Moduli spaces of AdS_5 vacua in N=2 supergravity

We determine the conditions for maximally supersymmetric AdS_5 vacua of five-dimensional gauged N=2 supergravity coupled to vector-, tensor- and hypermultiplets charged under an arbitrary gauge group. In particular, we show that the unbroken gauge group of the AdS_5 vacua has to contain an U(1)_R-factor. Moreover we prove that the scalar deformations which preserve all supercharges form a Kahler submanifold of the ambient quaternionic Kahler manifold spanned by the scalars in the hypermultiplets.


Introduction
Anti-de Sitter (AdS) backgrounds of supergravity are an essential part of the AdS/CFT correspondence [1] and have been studied in recent years from varying perspectives. On the one hand they can be constructed as compactifications of higher-dimensional supergravities as is the natural set up in the AdS/CFT correspondence. 1 Alternatively, one can investigate and, if possible, classify their appearance directly in a given supergravity without relating it to any compactification.
For a given AdS background it is also of interest to study its properties and in particular its moduli space M, i.e. the subspace of the scalar field space that is spanned by flat directions of the AdS background. This moduli space has been heavily investigated in Minkowskian backgrounds of string theory as it prominently appears in its low energy effective theory. For AdS backgrounds much less is known about M, partly because the defining equations are more involved and furthermore quantum corrections contribute unprotected.
In [5,6] supersymmetric AdS 4 vacua and their classical supersymmetric moduli spaces were studied in four-dimensional (d = 4) supergravities with N = 1, 2, 4 supersymmetry without considering their relation to higher-dimensional theories. 2 For N = 1 it was found that the supersymmetric moduli space is at best a real submanifold of the original Kähler field space. Similarly, for N = 2 the supersymmetric moduli space is at best a product of a real manifold times a Kähler manifold while N = 4 AdS backgrounds have no supersymmetric moduli space. This analysis was repeated for AdS 5 vacua in d = 5 gauged supergravity with 16 supercharges (N = 4) in [7] and for AdS 7 vacua in d = 7 gauged supergravity with 16 supercharges in [8]. For the d = 5, N = 4 theories it was shown that the supersymmetric moduli space is the coset M = SU(1, m)/(U(1) × SU(m)) while in d = 7 it was proven that again no supersymmetric moduli space exists.
In this paper we focus on supersymmetric AdS 5 vacua in d = 5 gauged supergravities with eight supercharges (N = 2) coupled to an arbitrary number of vector-, tensor-and hypermultiplets. A related analysis was carried out in [9] for the coupling of Abelian vector multiplets and hypermultiplets. We confirm the results of [9] and generalize the analysis by including tensor multiplets and non-Abelian vector multiplets. In particular, we show that also in this more general case the unbroken gauge group has to be of the form H × U(1) R where the U(1) R -factor is gauged by the graviphoton. This specifically forbids unbroken semisimple gauge groups in AdS backgrounds.
In a second step we study the supersymmetric moduli space M of the previously obtained AdS 5 backgrounds and show that it necessarily is a Kähler submanifold of the quaternionic scalar field space T H spanned by all scalars in the hypermultiplets. 3 This is indeed consistent with the AdS/CFT correspondence where the moduli space M is mapped to the conformal manifold of the dual superconformal field theory (SCFT). For the gauged supergravities considered here the dual theories are d = 4, N = 1 SCFTs. In [10] it was indeed shown that the conformal manifold of these SCFTs is a Kähler manifold.
The organization of this paper is as follows. In section 2 we briefly review gauged N = 2 supergravities in five dimensions. This will then be used to study the conditions for the existence of supersymmetric AdS 5 vacua and determine some of their properties in section 3. Finally, in section 4 we compute the conditions on the moduli space of these vacua and show that it is a Kähler manifold.

Gauged N = supergravity in five dimensions
To begin with let us review five-dimensional gauged N = 2 supergravity following [11][12][13]. 4 The theory consists of the gravity multiplet with field content where g µν is the metric of space-time, Ψ A µ is an SU(2) R -doublet of symplectic Majorana gravitini and A 0 µ is the graviphoton. In this paper we consider theories that additionally contain n V vector multiplets, n H hypermultiplets and n T tensor multiplets. A vector multiplet {A µ , λ A , φ} transforms in the adjoint representation of the gauge group G and contains a vector A µ , a doublet of gauginos λ A and a real scalar φ. In d = 5 a vector is Poincaré dual to an antisymmetric tensor field B µν which carry an arbitrary representation of G.
