Maximally Supersymmetric AdS4 Vacua in N=4 Supergravity

We study AdS backgrounds of N=4 supergravity in four space-time dimensions which preserve all sixteen supercharges. We show that the graviphotons have to form a subgroup of the gauge group that consists of an electric and a magnetic SO(3)_+ x SO(3)_-. Moreover, these N=4 AdS backgrounds are necessarily isolated points in field space which have no moduli.


Introduction
The maximally symmetric space-time backgrounds of supergravity theories which preserve some of the supercharges are either anti-de Sitter (AdS) or Minkowski (M) spaces. It is of interest to study such backgrounds and find the (model-independent) properties of the associated moduli spaces. In N = 2 supergravities in four space-time dimensions (d = 4) the fully supersymmetric AdS 4 backgrounds were determined in [1,2] while the structure of the moduli space of N = 1 and N = 2 AdS 4 backgrounds was given in [3]. It was found that generically a supersymmetric AdS 4 background of N = 1 and N = 2 supergravity has no moduli space. However, by appropriately tuning the mass parameters of the theory flat directions which preserve all supercharges may occur. In N = 1 they span a field space which is necessarily real and has at best half the dimension of the original field space. In N = 2 the moduli space is a Kähler manifold -again at best of half the dimension of the original field space. Both results are in agreement with the AdS/CFT correspondence which relates these backgrounds to superconformal field theories on the d = 3 boundary of AdS 4 with multiplets which are in representations of theories that have only half of the supercharges. 1 For N = 4 supergravity in d = 4 an analogous investigation is lacking so far and it is the purpose of this paper to close this gap. In contrast to gauged supergravities with eight or less supercharges, in N = 4 supergravity the mass parameters cannot be freely tuned and are determined by the choice of the gauge group. This in turn suggests that the dimension and structure of the moduli space is also fixed. From the AdS/CFT perspective one expects the moduli space to be hyper-Kähler -if it exists at all. 2 After the initial construction of (electrically) gauged N = 4 supergravity [5][6][7][8][9] it was shown that within this class of theories no supersymmetric AdS-backgrounds exist and any background preserving some supercharges has to be Minkowskian M 4 [10,11]. 3 However the same papers realized that supersymmetric AdS-backgrounds can occur when additional parameters are non-trivial. These de Roo-Wagemans angles gauge isometries with respect to dual magnetic vector multiplets. Generic gauged supergravities including magnetic vector multiplets have been constructed in [13] introducing what is now called the embedding tensor formalism. This was subsequently used in [14] to construct the most general gauged N = 4 supergravity coupled to vector multiplets in d = 4. It is within this framework that we conduct our analysis.
We find that the existence of a fully supersymmetric AdS 4 background imposes a set of constraints on the embedding tensor. They in turn imply that the complex scalar τ of the gravitational multiplet has to be uncharged and they also restrict the possible gauge groups G 0 . More precisely, the six graviphotons of N = 4 supergravity have to gauge an unbroken SO(3) + × SO(3) − inside the R-symmetry SO(6) R where one of the factors is electric while the other is magnetic. In general the two factors can be part of a larger gauge group with the structure is a separate factor. In the N = 4 AdS 4 vacuum the group G + × G − is spontaneously broken to its maximal compact subgroup, containing the two SO(3) ± factors. Furthermore, the potential has supersymmetric flat directions which, however, are precisely the Goldstone bosons of the spontaneous symmetry breaking. No further flat directions and thus no moduli space exists. This paper is organized as follows. In Section 2 we recall the properties of N = 4 gauged supergravity that we need for our analysis. In Section 3 we analyze N = 4 AdS 4 backgrounds and determine the constraints on the embedding tensor. We then show that an SO(3) + × SO(3) − subgroup of the R-symmetry group is necessarily gauged and we also determine the allowed structure of the full gauge group G 0 . In Section 4 we show that the conditions for an N = 4 AdS-background admit a set of flat directions corresponding to the Goldstone bosons of the spontaneously broken G 0 . However, no further flat directions do exist which indeed confirms that the backgrounds found in Section 3 are isolated points in the scalar field space. Some of the technical analysis is relegated to three appendices.
