore, the U (1) × U (1) gauge fields are given by (41) with M = 3 for the electricpart and M = 6 for the magnetic part, corresponding to the generators X and X ,respectively.Eqs. (43) together with the potential (49) yield the following dS × Σ solutions f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! ,φ = 0 (51)15ith κ = +1 for Σ = S and κ = − = H . After imposing (46), the solutionsbecome f = q − κ √ g ,g = −
12 ln h − κ + √ g i φ = 0 , (52)The cosmological solutions interpolating between the dS × Σ fixed points (52) and the dS solution (50) are plotted in Fig.1. - - t g ( t ) (a) Solution for g - - t f ′ ( t ) (b) Solution for ˙ f Figure 1: Cosmological solutions interpolating between the dS × Σ (52)fixed point (withsolid line for S and dashed line for H ) at early times and the dS (50) solution at latetimes in the first type dS gauging with gauge group SO (3) × SO (3) with g = 1. In addition to the SO (3) × SO (3) theory above, type I dS gauged theories include threemore non-compact gauge groups SO (3 , × SO (3 , SO (3) × SO (3 , SO (3) × SL (3 , R ) . These gauged theories cannot be realized with just pure N = 4 supergravity.Instead, they require coupled vector multiplets in order to be implemented, since their non-compact directions are embedded entirely in the fundamental representation of the mattersymmetry group SO ( n ) ⊂ SO (6 , n ). However, because the SO (3) × SO (3) subgroup ofthese three theories are the same as the SO (3) × SO (3) gauge theory, the analyses forthese three remaining gauge groups in the type I gauged theories yield identical resultsto the SO (3) × SO (3) case studied above. As already mentioned, when truncated to just χ, φ , the scalar potentials as well as the gauge ansatze of all four theories are the same.Consequently, we will not repeat these analyses but note only that dS × Σ fixed pointsolutions and cosmological solutions in these gauged theories are given by (51, 52) andFig. 1, respectively. 16 Type II dS gauged theories For this type of gaugings, the U (1) × U (1) gauge fields are given by (41) with M assumingthe values along the matter symmetry group SO ( n ) ⊂ SO (6 , n ) in 1 for both the electricand the magnetic parts. SO (2 , × SO (2 , From the embedding tensor given in Table 1, the compact part SO (2) × SO (2) ⊂ SO (2 , × SO (2 ,
1) is generated by X and X . Accordingly, we can turn on the U (1) × U (1) gauge fields (41) with M = 7 and M = 10 for the electric and magneticparts, respectively. The scalar potential for this gauge group is V = 12 (cid:0) e φ g + e − φ g (cid:1) (53)with a dS vacuum at φ = 0 , f = g √ , g = ± g . (54)The equations of motion (43) together with the potential (53) yield the following dS × Σ fixed point solutions f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! ,φ = 0 (55)which become f = q − κ √ g ,g = −
12 ln h − κ + √ g i φ = 0 (56)with κ = 1 for Σ = S and κ = − = H , after imposing (46).The cosmological solutions interpolating between the dS × Σ solutions (56) and the dS solution (54) are plotted in Fig. 2. We note that the dS × Σ fixed point solutions(55, 56) and the cosmological solutions Fig. 2 are identical to those of the type I gauging,Eqs. (51, 52) and Fig.(1). 17 - t g ( t ) (a) Solution for g - - t f ′ ( t ) (b) Solution for ˙ f Figure 2: Cosmological solutions interpolating between the dS × Σ (56) fixed point (withsolid line for S and dashed line for H ) at early times and the dS (54) solution at latetimes in the second type dS gauging with gauge group SO (2 , × SO (2 ,
1) with g = 1. SO (2 , × SO (2 , From the embedding tensor given in Table 1, the compact part SO (2) × SO (2) × SO (2) ⊂ SO (2 , × SO (2 ,
2) is generated by X , X and X , respectively for each of the three SO (2)’s. Correspondingly, we can turn on the U (1) × U (1) gauge fields (41) with M = 7or M = 8 for the electric part and M = 9 for the magnetic part. The scalar potential ofthis theory is V = 12 (cid:0) e − φ g + 2 g e φ (cid:1) (57)with a dS critical point at φ = 0 , f = r g , g = ± √ g . (58)The equations (43) with the potential V (57) admit the following dS × Σ fixed points f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! ,φ = 0 (59)which become f = q − κ √ g ,g = −