This gives rise to tensor multiplets which have the same field content as vector multiplets, but with a two-form instead of a vector. Since vector-and tensor multiplets mix in the Lagrangian, we label their scalars φ i by the same index i, j = 1, ..., n V + n T . Moreover, we label the vector fields (including the graviphoton) by I, J = 0, 1, ..., n V , the tensor fields by M, N = n V + 1, ..., n V + n T and also introduce a combined indexĨ = (I, M). Finally, the n H hypermultiplets contain 4n H real scalars q u and 2n H hyperini ζ α .
The bosonic Lagrangian of N = 2 gauged supergravity in five dimensions reads 5 [13] In the rest of this section we recall the various ingredients which enter this Lagrangian. First of all are the field strengths with g being the gauge coupling constant. The scalar fields in L can be interpreted as coordinate charts from spacetime M 5 to a target space T , Locally T is a product T V T × T H where the first factor is a projective special real manifold (T V T , g) of dimension n V + n T . It is constructed as a hypersurface in an (n V + n T + 1)dimensional real manifold H with local coordinates hĨ. This hypersurface is defined by P (hĨ(φ)) = CĨJKhĨhJ hK = 1, (2.5) where P (hĨ(φ)) is a cubic homogeneous polynomial with CĨJK constant and completely symmetric.
The generalized gauge couplings in (2.3) correspond to a positive metric on the ambient space H, given by where hĨ = CĨJKhJ hK . (2.7) The pullback metric g ij is the (positive) metric on the hypersurface T V T and is given by These quantities satisfy (see Appendix C in [13] for more details) where we raise and lower indices with the appropriate metrics aĨJ or g ij respectively. The metric g ij induces a covariant derivative which acts on the hĨ i via where T ijk := CĨJKhĨ i hJ j hK k is a completely symmetric tensor.
The second factor of T in (2.4) is a quaternionic Kähler manifold (T H , G, Q) of real dimension 4n H (see [14] for a more extensive introduction). Here G uv is a Riemannian metric and Q denotes a ∇ G invariant rank three subbundle Q ⊂ End(T T H ) that is locally spanned by a triplet J n , n = 1, 2, 3 of almost complex structures which satisfy J 1 J 2 = J 3 and (J n ) 2 = −Id. Moreover the metric G uv is hermitian with respect to all three J n and one defines the associated triplet of two-forms ω n uv := G uw (J n ) w v . In contrast to the Kählerian case, the almost complex structures are not parallel but the Levi-Civita connection ∇ G of G rotates the endomorphisms inside Q, i.e. (2.12) Note that ∇ differs from ∇ G by an SU(2)-connection with connection one-forms θ p . For later use let us note that the metric G uv can be expressed in terms of vielbeins U αA u as where C αβ denotes the flat metric on Sp(2n H , R) and the SU(2)-indices A, B are raised and lowered with ǫ AB .
The gauge group G is specified by the generators t I of its Lie algebra g and the structure (2.14) The vector fields transform in the adjoint representation of the gauge group, i.e. t K IJ = f K IJ while the tensor fields can carry an arbitrary representation. The most general representation for n V vector multiplets and n T tensor multiplets has been found in [12] and is given by We see that the block matrix t N IJ mixes vector-and tensor fields. However the t N IJ are only nonzero if the chosen representation of the gauge group is not completely reducible. This never occurs for compact gauge groups but there exist non-compact gauge groups containing an Abelian ideal that admit representations of this type, see [12]. There it is also shown that the construction of a generalized Chern-Simons term in the action for vector-and tensor multiplets requires the existence of an invertible and antisymmetric matrix Ω M N . In particular, the t N IJ are of the form The gauge group is realized on the scalar fields via the action of Killing vectors ξ I for the vector-and tensor multiplets and k I for the hypermultiplets that satisfy the Lie algebra g of G, (2.17) In the case of the projective special real manifold, one can obtain an explicit expression for the Killing vectors ξ i I given by [13] The second equality is due to the fact that [15] tK IJ hJ hK = 0 , and thus The Killing vectors k u I on the quaternionic Kähler manifold T H [12,14,16] have to be triholomorphic which implies Here µ n I is a triplet of moment maps which also satisfy and the equivariance condition Furthermore the covariant derivative of the Killing vectors obeys [16,17] where the L Iuv are related to the gaugino mass matrix and commute with J n . For later use we define where the S n Iuv are symmetric in u, v [16]. Before we proceed let us note that for n H = 0, i.e. when there are no hypermultiplets, constant Fayet-Iliopoulos (FI) terms can exist which have to satisfy the equivariance condition (2.24). In this case the first term on the right hand side of (2.24) vanishes which implies that there are only two possible solutions [13]. If the gauge group contains an SU(2)-factor, the FI-terms have to be of the form where the e n I are nonzero constant vectors for I = 1, 2, 3 of the SU(2)-factor that satisfy The second solution has U(1)-factors in the gauge group and the constant moment maps are given by where e n is a constant SU(2)-vector and I labels the U(1)-factors.