The field space M of the scalars is the coset where the first factor is spanned by τ while the second factor is spanned by the scalars φ am in the vector multiplets. Both cosets are conveniently parametrized by vielbein fields. For the first factor the vielbein is the complex vector ν α , α = +, −, which reads in terms of τ as and defines M αβ = Re (ν α (ν β ) * ) , ǫ αβ = Im (ν α (ν β ) * ) . (2. 3) The second factor in (2.1) is parametrized by the vielbein ν = (ν m M , ν a M ), M = 1, . . . , n + 6 which is an element of SO(6, n) and thus obeys where η M N = diag(−1, −1, −1, −1, −1, −1, +1, . . . , +1) is the flat SO(6, n) metric. The metric on the coset is then given by The couplings of N = 4 gauged supergravity depend on two field-independent SL(2) × SO(6, n)-tensors (called embedding tensors) denoted by ξ αM and f α[M N P ] . Their entries are real numbers and supersymmetry imposes a set of coupled consistency conditions on both tensors known as the quadratic constraints [14] ξ M α ξ βM = 0 , Their solutions parametrize the different consistent N = 4 theories and in particular determine the gauge group, the order parameters for spontaneous supersymmetry breaking and the potential.
The full bosonic Lagrangian is recorded in [14] but for the analysis in this paper we only need the potential V and the kinetic terms of the scalar fields which are given by The covariant derivative of M M N reads M is the matrix of gauge charges and A µ P + are n + 6 electric gauge bosons while A µ P − are their magnetic duals. 4 We see that a non-vanishing Θ − leads to magnetically charged scalar fields but the above mentioned quadratic constraint (2.6) also ensures mutual locality of electric and magnetic charges. D µ M αβ depends only on ξ αM which, as we will see shortly, vanish for N = 4 AdS backgrounds implying that τ is uncharged and D µ M αβ reduces to an ordinary derivative.
The conditions for a supersymmetric AdS-background can be concisely formulated in terms of the scalar components of the N = 4 supersymmetry transformations. For the four gravitinos ψ i µ , the four spin-1/2 fermions in the gravitational multiplet χ i and the gauginos λ i a they are given by [14] δψ where ǫ j are the four supersymmetry parameters and the dots indicate terms that vanish in a maximally symmetric space-time background. The fermion shift matrices read where the ν ij M are defined with the help of SO(6) Γ-matrices as In the embedding tensor formalism electric and magnetic gauge bosons are simultaneously introduced into the action and a global G =SL(2)×SO(6,n) is manifest as long as ξ M α and f αMN P transform as tensors under G. Any specific and consistent choice of ξ M α and f αMN P breaks that symmetry and determines the local gauge group G 0 ⊂ G.
We give more details on the Γ-matrices in Appendix A. In terms of the shift matrices the scalar potential is given by In this section we study N = 4 gauged supergravities that admit a fully supersymmetric AdS 4 background, that is, all sixteen supercharges are left unbroken. The latter requirement demands that the supersymmetry variations (2.9) of χ i and λ i a have to vanish in the AdS 4 background while the supersymmetry variations of the gravitinos have to be proportional to the cosmological constant. Inspecting (2.9) and (2.12) we see that this implies where V = − 4 3 |µ| 2 is the cosmological constant and · indicates that a quantity is evaluated in the AdS-background. In A 2 the first (second) term is anti-symmetric (symmetric) in i and j and thus they have to vanish independently. Similarly in A 2a the two terms correspond to a decomposition into the trace and a traceless part and thus they also have to vanish independently. We can immediately conclude that fully supersymmetric AdS 4 backgrounds can only occur in N = 4 supergravities which have This property considerably simplifies the following analysis and is also the reason why M αβ or similarly τ is uncharged in the Lagrangian (2.7). From (2.2) we also see that for purely electric gaugings, i.e. ξ M α = f −N P Q = 0, one has A 1 = A 2 and thus no supersymmetric AdS-background is possible [10,11].