12 ln h − κ + √ g i φ = 0 (60)18ith κ = 1 for Σ = S and κ = − = H , after imposing (46).The cosmological solutions interpolating between the dS × Σ solutions (60) and the dS solution (58) are plotted in Fig. 3. - - t g ( t ) (a) Solution for g - - t (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)3(cid:22)(cid:23) f ′ ( t ) (b) Solution for ˙ f Figure 3: Cosmological solutions interpolating between the dS × Σ (60) fixed point (withsolid line for S and dashed line for H ) at early times and the dS (58) solution at latetimes in the second type dS gauging with gauge group SO (2 , × SO (2 ,
2) with g = 1. SO (3 , × SO (3 , The compact part SO (3) × SO (3) ⊂ SO (3 , × SO (3 ,
1) is generated by X , X , X and X , X , X (see Table 1), so we can turn on the U (1) × U (1) gauge fields (41) with M = 9 for the electric part and M = 12 for the magnetic part. The scalar potential forthis gauge group is V = 32 (cid:0) e φ g + g e − φ (cid:1) (61)with a dS vacuum at φ = 0 , f = g , g = ± g . (62)The equations of motion (43) together with the potential (61) yield the following dS × Σ fixed point solution f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! ,φ = 0 (63)19here κ = 1 for Σ = S and κ = − = H . After imposing the condition (46),(63) become κ = 1 : f = 2 g , g = − log( √ g ) , φ = 0 κ = − f = 2 √ g , g = − log( √ g ) , φ = 0 . (64)The cosmological solutions interpolating between (64) at early times and (62) at late timesare numerically solved for and plotted in Fig.4. - - t g ( t ) (a) Solution for g - - t (cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" f ′ ( t ) (b) Solution for ˙ f Figure 4: Cosmological solutions interpolating between the dS × Σ (64) fixed point (withsolid line for S and dashed line for H ) at early times and the dS (62) solution at latetimes in the second type dS gauging with gauge group SO (3 , × SO (3 ,
1) with g = 1. SO (2 , × SO (3 , The compact part SO (2) × SO (3) ⊂ SO (2 , × SO (3 ,
1) is generated by X and X , X , X , respectively (see Table 1). Accordingly, we can turn on the U (1) × U (1)gauge fields (41) with M = 9 for the electric part and M = 10 for the magnetic part. Thescalar potential of this theory is V = 12 (cid:0) e φ g + g e − φ (cid:1) (65)with the following dS vacuum φ = 0 , f = g , g = ± √ g . (66)which has the same f as (62). The equations (43) with the potential (65) yield the same dS × Σ fixed-point solutions as (63) of the SO (3 , × SO (3 ,
1) theory, which become (64)after imposing (46). The cosmological solutions from this theory, interpolating between(64) and (65), are the same as and given by Fig.4.20 .5 SO (2 , × SO (4 , The compact part SO (2) × SO (4) ⊂ SO (2 , × SO (4 ,
1) is generated by X and X , . . . , X , respectively (see Table 1). Equivalently, the SO (4) factor can be writtenas SO (4) ∼ = SO (3) + × SO (3) − with the corresponding generators SO (3) ± : X ± X , X ∓ X , X ± X . (67)The U (1) × U (1) gauge fields in this case are given by (41) with M = 13 for the magneticpart corresponding to SO (2) ⊂ SO (2 , A φ = A φ = a √ ( cos θ Σ = S cosh θ Σ = H (68)for the electric part, corresponding to the Cartan of SO (3) + factor in (67).The scalar potential for this gauge group is V = 12 (cid:0) e φ g + g e − φ (cid:1) (69)with a dS vacuum at φ = 0 , f = √ g , g = ± √ g . (70)The equations of motion (43) together with the potential (69) yield the following dS × Σ fixed point solutions f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g − a ! ,φ = 0 (71)which, after imposing (46), become f = √ r(cid:16) − κ √ (cid:17) g g = −
