Supersymmetric AdS_7 backgrounds in half-maximal supergravity and marginal operators of (1,0) SCFTs

We determine the supersymmetric AdS_7 backgrounds of seven-dimensional half-maximal gauged supergravities and show that they do not admit any deformations that preserve all 16 supercharges. We compare this result to the conformal manifold of the holographically dual (1,0) superconformal field theories and show that accordingly its representation theory implies that no supersymmetric marginal operators exist.


Introduction
AdS backgrounds of gauged supergravities have been prominently studied in connection with the AdS/CFT correspondence [1]. In particular a large variety of explicit solutions of tenand eleven-dimensional supergravities of the form AdS d × Y 10/11−d have been constructed by now. Generalization of the original AdS 5 × S 5 started in refs. [2,3] and the more recent developments are summarized, for example, in [4]. Refs. [5,6] on the other hand studied AdS 4 backgrounds within four-dimensional (d = 4) supergravities without considering any explicit relation with solutions of higher-dimensional supergravities. It was found that the existence of AdS backgrounds imposes specific conditions on the couplings of the supergravity. For N = 1 these conditions are formulated in terms of the Kähler potential and the superpotential. For N > 1 AdS backgrounds can only appear in gauged supergravities and the necessary gaugings are conveniently expressed in terms of the embedding tensor [7,8]. Concretely refs. [5,6] studied N = 2 and N = 4 AdS backgrounds together with their deformations that preserve all supercharges and determined the structure and properties of this moduli space. For N = 4 it is even possible to classify the AdS 4 backgrounds in that the structure of possible gauge groups can be given. In particular a specific subgroup of the R-symmetry group always has to be gauged and has to be unbroken in the AdS background. Furthermore it was shown that no deformations preserving all 16 supercharges exist and one can only have isolated vacua. In [9] the analysis was carried over to AdS 5 backgrounds of five-dimensional gauged supergravities with 16 supercharges where a similar classification is possible but in this case a moduli space does exist.
In this paper we extend these studies to seven-dimensional supergravities with sixteen supercharges (half-maximal) coupled to an arbitrary number of vector multiplets and determine their AdS 7 backgrounds. Unfortunately, the most general Lagrangian of these supergravities formulated in the embedding tensor formalism is not yet know. The original papers [10,11] take some of the embedding tensor components into account but not all. This has been partly remedied in [12][13][14] where all embedding tensor components have been identified. However, so far only certain terms of the full Lagrangian for these additional embedding tensor components have been given. Luckily we can show that in supersymmetric AdS 7 backgrounds only specific embedding tensor components can be non-trivial and for these the Lagrangian is known.
While we work solely within the framework of seven-dimensional gauged supergravity, supersymmetric AdS 7 solutions can be also discussed from the perspective of higherdimensional supergravity. The only half-maximally supersymmetric solutions in M-theory are of the form AdS 7 × S 4 /Z k [15,16]. There are no supersymmetric AdS 7 solutions in type IIB supergravity, but solutions of massive type IIA supergravity have been explicitly constructed and classified in [17][18][19]. All these solutions can be truncated consistently to minimal gauged supergravity in seven dimensions [20] and should hence describe a possible higher-dimensional origin for the solutions discussed in this paper. 1 In our analysis we find that supersymmetric AdS 7 backgrounds require the gauge group G to be of the general form where n is the number of vector multiplets, H is a compact semi-simple factor while G 0 needs to contain an SO(3) subgroup which has to coincide with the unbroken gauge group in the vacuum. The unbroken SO(3) is the R-symmetry of the supergravity or an admixture of the R-symmetry with an appropriate SO(3) factor associated with the vector multiplets. 2 If we assume the gauge group to be semi-simple, we can further restrict G 0 to be either SO(3), SO(3, 1) or SL(3, R). A related result has been obtained in [22,23] from a different approach.