Inserting (3.2) into (2.10) the conditions (3.1) simplify and read where P ij is a constant matrix obeying P ik P kj = δ i j but is otherwise arbitrary. The conditions (3.3)-(3.5) have to be solved subject to the quadratic constraints (2.6) which for ξ M α = 0 also simplify and are given by Due to the homogeneity of the N = 4 field space (2.1) one can translate any point of M to its origin and perform the analysis there. 5 Here we prefer to perform the analysis at some arbitrary but fixed vacuum expectation value of the scalar fields corresponding to an AdS 4 background. This leads us to redefine the components of the embedding tensor and introduce the complex quantities where f 1,2 QRS are the real and imaginary parts, respectively. Using (2.2) they are given by We see that f 2 is directly related to the magnetic components f − of the embedding tensor while f 1 is an admixture of electric and magnetic components. Note that at the origin of M the vielbeins are unit matrices, Re τ vanishes and f 1 is purely electric. Before we proceed let us also give the quadratic constraint (3.6) in terms of f. Using (2.2) one finds Let us now turn to the solution of the conditions (3.3)-(3.5) and start by analyzing the gaugino variation (3.5). Using (2.11), (3.7) and the antisymmetry of the f αM N P we can rewrite (3.5) as f amn (Γ mn ) ij = 0 , (3.10) where Γ mn are generators of SU(4) defined in Appendix A. Since the Γ mn are linearly independent generators we immediately conclude which, using (3.8), also implies f αamn = 0.
We employ the same strategy to analyze the variations of the fermions in the gravitational multiplet, i.e. (3.3) and (3.4). Using (2.11) and (3.7) they are equivalent to Since the antisymmetric products of three Γ-matrices are linearly independent up to the relation (A.3), we can further rewrite (3.12) as We learn that f mnp (f * mnp ) is imaginary self-dual (anti-self-dual) with a norm related to the cosmological constant. 6 In addition to (3.13) the f mnp have to satisfy the quadratic constraints. Due to (3.11) the mnpq-component of (3.9) simplifies and reads In Appendix B we show that (3.13) and (3.14) together have a unique solution which can always be put into the form 15) or, in terms of real and imaginary part and for µ real 7  17) and the scalar fields are only charged with respect to the six graviphotons while they remain neutral with respect to all other n Abelian vector fields. The number of vector multiplets n in this solution is arbitrary including n = 0 in which case τ is the only scalar field. 8 However, as we will show now, fully supersymmetric AdS-backgrounds with larger gauge groups G 0 can also exist for f mab = 0 and/or f abc = 0. 9 In this case the solution (3.15) of (3.3)-(3.5) is unaffected but the quadratic constraints (3.9) change and have to be reanalyzed. In particular they couple different components of the embedding tensor.
Let us first consider supergravities with f mab = 0, f abc = 0. In this case the quadratic constraints (3.9) split into two disjoint set of conditions and give a standard Jacobi-identity for f abc . Thus the gauge group is

18)
6 Note that at the origin the f nmp are related to the f (±) defined in [16]. Furthermore, (3.13) forbids any real f mnp which corresponds to the observation that a purely electric gauge theory does not admit an AdS-background. 7 By an appropriate rotation of the gravitinos µ can always be chosen real. 8 For n = 0, 1 this solution was first found in [9] and it is also discussed in [15]. 9 Physically the f αmab determine the supersymmetric fermionic and bosonic mass matrices while f αabc only contributes to mass terms when supersymmetry is broken [16].
where G v 0 ⊂ SO(n) is the gauge group with structure constants f αabc which only acts among the gauge bosons of the vector multiplets.
For f mab = 0, f abc = 0 the situation is slightly more involved. First of all there can be a split within the f abc into two disjoint sets so that one subset of them has no common indices with f mab and thus satisfies a standard Jacobi-identity with no further interference terms in (3.9). As before this corresponds to a separate factor G v 0 ⊂ SO(q), q ≤ n in the gauge group G 0 . Now let us turn to the f mab , f abc which do share common indices a, b, c. Considering the mnab-component of the last constraint in (3.9) we learn that in the basis where (3.16) holds, f 1 mab can only be non-zero for m = 1, 2, 3 while f 2 mab can only be non-zero for m = 4, 5, 6. Furthermore the same equation also says that f 1 mab and f 2 mab cannot share any ab indices. Or in other words the f mab decompose into two disjoint sets for f 1 mab , m = 1, 2, 3 and f 2 nab , n = 4, 5, 6. With this observation all other quadratic constraints turn into standard Jacobi-identities of three separate group factors G + , G − , G v 0 so that the total gauge group is of the form where The maximal compact subgroups for each factor are SO(3) ± × H ± with H ± ⊂ SO(m ± ) and m + + m − + q = n. Special cases of this solution have been discussed in [15]. As we will see in the next section in the N = 4 AdS background G 0 is spontaneously broken to its maximal compact subgroup.