12 log (cid:16) (cid:16) √ − κ (cid:17) g (cid:17) ,φ = 0 (72)where κ = +1 for Σ = S and − = H . The cosmological solutions connecting(72) at early times to (70) at late times are numerically solved for and plotted in Fig. 5.21 - t g ( t ) (a) Solution for g - - t ’()*+,-./0456789:; f ′ ( t ) (b) Solution for ˙ f Figure 5: Cosmological solutions interpolating between the dS × Σ (72) fixed point (withsolid line for S and dashed line for H ) at early times and the dS (70) solution at latetimes in the second type dS gauging with gauge group SO (2 , × SO (4 ,
1) with g = 1. SO (2 , × SU (2 , The compact part SO (2) × SU (2) × U (1) ⊂ SO (2 , × SU (2 ,
1) is generated by X , X , X , X , and X , respectively (see Table 1). Hence, we can turn on the U (1) × U (1)gauge fields (41) with M = 11 for the magnetic part and M = 9 or M = 10 for theelectric part. The scalar potential for this gauge group is V = 12 (cid:0) g e φ + g e − φ (cid:1) (73)with a dS vacuum at φ = 0 , f = 2 g , g = ± √ g . (74)The equations of motion (43) together with the potential (73) yield the following dS × Σ fixed point solutions f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! ,φ = 0 (75)which, after imposing (46), become f = √ r(cid:16) − κ √ (cid:17) g ,g = −
12 log (cid:16) (cid:16) √ − κ (cid:17) g (cid:17) ,φ = 0 (76)22here κ = +1 for Σ = S and − = H . The cosmological solutions connecting(76) at early times to (74) at late times are numerically solved for and plotted in Fig. 6. - - t <= g ( t ) (a) Solution for g - - t > f ′ ( t ) (b) Solution for ˙ f Figure 6: Cosmological solutions interpolating between the dS × Σ (76) fixed point (withsolid line for S and dashed line for H ) at early times and the dS (74) solution at latetimes in the second type dS gauging with gauge group SO (2 , × SU (2 ,
1) with g = 1. In this section, we summarize all solutions of the type II dS gauged theories. For allgauge groups, excluding SO (3 , × SO (3 , g /g ratios given in Table 2. We list all the rewritten solutionsin Table 3. Furthermore, we collect all cosmological solutions given Figs. 2, 3, 4, 5, 6 ina single plot Fig. 7. 23ype II gauge group dS × Σ solutions SO (2 , × SO (2 , SO (2 , × SO (2 , SO (2 , × SO (3 , SO (2 , × SO (4 , SO (2 , × SU (2 , f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! ,SO (3 , × SO (3 , f = s a g + κ p − a g − a ,g = −
12 log κ − p − a g a ! , Table 3: Summary of all fixed-point dS × Σ solutions in type II dS gauged theories withthe six gauge groups. All solutions have φ = 0, and κ = +1 for Σ = S and κ = − = H . 24 - t f ′ ( t ) (a) Solution for ˙ f - - t g ( t ) S? ( ) xSO ( ) SO ( ) xSO ( ) SO ( ) xSO ( ) SO ( ) xSO ( ) SO ( ) xSU ( ) (b) Solution for g Figure 7: All cosmological solutions from dS × Σ at the infinite past to dS in the infinitefuture with g = 1 from the type II dS gauged theories. Solid lines represent Σ = S solutions and dashed lines represent Σ = H solutions. Note that the solutions of the SO (2 , × SO (3 ,
1) theory are given by those of SO (3 , × SO (3 ,
1) theory.25
First-order systems
In this section, we will derive the first-order equations, containing some relevant pseudo-superpotential W [25], [26], [27], that can solve the second order equations of motion43. Next, we check whether these first-order equations admit the dS vacua and theirassociated cosmological solutions found in sections 5 and 6. Although first-order equa-tions are most often linked to the supersymmetric AdS case in which domain walls orholographic RG flow solutions can arise as solutions of some first-order BPS equationsobtained by setting to zero the supersymmetry transformations of the fermionic fields, itmust be noted that first-order equations can and do arise completely independently fromsupersymmetric systems in the so-called fake supersymmetric case. In this scenario, theHamilton-Jacobi (HJ) formalism has been shown to produce first-order equations con-taining some fake superpotential obtained from the factorization of the HJ characteristicfunction [29]. More details on the relation between the HJ formalism and fake supersym-metry can be found in [28], [29].As mentioned in section 1, the fact that dS vacua