Furthermore, we study the scalar deformations of the AdS backgrounds which preserve all supercharges and show that they all are Goldstone bosons of the spontaneously broken 1 In [21] it was however noted that certain solutions of ten-dimensional type IIA string theory (including localized and smeared branes) do not seem to have a description within seven-dimensional gauged supergravity.
2 Contrary to AdS 4 and AdS 5 backgrounds with 16 supercharges, in d = 7 the entire R-symmetry group SO(3) has to be gauged and unbroken. gauge group G and therefore do not count as physical moduli. Consequently there is no supersymmetric moduli space exactly as for d = 4, N = 4 [6].
In the second part of the paper we consider the holographically dual six-dimensional N = (1, 0) superconformal field theories (SCFT) and their possible exactly marginal deformations. This deformation space is known as the conformal manifold and according to the AdS/CFT correspondence it should coincide with the moduli space of the AdS solutions. In agreement with our previous results we can indeed show that there is no N = (1, 0) SCFT that can have any supersymmetric marginal deformations and thus no conformal manifold exists. This follows solely from the representation theory of the superconformal algebra in that any possible marginal operator violates the unitarity bounds and therefore is forbidden. Here we essentially follow a similar analysis for N = 1 SCFTs in d = 4 performed in [24] and use the N = (1, 0) representation theory determined in [25][26][27]. When this manuscript was being completed we learned about ref. [28] which has considerable overlap with the second part of this paper. This paper is organized as follows. In section 2.1 we briefly review the half-maximal supergravities in D = 7. In 2.2 we show that supersymmetric AdS 7 backgrounds imply the gauge group given in (1.1). In 2.3 we show that the resulting scalar potential has flat directions but all of them correspond to Goldstone bosons of a spontaneously broken gauge group in the AdS 7 vacuum. In section 3 we turn to the dual superconformal theories. After discussing some general properties in 3.1 we show in section 3.2 that there are no marginal operators in six-dimensional N = (1, 0) SCFTs. In Appendix A we review the six-dimensional N = (1, 0) superconformal algebra, and in Appendix B we discuss the group theoretical restrictions on the level of a Lorentz invariant descendant operator.
2 AdS 7 backgrounds of seven-dimensional half-maximal supergravity
The field space M of the scalars is given by where the 3n dimensional coset manifold is spanned by the scalars φ ri in the vector multiplet while the R + factor corresponds to σ. The coset can be conveniently parametrized by a coset representative 3 L = L i I , L r I , I = 1, . . . , n + 3 . L is an SO(3, n) matrix and hence satisfies where η IJ = diag(−1, −1, −1, +1, . . . , +1) is the canonical SO(3, n) metric. The scalar manifold M can be described by the metric The (n + 3) vector fields are combined into A I = (A i , A r ) and can be rotated into each other by the global symmetry group SO(3, n). A subgroup G ⊂ SO(3, n) can be made local provided that the structure constants f IJ K of G are completely antisymmetric, i.e. they satisfy the linear constraint Clearly the dimension of G is restricted by the number of vectors fields to be not larger than n + 3. As explained in [29] the condition (2.7) restricts the choice of possible (non-compact) gauge groups G strongly. Since η IJ has signature (3, n) any semi-simple G can be either generated by at most three compact or three non-compact generators and the allowed semisimple gauge groups are cataloged in [12]. In the next section we will determine which of the gauge groups can give rise to AdS vacua.
To construct a gauge invariant action it is convenient to introduce the gauged Maurer-Cartan one-forms where L Ir denotes the inverse coset representative. The gauge covariant field strengths are defined by Furthermore, for the existence of AdS vacua it turns out to be necessary to dualize the two-form B 2 into a three-form G 3 and to add to the action the topological mass term [12] where h is a real constant and H 4 = dG 3 is the four-form field strength.