N = AdS moduli space
After having determined the N = 4 AdS-backgrounds we turn to the question to what extent they are isolated points in field space or if they can have flat directions (a moduli space) which preserve all supercharges. We use the same method as in [3] in that we vary the supersymmetry conditions (3.1) and then find all possible directions in the scalar field space M which are left undetermined by (3.1). More concretely, we look for continuous solutions of  Similarly, for the SL(2)/SO(2) factor of M we have Using (4.4) and (4.6) we can also determine the variations of f to be 10 Im τ Im f npb δRe τ − 1 2Im τ f * npb δIm τ .

(4.7)
With the help of these variations we can now discuss the variations of A 1 and A 2 . Starting from (3.12) and using that P ij is constant we obtain δf * mnp = δf mnp = 0 . We are left with the variation of A 2a or in other words the variation of (3.11). Using (4.7) and (4.9) we find In Appendix C we show that all solutions of (4.10) have to be of the form where λ b 1,2 or equivalently λ b α are arbitrary real parameters.
The deformations (4.11) also have a geometrical meaning. The Lagrangian and the background are invariant under global symmetry transformations inside SO(6, n) that leave the gauge group G 0 = G + ×G − ×G v 0 (and the associated structure constants) invariant. This global symmetry is G 0 itself times its maximal commutant H c inside SO(6, n). 11 Since H c commutes with G ± , it commutes with SO(3) ± and therefore must be inside SO(n), i.e. H c is a compact group. Thus, the scalar deformations in (2.1) which preserve supersymmetry should correspond to the non-compact directions in G 0 × H c . Since G v 0 and H c are compact, the supersymmetric scalar deformations span the coset where H ± are the compact subgroups of G ± and contain an SO(3) ± factor. If we linearize the scalars in this coset, we indeed find the deformations (4.11).
Let us confirm the masslessness of the deformations (4.11) by computing their scalar mass matrix. The mass matrices of N = 4 gauged supergravity have been given and analyzed for example in [15,16]. Nevertheless let us spend a few steps to derive them in an N = 4 background in our notation. Inserting (2.10) into (2.12) using (3.7) one finds for ξ M = 0 Computing the second derivative of V with respect to δφ am , we find for an N = 4 background (where (3.11) and (3.13) hold) the mass matrix (4.14) We see that scalars of the form (4.11) indeed fulfill M am,bn δφ bn = 0 and thus are massless. This confirmation can also be viewed as a consistency check of (4.10). 12 It is easy to see from (4.13) that there are no mass terms mixing δτ and δφ am for an N = 4 vacuum.
Let us finally show that all flat directions in (4.11) correspond to Goldstone bosons which are eaten by massive vector fields. To do so we inspect the covariant derivative of δφ ma and find from (2.8) and (4.5) First of all we note that in the AdS-background, i.e. for δφ am = 0, all six graviphotons A αm are massless and thus, as expected, the SO(3) + × SO(3) − part of the gauge symmetry is unbroken. From the last term we see that there is a mass term for the gauge bosons A αa µ in the vector multiplets given byM 2 αaβb ∼ f αacm f βbcm . This gives mass to rk(M ) gauge bosons whereM am αb ∼ f αmab . This coincides with the number of flat directions determined in (4.11) as the same matrixM appears. This is no coincidence: When the gauge group is spontaneously broken from the Goldstone bosons form the coset (4.12). Therefore the supersymmetric directions in (4.12) are precisely the Goldstone bosons eaten by the massive vectors. Thus we showed that all supersymmetric scalar deformations are Goldstone bosons and therefore any AdS 4 background that preserves all supercharges is an isolated point in field space with no further moduli.

Appendix A SU(4) Γ-matrix properties
The fermions of N = 4 gauged supergravity transform in the fundamental representation of the SU(4) R-symmetry. On the scalars the R-symmetry acts as SO(6) ∼ SU(4)/Z 2 rotations. The two representations are linked via the Γ-matrices Γ ij m = Γ [ij] m given by [16] The antisymmetric products of two Γ-matrices (Γ mn ) i j = 1 2 (Γ m ) ik (Γ * n ) kj are the (linearly independent) generators of SU(4). On the other hand the antisymmetric products of three Γ-matrices obey the relation In (2.11) we use the Γ-matrices to convert the SO(6, n) vielbein components ν m M into objects with spinor indices, i.e. we define ν ij M = ν m M Γ ij m .