of 4D N = 4 supergravity areunstable is a strong indication of the lack of pseudosuperymmetry that ensures the ex-istence of relevant pseudosuperpotentials and corresponding first-order equations. It isthen the objective of this section to characterize the extent to which there fail to existrelevant pseudosuperpotentials for the type I and type II dS theories. We will proceed asfollows. For the purpose of deriving the first-order equations in the dS gauged theories,our analysis will be based on the AdS case since their equations of motion are almostidentical up to various signs. The form of the first-order equations for the dS case will beinferred from that of the BPS equations in the AdS case. To derive the field equationsfor either
AdS and dS solutions, we will work with the minimally required field contentconsisting only of the metric and dilaton φ . The axion χ can be truncated out since itvanishes in either AdS or dS vacuum. The Lagrangian reads e − L = 12 R − ∂ µ φ ∂ µ φ − V ( φ ) . (77)In the cases of the AdS × Σ and dS × Σ , we will use (38). AdS case
AdS The ansatz for
AdS metric is ds = dr + e f ( r ) (cid:2) − dt + dx + dy (cid:3) . (78)The field equations resulting from using (85) in (77) are0 = 2 f ′′ + 3 f ′ + 14 φ ′ + V, φ ′′ + 32 f ′ φ ′ − (cid:18) ∂V∂φ (cid:19) , (79)26he equations (79) are solved by the following set of first-order equations f ′ = √ W,φ ′ = − √ (cid:18) ∂W∂φ (cid:19) , (80)subject to the following condition on the scalar potential and the superpotential V = 89 (cid:18) ∂W∂φ (cid:19) − W . (81)Recall that there are four gaugings which admit fully supersymmetric AdS solutions.These are the exact ones given in the type I dS gaugings (see Table 1). With the vectormultiplet scalars truncated out, the scalar potentials from these four gaugings are identicaland can be obtained from the ones in Table 2 by setting χ = 0. V = 2 g g − e − φ g − e φ g (82)To obtain AdS solutions we need to set g = − g so that the potential (82) becomes V = − g e − φ (cid:0) e φ + e φ (cid:1) (83)There exists the following superpotential WW = 3 g √ (cid:0) e − φ/ + e φ/ (cid:1) (84)such that the potential (83) can be written in terms of W as in the required relation(81). Note that the equations (80) can be derived by setting to zero the supersymmetrytransformations of the gravitino and dilatino fields, δψ µ i = 0 , δχ i = 0, as were done in[23] . Before moving on, we remark that the BPS equations (80) with the superpotential W (84) and the equations of motion (79) with the scalar potential V (83) admit the same AdS vacuum f = g , φ = 0 , (85)as should be the case. In [23], the derivation of the BPS equations, from δψ µ i = 0 , δχ i = 0, involves the axion χ . As such,the superpotential and first-order equations are more general. Another difference between [23] and thiswork is a factor of 2 in V and the first-order equations given in [23]. .1.2 First-order equations for AdS × Σ The metric ansatz for
AdS × Σ is ds = − e f ( r ) dt + dr + e g ( r ) d Ω , (86)with d Ω being the line element for Σ = S , H . The gauge field ansatz is given by(41), with A Me ( m ) φ being the U (1) × U (1) gauge fields in the R-symmetry directions where M = 3 for α = e and M = 6 for α = m . This is exactly the same as the gauge ansatzfor the type I dS gauged theories. The equations of motion for AdS × Σ resulting fromusing the ansatze (86, 41) in (38) are0 = − λe − g + 12 a e − g (cid:0) e − φ + e φ (cid:1) + 2 g ′′ + 3 g ′ + V ( φ ) + 14 φ ′ , − a e − g (cid:0) e − φ + e φ (cid:1) + f ′′ + f ′ g ′ + f ′ + g ′′ + g ′ + V ( φ ) + 14 φ ′ , − a e − g (cid:0) e − φ − e φ (cid:1) − f ′ φ ′ − g ′ φ ′ + 2 V ′ ( φ ) − φ ′′ (87)with λ = +1 for Σ = S and λ = − = H . It is instructive to compare thefield equations (87) in this case to those from the dS case (43). The two sets are almostidentical except for the opposite signs for the non-derivative terms ( V, λe − g , and gaugefield strength terms). The equations of motion for the AdS × S (87) case