With these ingredients the total bosonic Lagrangian of gauged N = 2 supergravity reads [11,12] where the Chern-Simons three-form ω 3 is given by The potential V takes the form where we abbreviated (2.14) Finally, to find the background solutions that preserve supersymmetry we need the supersymmetry variations of all fermionic fields. They are given by where we suppressed the R-symmetry index A and the ellipses denote terms which vanish in a maximally symmetric space-time background.
So far we used the supergravity as determined in [11,12]. However, ref. [13] pointed out that this is not the most general formulation of gauged N = 2 supergravity because apart from the totally antisymmetric structure constant f [IJK] there can also be another gauge parameter ξ I which transforms in the vector representation of SO (3, n). Denoting the generators of SO(3, n) by t [IJ] and the generator of the R + shift symmetry of σ by t 0 , the full embedding tensor is then given by [13] Θ 16) and the general covariant derivative reads With this information one can determine the Lagrangian of the supergravity. A partial answer has been obtained recently in [14] but the full Lagrangian has not been given yet.
Luckily, we will see in the next section that supersymmetric AdS solutions can only occur for ξ I = 0, so that in fact we do not need to use the most general formulation. In order to show this we will need the additional ξ I dependent terms in the supersymmetry variations given in (2.15). They are of the form [14] δχ where ξ i = L i I ξ I , ξ r = L r I ξ I . These variations in turn induce an additional term in the potential given by

Supersymmetric AdS backgrounds
In this section we derive conditions on the gauge group G such that the theory admits fully supersymmetric AdS vacua. Unbroken supersymmetry implies that the supersymmetry variations of the fermions (2.15) and (2.18) vanish in the AdS background and therefore we need to have As promised we find ξ I = 0 which follows from the fact that (1, σ i ) forms a basis of twodimensional Hermitian matrices and thus the terms given in (2.18) cannot cancel against terms in (2.15). Using the "dressed" structure constants defined in (2.14) the first two conditions in (2.20) read where the coupling constant g can be chosen arbitrarily and dictates together with h the value of the cosmological constant. Inserted into (2.13), the cosmological constant is and we indeed see that the background is AdS if and only if a topological mass term with coupling h is included into the action [23,30]. We also see from (2.20) and (2.21) that the scalar σ from the gravity multiplet has to take the background value . (2.23) The conditions (2.21) on the structure constants are very similar to those derived in [6] so that we can essentially follow their analysis for determining the gauge group. The simplest situation occurs when in addition to (2.21) there are no mixed index components of the structure constants, i.e. f ist = 0. In this case the gauge group is where the SO (3) factor is related to the unbroken R-symmetry and H ⊂ SO(n) has dimension dim H ≤ n and is specified by f rst . 4 Since G is compact it automatically satisfies the condition (2.7) and therefore is an allowed gauge group.
The generic case f ist = 0 is most conveniently analyzed if we go to a specific basis for the vector multiplet index r where we can split r intor andr such that the only non-vanishing components of the structure constants involving anr index are frst. These components thus correspond to a group H ⊂ SO(q), q ≤ n. The remaining components are f ijk , f irŝ and frŝt and they describe a non-compact group G 0 ⊂ SO(3, m), with m + q = n and SO(3) ⊂ G 0 . The total gauge group then is (2.25) If we furthermore assume that the gauge group is semi-simple, we can list all possible options for G 0 explicitely. From (2.7) we know that G 0 can have either at most three compact or at most three non-compact generators and the only non-compact semi-simple groups satisfying this condition and containing SO(3) as a subgroup are SO(3, 1) and SL(3, R). Therefore G has to be of the form where H is an arbitrary semi-simple compact group. This is in agreement with the results from [23], where however the compact factor H was not taken into account for the analysis of AdS vacua.

Moduli spaces of AdS backgrounds
Let us now compute the moduli space of the AdS backgrounds determined in the previous section. The moduli are the directions in the scalar manifold M given in (2.3) which are undetermined by the conditions (2.20). Or in other words we are looking for continuous solutions of the variations The resulting scalar fields are automatically flat directions of the potential (2.13) and thus can be viewed as the scalar degrees of freedom that remain massless in an AdS background.