B Classification of structure constants
In this appendix we supply the details of the solution of (3.13) and (3.14) given in (3.15). We know from (3.13) that f is the coefficient of an imaginary self-dual three-form, which means that the form is of type (2, 1) ⊕ (0, 3) with respect to a given complex structure I on the six-dimensional space parametrized by the index m. In addition we have to solve the quadratic constraints (3.14) for such a three-form. If we write f with holomorphic and anti-holomorphic indices u,ū = 1, 2, 3, the quadratic constraints can be rewritten as In terms of a more convenient parametrization As we will now show these constraints imply that α is symmetric. To see this, assume that the antisymmetric part of α is non-zero and parametrize it by α [uv] = ǫ uvw a w . From the last two equations of (B.3) we then find which after contraction withāū and using the second equation of (B.3) shows that a u = 0 and α therefore is symmetric. With this simplification the remaining conditions of (B.3) imply α uw (α * ) wv = |β| 2 δ u v , (B.5) where we lowered the indices with δ uū . Since (B.5) says that the symmetric matrix α is also normal, we can diagonalize it as a complex matrix, and the diagonal entries are β times some phase factors. Rotating the holomorphic coordinates by a phase corresponds to SO(2) 3 ⊂ SO(6) rotations, and together with electromagnetic SO(2) ⊂ SL(2) rotations that only affect the overall phase of f, we can bring α and β into the final form where we also used the second equation in (3.13). If we choose the complex structure as only need to consider vector-like representations where s is an integer. The representation m can then be understood as a totally symmetric and traceless tensor of degree s. They can be generated by tensor products of the form which are the symmetric traceless, anti-symmetric and trace components of this tensor product, respectively. A vector vã in the m-representation can be written as v n 1 ...ns = lã n 1 ...ns vã, n 1 , . . . , n s = 1, 2, 3 where lã n 1 ...ns is a constant symmetric, traceless tensor, i.e. it obeys lã n 1 ...ns = lã (n 1 ...ns) and lã pqn 1 ...n s−2 δ pq = 0. The SO (3) .15) we also find f mãc f * mcb = 1 6 s(s + 1) |µ| 2 δãb . (C.7) In this notation the representations (C.5) are then given by (w m vã) s+1 = lã n 1 ...ns lb (n 1 ...ns w m) vb − 1 3 s lã mn 1 ...n s−1 lb n 1 ...n s−1 p w p vb , (w m vã) s = 1 s(s+1) f mn 1 p lã n 2 ...nsp f * qr(n 1 lb n 2 ...ns)r w q vb = 1 s(s+1) f mãc f * ncb w n vb , (w m vã) s−1 = 1 3 s lã mn 1 ...n s−1 lb n 1 ...n s−1 p w p vb . (C.8) After this interlude let us return to the solution of (C.4). If the representation is trivial, i.e. we have m i = 1 and f mãb = 0, we find that (C.4) implies δφ mã = 0, or in other words we do not find any moduli in this subspace. Let us therefore assume in the following that the representation m i is non-trivial. From their index structure we see that the scalars δφ mã are in the tensor product (C.5). We thus evaluate the condition (C.3) for each representation in this tensor product individually. If one uses (C.8) and (C.2), one can easily show that the anti-symmetric m i -representation obeys (C.3). If δφ am is in the m i + 2 ⊕ m i − 2 representation we contract (C.4) with f mnq and find from (3.15) and (C.2) that Re (f mãc f * pcb − f pãc f * mcb ) δφ pb = 1 6 |µ| 2 δφ mã . (C.9) From (C.8) we see that a δφ mã in the m i + 2 ⊕ m i − 2 representation can be written as δφ mã = lã n 1 ...ns δφ mn 1 ...ns , (C.10) whereφ is totally symmetric. Using (C.6) one can then show that f pcb δφ pb = 0 and that f pãc f mcb φ pb = − 1 6 s|µ| 2 δφ mã . Thus, we can conclude that a δφ ma in the m i + 2 and m i − 2 representations cannot fulfill (C.4). Or in other words, the condition (C.4) projects onto the m i -representation and we can parametrize δφ mã = f mãb λb with λb being any element in that representation.
To summarize, we just showed that the scalar deformations δφ mã that preserve N = 4 supersymmetry must be of the form corresponding to rk(M) = i m 1 i + i m 2 i massless degrees of freedom, whereM am αb ∼ f αmab .