Finally, the covariant derivatives of the scalars in (2.3) are given by The scalar potential is defined in terms of the couplings 7 (2.32) Here σ n AB are the Pauli matrices with an index lowered by ǫ AB , i.e.
As usual the couplings (2.32) are related to the scalar parts of the supersymmetry variations of the fermions via Here ǫ A denote the supersymmetry parameters. This concludes our review of d = 5 supergravity and we now turn to its possible supersymmetric AdS backgrounds.

Supersymmetric AdS 5 vacua
In this section we determine the conditions that lead to AdS 5 vacua which preserve all eight supercharges. This requires the vanishing of all fermionic supersymmetry transformations, i.e.
where denotes the value of a quantity evaluated in the background. Using the fact that W iAB and K i are linearly independent [11] and (2.34), this implies the following four conditions, Here Λ ∈ R is related to the cosmological constant and U AB = v n σ n AB for v ∈ S 2 is an SU(2)-matrix. U AB appears in the Killing spinor equation for AdS 5 which reads [18] As required for an AdS vacuum, the conditions (3.2) give a negative background value for the scalar potential V (φ, q) < 0 which can be seen from (2.31). Using the definitions (2.32), we immediately see that the four conditions (3.2) can also be formulated as conditions on the moment maps and Killing vectors, Note that due to (2.5), (2.8) we need to have h I = 0 for some I and hĨ i = 0 for every i and someĨ. 8 In order to solve (3.4) we combine the first two conditions as Let us enlarge these equations to the tensor multiplet indices by introducing µ ñ I where we keep in mind that µ n N ≡ 0. Then we use the fact that the matrix (hĨ, hĨ i ) is invertible in special real geometry (see Appendix C of [13]), so we can multiply (3.5) with (hĨ, hĨ i ) −1 to obtain a solution for both equations given by Note that this condition is non-trivial since it implies that the moment maps point in the same direction in SU (2)-space for all I. Furthermore, using the SU(2) R -symmetry we can rotate the vector v n such that v n = vδ n3 and absorb the constant v ∈ R into Λ. Thus only µ I := µ 3 I = 0, ∀I in the above equation. Since by definition µ n N = 0, this implies In particular, this means that the first two equations in (2.10) hold in the vacuum for only the vector indices, i.e.
Moreover due to the explicit form of the moment maps in (3.7), the equivariance condition (2.24) reads in the background Since (2.31) has to hold in the vacuum, h I = 0 for some I and thus the background necessarily has non-vanishing moment maps due to (3.7). This in turn implies that part of the R-symmetry is gauged, as can be seen from the covariant derivatives of the fermions which always contain a term of the form A I µ µ 3 I [13]. More precisely, this combination gauges the U(1) R ⊂ SU(2) R generated by σ 3 . From (3.7) we infer A I µ µ 3 I = ΛA I µ h I which can be identified with the graviphoton [15].
We now turn to the last two equations in (3.4). Let us first prove that the third equation h I k u I = 0 implies the fourth h I ξ i I = 0. This can be shown by expressing ξ i I in terms of k u I via the equivariance condition (3.9). Note that we learn from (2.18) that the background values of the Killing vectors on the manifold T V T are given by where we used (2.15) and (3.7). Inserting (3.7), (3.9) into (3.10) one indeed computes But then h I ξ i I = 0 is always satisfied if h I k u I = 0. Moreover this shows that ξ i I = 0 is only possible for k u I = 0. Note that the reverse is not true in general as can be seen from (3.10). We are thus left with analyzing the third condition in (3.4).