We proceed along the lines of [6] and parametrize the variations of the coset representatives as where δφ ir are the fluctuations of the 3n scalar fields around their background value. Using (2.5) this implies δL r I = L i I δφ ir , (2.29) while the variations of the inverse coset representatives are given by (2.30) To simplify the notation we will from now on suppress the brackets and assume that all field dependent quantities are evaluated in the background whenever this is appropriate. Since ξ I = 0 it follows directly that δξ i = δξ r = 0 are satisfied without imposing any conditions on the variations of the scalar fields. From (2.14) and (2.21) we learn where λ s are arbitrary real parameters. Hence the number of independent moduli is given by the rank of the (3n × n) matrix f ir s . 5 Adopting the notation of the previous section we should denote them by λŝ and (2.33) becomes δφ ir = f irŝ λŝ, δφ ir = 0. 5 The notation should be understood in such a way that the pair of indices ir labels the rows of the matrix f ir s while s labels its columns.
The structure constants f irŝ precisely correspond to the non-compact generators of G 0 . Since the maximally compact subgroup of G 0 is in every case given by SO (3), we see that the scalar deformations span the coset manifold . (2.34) Let us denote byG the maximal subgroup of SO(3, n) that leaves the gauge group G and hence the structure constants invariant. Therefore, acting withG on a solution of (2.21) gives a rotated solution. It is therefore not unexpected that M δφ is of the form of an orbit ofG acting on the scalar manifold M given in (2.3).
We will now argue that all scalars given in (2.33) are in fact Goldstone bosons eaten by massive vector fields and thus no physical moduli. For this purpose we evaluate the gauged Maurer-Cartan form (2.8) in the AdS background to find where A s = L s I A I . This expression appears quadratically in the Lagrangian (2.11) and thus gives a mass term for every vector field A s in the preimage of the matrix f ir s . Adopting again our previous notation, (2.35) reads P ir = L I i dLr I + f irŝ Aŝ, P ir = L I i dLr I and we see that there is precisely one massive vector field Aŝ for every scalar λŝ. So no physical massless direction is left and the moduli space can only consist of isolated points.
We can also understand this result directly without analyzing the condition (2.32). The vectors that obtain a mass in the AdS vacuum are in one-to-one correspondence with the non-compact generators of the gauge group G. Therefore the mass term (2.35) breaks the gauge group spontaneously to its maximally compact subgroup, i.e. (2.36) Breaking G 0 to SO(3) in (2.34) indeed reduces M δφ to a single point.

The conformal manifold of the dual SCFT
In this section we study six-dimensional N = (1, 0) superconformal field theories (SCFTs) which can serve as holographic duals of the AdS backgrounds studied in the previous section. In particular we focus on possible marginal deformations of such SCFTs which preserve all supercharges. We will however show that the representation theory of the N = (1, 0) superconformal algebra forbids any such operators and thus no exactly marginal supersymmetric deformations exist. This is equivalent to the statement that there is no conformal manifold C. The AdS/CFT dictionary relates C to the moduli space of the dual AdS backgrounds which we studied in the previous section. As on both sides we only find vanishing deformation spaces our results show perfect agreement.

Preliminaries
Given a SCFT we can deform it by adding conformal operators O i to the theory L denotes the Lagrangian but this notation is somewhat symbolic as we also consider SCFTs which do not have a Lagrangian description. Operators O i that do not break (super-) conformal invariance are called exactly marginal operators. The space spanned by the corresponding exactly marginal couplings λ i is called the conformal manifold C.