are solved bythe following set of first-order system f ′ = √ W (cid:18) λae − g g + 1 (cid:19) ,g ′ = √ W (cid:18) − λae − g g (cid:19) ,φ ′ = − √ (cid:18) − λae − g g (cid:19) ∂W∂φ , (88)subject to the condition (81) on the scalar potential and the superpotential, and thefollowing condition on the gauge flux a and gauge coupling g ag = − . (89)For the scalar potential V given by (83), the relation (81) was shown to be satisfied withthe superpotential given by (84). The BPS equations (88) subject to (89) with W givenby (84) admit the same AdS × Σ solutions as the equations of motion (87) with thepotential (83). However, not all solutions are real and thus physically acceptable. Inparticular, only in the case λ = −
1, there exists the following real
AdS × H solution toboth (88) and (87) f = 2 g , g = −
12 log (cid:2) g (cid:3) , φ = 0 (90)28he domain wall solution interpolating between the AdS × H solution (90) and the AdS solution (85) can be obtained by either solving (87) or (88) numerically. Finally,we note that the first-order equations (88) with W given by (84) and the twist condition(89) are essentially identical to the BPS equations that are given in [22] for the 4D N = 4 AdS × Σ holographic RG flow solutions. dS case dS The metric ansatz for dS is ds = − dt + e f ( t ) ( dx + dy + dz ) . (91)Using this ansatz in (77) gives the following set of equations of motion0 = 2 ¨ f + 3 ˙ f + 14 ˙ φ − V, φ + 32 ˙ f ˙ φ + (cid:18) ∂V∂φ (cid:19) . (92)These equations are almost identical to the ones for AdS (79), save for the opposite signsin front of the terms involving the scalar potential and its derivative. Eqs. (92) can besolved by the same set of first-order equations (80) as in the AdS case˙ f = √ W, ˙ φ = − √ (cid:18) ∂W∂φ (cid:19) , (93)but with the relation (81) replaced by V = − " (cid:18) ∂W∂φ (cid:19) − W , (94)where V has an opposite sign to (81). dS × Σ The equations of motion (43) are solved by the same first-order equations (88) as in the
AdS case ˙ f = √ W (cid:18) λae − g g + 1 (cid:19) , ˙ g = √ W (cid:18) − λae − g g (cid:19) , ˙ φ = − √ (cid:18) − λae − g g (cid:19) ∂W∂φ , (95)29here λ = 1 for Σ = S and − = H , subject to the following constraint betweenthe gauge flux a and gauge coupling constant g ag = 12 , (96)and the relation (94) V = − " (cid:18) ∂W∂φ (cid:19) − W . Note that unless an explicit pseudo-superpotential is substituted in the first-order equa-tions (95), the relation (94) and the constraint (89) are not enough to solve the equationsof motion (43).Having established the first-order equations that solve the second-order field equationsfor both the dS and dS × Σ cases, we now move on to check whether there exists anypseudo-superpotential W that satisfies the required relation (94) for both the type I andtype II gauged theories. With the vector multiplet scalars truncated out, the scalar potential from the four type Igaugings that admit dS solutions is given in (49), V = − g e − φ (cid:0) − e φ + e φ (cid:1) . This scalar potential appears in the equations of motion for both the cases of dS (92)and dS × Σ (43). The pseudo-superpotential that satisfies the relation (94) with V givenby (49) reads W = − i √ g h e φ − e − φ i . (97)The first-order equations (93) and (95) with W given in (97) solve the equations of motion(92) and (43), respectively. Although this is the case, these first-order equations donot give rise to either the dS solution (50) nor the comoslogical solutions interpolatingbetween this dS and the dS × Σ fixed-point solutions (52). We elaborate more on thisbelow. • dS solution . The dS solution (50) admitted by V (49) φ = 0 , f = g √ , is not a critical point of the pseudo-superpotential (97). This is because given (94),the extremization of V , V ′ = 0, implies either one of the two following conditions V ′ = 0 ⇒ W ′ = 0 , or 3 W − W ′′ = 0 . (98)30t the dS point φ = 0, although W ′ = 0,(3 W − W ′′ ) = 0 , (99)leading to V ′ = 0. As such, the dS solution (50) cannot