Let us first note that for n H = 0 there are no Killing vectors (k u I ≡ 0) and the third equation in (3.4) is automatically satisfied. However (3.7) can nevertheless hold if the constant FI-terms discussed below (2.26) are of the form given in (2.29) and thus only gauge groups with Abelian factors are allowed in this case. Now we turn to n H = 0. Note that then h I k u I = 0 has two possible solutions: i) k u I = 0 , for all I ii) k u I = 0 , for some I with h I appropriately tuned. (3.12) By examining the covariant derivatives (2.30) of the scalars we see that in the first case there is no gauge symmetry breaking by the hypermultiplets while in the second case G is spontaneously broken. Note that not all possible gauge groups can remain unbroken in the vacuum. In fact, for case i) the equivariance condition (3.9) implies This can only be satisfied if the adjoint representation of g has a non-trivial zero eigenvector, i.e. if the center of G is non-trivial (and continuous). 9 In particular, this holds for all gauge groups with an Abelian factor but all semisimple gauge groups have to be broken in the vacuum.
In the rest of this section we discuss the spontaneous symmetry breaking for case ii) and the details of the Higgs mechanism. Let us first consider the case where only a set of Abelian factors in G is spontaneously broken, i.e. k u I = 0 for I labeling these Abelian factors. From (3.10) we then learn ξ i I = 0 and thus we only have spontaneous symmetry breaking in the hypermultiplet sector and the Goldstone bosons necessarily are recruited out of these hypermultiplets. Hence the vector multiplet corresponding to a broken Abelian factor in G becomes massive by "eating" an entire hypermultiplet. It forms a "long" vector multiplet containing the massive vector, four gauginos and four scalars obeying the AdS mass relations. Now consider spontaneously broken non-Abelian factors of G, i.e. k u I = 0 for I labeling these non-Abelian factors. In this case we learn from (3.11) that either ξ i I = 0 as before or ξ i I = 0. However the Higgs mechanism is essentially unchanged compared to the Abelian case in that entire hypermultiplets are eaten and all massive vectors reside in long multiplets. 10 However there always has to exists at least one unbroken generator of G which commutes with all other unbroken generators, i.e. the unbroken gauge group in the vacuum is always of the form H × U(1) R . To see this, consider the mass matrix M IJ of the gauge bosons A I µ . Due to (2.30) and (3.11), this is given by (3.14) Here K uv is an invertible matrix which can be given in terms of G uv and S uv defined in (2.26) as Since h I k u I = 0 the mass matrix M IJ has a zero eigenvector given by h I , i.e. the graviphoton h I A µ I always remains massless in the vacuum. In the background the commutator of the corresponding Killing vector h I k u I with any other isometry k J is given by This vanishes for k u J = 0 and thus the R-symmetry commutes with every other symmetry generator of the vacuum, i.e. the unbroken gauge group is H × U(1) R . In particular, every gauge group G which is not of this form has to be broken G → H × U(1) R .
Let us close this section with the observation that the number of broken generators is determined by the number of linearly independent k u I . This coincides with the number of Goldstone bosons n G . In fact the k u I form a basis in the space of Goldstone bosons G and we have G = span R { k u I } with dim(G) = rk k u I = n G . In conclusion, we have shown that the conditions for maximally supersymmetric AdS 5 vacua are given by Note that the tensor multiplets enter in the final result only implicitly since the h I and its derivatives are functions of all scalars φ i . The first equation implies that a U(1) R -symmetry is always gauged by the graviphoton while the last equation shows that the unbroken gauge group in the vacuum is of the form H×U(1) R . This reproduces the result of [9] that the U(1) R has to be unbroken and gauged in a maximally supersymmetric AdS 5 background. In the dual four-dimensional SCFT this U(1) R is defined by a-maximization. Moreover we discussed that if the gauge group is spontaneously broken the massive vector multiplets are long multiplets. Finally, we showed that space of Goldstone bosons is given by G = span R { k u I } which will be used in the next section to compute the moduli space M of these vacua.

Structure of the moduli space
We now turn to the computation of the moduli space M of the maximally supersymmetric AdS 5 vacua determined in the previous section. Let us denote by D the space of all possible deformations of the scalar fields φ → φ + δφ, q → q + δq that leave the conditions (3.4) invariant. However, if the gauge group is spontaneously broken the corresponding Goldstone bosons are among these deformations but they should not be counted as moduli. Thus the moduli space is defined as the space of deformations D modulo the space of Goldstone bosons G, i.e. M = D/G. In order to determine M we vary (3.4) to linear order and characterize the space D spanned by δφ and δq that are not fixed. 11 We then show that the Goldstone bosons also satisfy the equations defining D and determine the quotient D/G.