A necessary condition for unbroken conformal invariance is that the λ i are dimensionless or equivalently that the operators O i have conformal dimension ∆ = 6, i.e. are marginal operators. This criterion is however not sufficient since higher-order corrections in λ i can perturb ∆. In the following analysis we only consider marginal operators which do not break the N = (1, 0) supersymmetry. Thus the O i of interest have to be the highest component of a supermultiplet or in other words have to be annihilated by all supercharges. In addition they should be singlets of the R-symmetry group. The superconformal group of six-dimensional N = (1, 0) SCFTs is the group OSp(6, 2|2) and its representations have been described in detail in [25][26][27]. Let us briefly recall some of their results which we need for the following discussion.
The bosonic subalgebra of OSp(6, 2|2) is SO(6, 2) × SU(2) R , where SO(6, 2) is the six-dimensional conformal algebra and SU(2) R is the R-symmetry. The fermionic part of OSp(6, 2|2) is generated by the supercharges (Q i α , S α i ) where Q i α is an R-doublet of chiral spinors with conformal dimension ∆ = + 1 2 , while S α i is an R-doublet of antichiral spinors with ∆ = − 1 2 . Here α = 1, . . . , 4, denotes the fundamental representation of SU(4) = Spin(6) and i = 1, 2 labels the fundamental representation of the SU(2) R . The representation theory of the superconformal algebra is most conveniently analyzed for the Euclidean theory, where one has the Hermiticity relation Q † = S, so that Q i α and S α i can be interpreted as ladder operators, raising and lowering the conformal dimension ∆ by 1 2 . As a consequence the unitary irreducible representations of OSp(6, 2|2) decompose into direct sums of representations of the maximally compact subgroup SO(2) × SO(6) × SU(2) R of the bosonic subgroup. Each representation can be built from a lowest weight state (conventionally called superconformal primary), which is characterized by the requirement that it is annihilated by all superconformal charges S α i . Each primary is labeled by its conformal dimension ∆ 0 , three half-integer SO(6) weights h i = (h 1 , h 2 , h 3 ) and a half-integer SU(2) weight k. 6 The corresponding supermultiplet is then obtained by successively acting with the supercharges Q i α on a superconformal primary. A state obtained by the action of l supercharges is called a level-l descendant and it has conformal dimension ∆ = ∆ 0 + l 2 . Notice that ∆ 0 , h i and k can be used to label the entire supermultiplet. It is however not possible to pick arbitrary combinations of values since unitarity imposes certain constraints. Using the superconformal algebra (see Appendix A) one can compute the norm of the descendant states. Requiring then that all states in a given representation have non-negative norm implies bounds on the conformal dimension ∆ 0 of the primary operators which have the generic form The function f is explicitly determined in [25,26] and we recall the results relevant for our analysis in the following section. Representations that saturate the bound (3.2) are short, as in this case some states have vanishing norm and are no longer part of the irreducible representation.

Classification of marginal operators
After these preliminaries let us go in detail through all possible candidates for supersymmetric marginal operators. As we discussed, they must be part of a unitary representation of the superconformal algebra and therefore are either primary operators or descendant operators that are obtained by acting with l supercharges Q i α on a primary operator. However, the primary operators that are invariant under Lorentz-symmetry, R-symmetry and supersymmetry have been shown to be proportional to the identity operator [24]. Therefore we can restrict our further analysis to descendant operators. Among the descendant operators we should also discard those operators where two of the supercharges can be traded for a momentum operator by means of the supersymmetry algebra. These operators add in (3.1) only a total derivative to the Lagrangian and hence do not deform the theory. For the same reason the order of supercharges in a descendant operator does not matter for our analysis.