arise from the first-orderequations (93). It was pointed out in [26] that in general, for a cosmological solution,if a scalar potential can be written in terms of a pseudo-superpotential as V = − (cid:0) W ′ − α W (cid:1) , (100)where α is a constant, then the critical point at which V ′ = 0 can arise from either W ′ = 0 or W ′′ − α W = 0 . (101)When the former condition is satisfied then the solution is pseudosupersymmetricand W is the corresponding pseudo-superpotential. When the latter condition issatisfied then the solution is not pseudo-supersymmetric. These conditions are tiedto the Breitenlohner-Freedman (BF) bound of the solutions, which, in the case ofcosmology, reads m ≤ ( D − L , (102)where D is the spacetime dimension and L = 1 /f is the radius of dS space. Solu-tions admitted by the pseudo-superpotential W (corresponding to the case W ′ = 0)do not violate the BF bound, while those not admitted by the pseudo-superpotential W (corresponding to the case W ′′ − α W = 0) do [26]. In this case, for D = 4 and L = q V = 1 /f , it can be verified from [21] that dS solutions of the four gaugingsin the type I theories do violate the BF bound. We explicitly list the mass valuesviolating this BF bound for each of the four type I gauged theories in Table 4. • dS × Σ cosmological solutions . Although the first-order equations (95) with W (97)and the second-order equations (43) with V (49) both admit the fixed-point solutions(51), the cosmological solutions interpolating between this fixed point solutions andthe dS solution (50) as given in Fig.1 are not solutions of (95). The reason forthis is twofold. Firstly, the pseudo-superpotential (97) does not admit dS solution(50) as V (49), as pointed out in the previous section. Secondly, the constraint(96) is needed so that solutions of the equations of motion (43) become solutions of(95). However, once (96) is imposed, f in (51) vanishes (for both κ = 1 and − dS solution (50)nor the fixed-point solutions (52) can be admitted by the first-order equations (43).Instead, the fixed point solutions (52) that are real and their associated cosmologicalsolutions Fig.1 can only arise from the second-order equations of motion (43) withthe constraint (46). 31 .2.4 Type II gauged theories For the scalar potentials given in Table 2, no suitable W can be found such that (94) issatisfied. Instead, we found the following WW = c g (cid:0) e φ/ − e − φ/ (cid:1) , c = ± √ , SO (2 , × SO (2 , ± , SO (2 , × SO (2 , ± (cid:18) (cid:19) / , SO (3 , × SO (2 , SO (3 , × SO (3 , ± √ , SO (2 , × SO (4 , ± r , SO (2 , × SU (2 ,
1) (103)that satisfies the following relation V = 89 (cid:18) ∂W∂φ (cid:19) + 29 W ( φ ) . (104)Consequently, there do not exist any systems of first-order equations that solve the second-order equations (92) for the dS case or (43) for the dS × Σ case for the type II dS solutions. It is worth recalling that since the type II dS gauged theories in 4D are directlyrelated to 5D dS gauged theories via dimensional reduction as shown in [30], this situationagrees with the result from the five-dimensional analysis [7] which shows there do not existany suitable pseudo-superpotentials (and systems of first-order equations) for the dS and dS , × Σ , cosmological solutions.We also note that the pseudo-superpotentials W as given in (103) do not admit the dS solutions at φ = χ = 0 that are admitted by the scalar potentials V given in (2).The reason for this is the same as the type I case above. Given the relation (104), theextremization of V , V ′ = 0, can be obtained from either W ′ = 0 , or ( W + 4 W ′′ ) = 0 (105)At the dS point φ = 0, W ′ = 0, but rather W + 4 W ′′ = 0 . (106)It can again be verified from [21] and [30] that dS solutions of the six gaugings in thetype II theories violate the BF bound (102). The mass values violating the BF bound(102) for each of the six type II gauged theories are listed in Table 4.32 .2.5 BF bounds for dS solutions For D = 4, the BF bound (102) reads m ≤ L , or m L ≤