Let us start by varying the second condition of (3.4). This yields where we used (3.4) and (2.23). Since this variation vanishes automatically, no conditions are imposed on the scalar field variation.
The variation of the first condition in (3.4) gives where in the second step we used (2.11), (2.23) while in the third we used (3.4). For n = 1, 2 (4.2) imposes while for n = 3 the deformations δφ i can be expressed in terms of δq u as Thus all deformations δφ i are fixed either to vanish or to be related to δq u . In other words, the entire space of deformations can be spanned by scalars in the hypermultiplets only, i.e. D ⊂ T H . Note that this is in agreement with (3.11) and also G ⊂ T H .
Finally, we vary the third condition in (3.4) to obtain Inserting (4.4) and using (2.10), (3.4) we find Thus we are left with the two conditions (4.3) and (4.6) whose solutions determine D. For a generic supergravity we will not solve them here in general. However the conditions alone suffice to prove that the moduli space is a Kähler submanifold of T H as we will now show.
As a first step we prove that the Goldstone bosons satisfy (4.3) and (4.6). We know from section 3 that the Goldstone directions are of the form δq u = c I k u I where c I are constants. Inserted into (4.3) we find where we used (3.9) and the fact that µ 1,2 K = 0. To show that the Goldstone bosons also satisfy (4.6) we first observe that where in the first step we used (3.4), added a term which vanishes in the background and then in the second step used (2.17). In addition we need to show Indeed, using (2.10) and h I k u I = 0 we find which proves (4.9) as promised.
Turning back to (4.6), we insert δq u = c I k u I and use (3.9) and (4.8) to arrive at Using again that µ I = Λ h I and applying (4.9), this yields  (4.3) this follows from the fact that J 3 interchanges the two equations. This can be seen by substituting δq ′u = (J 3 ) u v δq v and using that J 1 J 2 = J 3 on a quaternionic Kähler manifold. Turning to (4.6), we note that since only µ 3 I = 0 the covariant derivative (2.22) of the Killing vectors k u I commutes with J 3 in the vacuum, i.e.
This implies that the second term in (4.6) is invariant under J 3 and we need to show that this also holds for the first term. In fact, we will show in the following that this term vanishes on the moduli space and is only nonzero for Goldstone directions.
Let us first note that in general rk k u I ω 3 vw k wI ≤ rk k u I = n G . However, k u I ω 3 vw k wI k v J = 0 (as we saw in (4.12)) implies that the rank of the two matrices has to coincide. This in turn says that the first term in (4.6) can only be nonzero in the Goldstone directions and thus has to vanish for the directions spanning M. Thus the whole equation  In particular, this condition is satisfied if the associated fundamental two-form ω uw = G uw I w v on M vanishes. Now let us show that the moduli space M actually is totally complex and hence Kähler. To do so, we use (2.25) and (2.26) to note that in the vacuum (3.7) ω 3 uv is given by We just argued that k u I ω 3 vw k wI vanishes on M and thus (4.6) projected onto M also implies h I ∇ u k vI | M = 0 . Since ω 1 uv = − ω 3 uw (J 2 ) w v , we can multiply (4.16) with −(J 2 ) w v from the right and obtain where in the first step we used (2.26). This expression vanishes due to (4.17) and the fact that S 2 uv is symmetric while ω 1 uv is antisymmetric. Thus M is totally complex and in particular (M,G,J) is a Kähler submanifold.
As proved in [19] a Kähler submanifold can have at most half the dimension of the ambient quaternionic Kähler manifold, i.e. dim(M) ≤ 2n H . 12 Note that in the case of an unbroken gauge group we have G = {∅} and thus D = M. This is the case of maximal dimension of the moduli space. If the gauge group is now spontaneously broken then additional scalars are fixed by (4.3). Since M is J 3 -invariant, every δq u ∈ M can be written as δq u = (J 3 ) u v δq ′v for some δq ′u ∈ M. Combined with the fact that J 1 J 2 = J 3 this implies that the two conditions in (4.3) are equivalent on M. Furthermore we have rk h I i ω 1 uv k v I = rk k I u = n G and thus n G scalars are fixed by (4.3). In conclusion, we altogether have so the moduli space has at most real dimension 2n H − 2n G .