If we start with a primary operator with SO(6) weights (h 1 , h 2 , h 3 ) we can only find Lorentz invariant descendant operators at level with n being an arbitrary non-negative integer. In Appendix B we give a proof of this statement. Thus the conformal dimension of the primary operator needs to be Moreover, we will use in the following that k = 0 is only possible if l is even as descendants with an odd number of supercharges cannot be R-singlets. The general bound from [25,26] for a unitary representation reads A primary operator with these properties carries no R-symmetry indices and has to be an antisymmetric SU(4)-tensor (which is isomorphic to the six-dimensional vector representation of SO (6)). Thus the corresponding candidate descendant operator has to take the form Denoting the primary operator for the first solution (3.10) by U i α , it is indeed possible to identify a Lorentz and R-symmetry invariant descendant operator O 3 at level l = 3 Computing the norm yields O 3 ∼ ∆ 0 − 9 2 ∆ 0 + 7 2 and hence vanishes at the critical value ∆ 0 = 6 − l 2 = 9 2 . Consequently O 3 itself vanishes and cannot be considered as a possible marginal operator. Notice that it is in principle possible to contract the R-symmetry indices in a different fashion but all such operators differ from O 3 only by a total derivative. Moreover, we have checked that all these other combinations also have vanishing norm.
For the second solution (3.11) the primary operator has the index structure U (αβ) (with h = 1 and k = 0) and we can build a Lorentz and R-symmetry invariant descendant operator O 6 at level l = 6, There are also other possibilities to contract the indices within O 6 , which would however lead to total derivatives. In any case all these l = 6 operators are descendants of the operator [Q i[α , U (β]γ) ], whose norm is (∆ 0 − 3) and hence vanishes. c) Finally for h 1 = h 2 = h 3 = 0 there are short representations for (3.14) Since we have eight distinct supercharges, a descendant operator at level l > 8 is always zero by means of (A.2a), so according to (3.3) the only levels at which we should look for suitable candidate operators are l = 4, 8.
At level l = 4 we need ∆ 0 = 4 and there is one operator with k = 0, The only possibility for non-vanishing k is k = 1 as (3.14) implies for k > 1 that ∆ 0 > 4 while for k = 1 2 the level l cannot be even. The operator with k = 1 reads We can compute O ′ 4 ∼ (∆ 0 − 4)(∆ 0 + 6)(∆ 0 + 8), and thus this operator is ruled out as well. Clearly it is again also a total derivative. At level l = 8 we need ∆ 0 = 2. Using the same argument as above there is no operator with k = 0. Hence a Lorentz invariant level l = 8 operator is (up to total derivatives) always a descendant of the l = 2 operator If we antisymmetrize also in the R-symmetry indices i and j, we find O [ij] αβ ∼ ∆ 0 , but this operator is symmetric under the exchange of the two supercharges and we end up with a total derivative. On the other hand we find for the symmetric component that O To conclude we have thus shown that all candidates for marginal operators either have zero norm or are not supersymmetric. Notice that most of the operators are also total derivatives but we did not have to use this fact in our argument. Let us close with the observation that the above analysis can be easily extended to relevant operators with conformal dimension ∆ < 6. In this case the dimension of the primary operator needs to satisfy which is clearly also not compatible with the general bound (3.5). Moreover for generic ∆ < 6 all isolated short representations are ruled out as well. Only for ∆ = 4 the operators from c) with k = 0 remain possible candidate operators, but we have shown that their norms are negative at the appropriate dimensions.
Schweigert, Alessandro Tomasiello and especially Hagen Triendl and Marco Zagermann for pointing out an error in a previous version of this paper.
The problem is conveniently analyzed in the language of SU(4) Young tableau, since here N corresponds to the number of boxes that need to be added to the diagram to fill up every of its columns to the maximal length four. More generally if we switch to an arbitrary SU(n) Young tableau and call the length of its i th row r i and the length of its i th column l i , N is given by where the sum runs over all columns. If we use the fact that the lengths of the columns and rows are related via l i = p for r p+1 < i ≤ r p , p = 1, . . . , n − 1 , (B.2) and that the Dynkin labels a i can by read off from the tableau by where r n ≡ 0, we find Going back to the relevant case n = 4 and using that a 1 = h 2 −h 3 , a 2 = h 1 +h 2 , a 3 = h 2 +h 3 , the result reduces to N = 2 (h 1 + h 2 + h 3 ) . (B.5)