94 (107)Given that the dS length L is related to the extremized value V of V as L = 3 V , (108)the bound (107) can also be written in terms of V m ≤ V . (109)The mass spectra of dS solutions in all dS gauged theories, excluding SO (2 , × SO (4 , dS solution of the SO (2 , × SO (4 , V and the mass values violatingthe BF bound as given in [21] and [30]. Only one mass value for each gauged theory issufficient to show that the BF bound is violated. Due to the different notations used in[21] and [30], we will use (109) for [21] and (107) for [30].Some comments regarding these notations are in order to avoid any potential confu-sion. In [21], V is given in terms of a ij that is defined as a ij = g i g j sin( α i − α j ) , i, j = 1 , , . . . (110)where indices i, j label the various individual gauge factors G × G × . . . constituting thegauge group G , g i , g j are the corresponding gauge couplings, and α i are the SU (1 , ∼ = SL (2 , R ) angles. Although there can be more than two gauge factors in G , all cases wereshown to be reduced to just two factors (see Table 2). The mass spectra are also given interms of a ij , making it convenient to check the BF bound using (109). In [30], V is givenin terms of g after applying the scaling ratio g /g to bring the dS critical point φ = 0to the origin of the scalar manifold at φ = 0, while the mass spectrum is given in unitsof m L . Accordingly, it is convenient to use (107).33auge group G V Mass value m violating (109)( × multiplicities)Type I SO (3) × SO (3) −| a | − a a < − a ( × SO (3 , × SO (3) −| a | − a a < | a | − a ( × SO (3 , × SO (3 , −| a | − a a < | a | − a ( × SL (3 , R ) × SO (3) −| a | − a a < − a ( × SO (2 , × SO (2 , | a | | a | ( × SO (2 , × SO (2 , √ ∆ = p a + a + a √ ∆ ( a + a ) + 2 a a = a = a = r ∆ × SO (3 , × SO (2 , √ | a | √ | a ( × SO (3 , × SO (3 ,
1) 3 | a | | a | ( × SU (2 , × SO (2 ,
1) 2 √ | a | √ | a | ( × SO (2 , × SO (4 ,
1) BF bound is (107) m L = 6 ( × dS gauged theories taken from [21] with the exception of the gauge group SO (2 , × SO (4 , .3 Summary For both types of dS gauged theories, we checked whether the first-order systems ofequations, to which the second-order field equations reduce, can give rise to the dS andcosmological solutions of sections 5, 6. Following the domain wall/cosmology correspon-dence established in [26], this analysis was performed using the AdS case as a reference,since the equations of motion for the
AdS case are almost identical to those of the dS case, except for a few opposite signs in front of some non-derivative terms. The first-orderequations for the dS and dS × Σ solutions, if they exist, should be identical in form tothe BPS equations for AdS and AdS × Σ solutions, respectively. The reducibility of thesecond-order field equations to the first-order equations hinges on there being a requiredrelation V ( W ) between V and W such that V can be written in terms of the squares of W and its derivatives. This V ( W ) relation in the dS case should be identical, save foran overall opposite sign, to the V ( W ) relation in the AdS case. While supersymmetryautomatically ensures the existence of the superpotential W in the AdS case, there is noguarantee that a suitable pseudo-superpotential W can be found in the dS case. Thus,while the general form of the first-order equations in the dS case is settled, its validity isonly confirmed if a suitable W exists. The suitability of W is decided by two conditions,the first one being whether it satisfies the required V ( W ) relation, and the second onebeing whether it admits the same dS critical point as V .For the type I dS gauged theories, we did find a pseudo-superpotential W satisfyingthe required V ( W ) relation, but this W does not admit the dS solution that is admittedby the scalar potential V . Hence, the first suitability condition is satisfied but the sec-ond is not. This eliminates the possibility of the type I dS solution and its associated dS × Σ cosmological solutions arising from the first-order equations that have the sameform as the BPS equations in the AdS case. For the type II dS gauged theories, on theother hand, there does not exist any W that satisfies the required V ( W ) relation. Thus,even the first suitability condition fails. This is almost the same as the result of a similaranalysis in the five-dimensional case [7] where there is only one type of dS gaugings thatcorresponds to type II gaugings in four dimensions. In five dimensions, although the firstsuitability condition is not fulfilled in an exact manner as in the type II theories in fourdimensions, the second one is. Regardless of the exact way the pseudo-superpotentialfails to be suitable in either the 4D type I or type II theories, and even in the 5D dS gauged theories, this failure can be traced to the fact that firstly these dS solutions areunstable, and secondly they violate the BF bound that serves as a means to guaranteepseudo-supersymmetry. In this work, we have studied cosmological solutions interpolating between a dS × Σ spacetime, with Σ = S and H , and a dS spacetime from N = 4 four-dimensionalgauged supergravity. We emphasize that our motivation for the study of these solutionsis completely decoupled from any holographic contexts. Instead, we are solely motivated35y the question of whether cosmological solutions exist in 4D N = 4 supergravity, giventhe existence of dS vacua [30]. Consequently, the cosmological solutions found were ob-tained by solving the second-order field equations in theories with gauge groups capable ofadmitting dS solutions, of which there are two types. Type I dS gauged theories consistof four theories with gauge groups SO (3) × SO (3), SO (3 , × SO (3 , SO (3) × SO (3 , SO (3) × SL (3 , R ) whose common compact subgroup SO (3) × SO (3) is fully embeddedin the SO (6) R-symmetry directions. These type I theories can admit both dS and AdS solutions. The cosmological solutions in the type I theories require an Abelianvector field that corresponds to the diagonal U (1) of the U (1) × U (1) subgroup of theaforementioned SO (3) × SO (3) compact part. Type II dS gauged theories consist of sixtheories with gauge groups SO (2 , × SO (2 , SO (2 , × SO (2 , SO (2 , × SO (3 , SO (3 , × SO (3 , SO (2 , × SO (4 ,
1) and SO (2 , × SU (2 ,
1) whose compact sub-groups are entirely embedded in the SO ( n ) matter symmetry directions. These type IItheories can only admit dS solutions without the possibility of admitting AdS solutions.To obtain cosmological solutions in the type II dS theories, we needed to turn on anAbelian gauge field corresponding to the diagonal of the U (1) × U (1) subgroup of thecompact part along the SO ( n ) matter directions.Furthermore, we also characterized the extent to which these nonsupersymmetric dS and their associated cosmological solutions fail to arise from the relevant first-order equa-tions that solve the second-order field equations by studying the lack of suitable pseudosu-perpotentials in the type I and II dS gauged theories. Finally, we note that cosmologicalsolutions arising from either type I or type II theories require the square of the product ofthe gauge flux a and gauge coupling g to be negative in order to be real. This feature isalready encountered in the five-dimensional cosmological solutions [7] and resembles thesituation in the dS supergravities, arising from the dimensional reduction of M ⋆ /IIB ⋆ -theories, with the wrong sign for the gauge kinetic terms. Instead of regarding thesesolutions as pathological due to this particular feature, we interpret this as an additionalcharacterization of the dS vacuum structure of half-maximal supergravity. The implica-tions of this remain to be understood and we hope that more work will elucidate thismatter further. Acknowledgements : HLD is supported by the RSB grant C-144-000-207-532/C-141-000-777-532 for postdoctoral research.
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