Supersymmetric solutions from matter-coupled 7D N=2 gauged supergravity

We study supersymmetric solutions within seven-dimensional $N=2$ gauged supergravity coupled to three vector multiplets in seven dimensions. The gauged supergravity contains six vector fields that gauge the $SO(4)\sim SO(3)\times SO(3)$ symmetry and admits two $N=2$ supersymmetric $AdS_7$ vacua with $SO(4)$ and $SO(3)_{\textrm{diag}}\subset SO(3)\times SO(3)$ symmetries. We consider solutions interpolating between two asymptotically locally $AdS_7$ geometries in the presence of a three-form field. For a particular value of the two $SO(3)$ gauge coupling constants, the $SO(3)_{\textrm{diag}}$ supersymmetric $AdS_7$ vacuum does not exist, but the solutions can be uplifted to eleven dimensions by a known reduction ansatz. We also study solutions of this type and their embedding in M-theory. We further extend these solutions to include the $SO(3)_{\textrm{diag}}$ gauge fields and argue that, in general, this generalization does not lead to supersymmetric solutions.


Introduction
Over the past twenty years, the AdS/CFT correspondence has been widely tested and confirmed by a large number of interesting results. It has been applied to holographic studies of strongly coupled field theories in various space-time dimensions. One of the interesting cases is AdS 7 /CFT 6 correspondence which has been argued to describe the dynamics of M5-branes in M-theory since the first proposal of the correspondence in [1].
As in other cases, AdS 7 /CFT 6 correspondence can be efficiently investigated by using gauged supergravities in seven dimensions. For example, AdS 7 ×S 4 geometry of M-theory, dual to N = (2, 0) superconformal field theory (SCFT) in six dimensions, can be described by SO(5) N = 4 gauged supergravity [2,3]. In this paper, we are interested in the case of half-maximal N = 2 gauged supergravity. The corresponding AdS 7 vacua are dual to N = (1, 0) SCFTs, see for example [4,5]. Some of these SCFTs can be obtained from an orbifold of the N = (2, 0) SCFTs [6,7], and recently, an interest in N = (1, 0) SCFTs has been increased by many new results, see [8,9,10,11,12] for an incomplete list. N = 2 gauged supergravity has been constructed for a long time in [13,14,15]. These theories, however, do not admit any AdS 7 vacua. The existence of an AdS 7 vacuum requires an additional deformation in the form of a mass term for the three-form field, dual to the two-form field in the N = 2 gravity multiplet. The N = 2 gauged supergravity including both types of deformations has been given in [16], see also [17]. An extension of this N = 2 gauged supergravity to include vector multiplets has been given in [18]. A number of supersymmetric AdS 7 vacua and various types of holographic solutions within this gauged supergravity have been studied in [19,20,21]. A classification of possible gauge groups that can give rise to maximally supersymmetric AdS 7 vacua has also been given in [22]. Most of the previously known solutions of pure and matter-coupled N = 2 gauged supergravity only involve the metric and scalar fields although the results of [20] and [21] do include solutions with non-vanishing gauge fields. Supersymmetric solutions of pure N = 2 gauged supergravity with all bosonic fields, including the three-form and gauge fields, non-vanishing have appeared recently in [23] along with the embedding in M-theory by using the result of [24]. The solution without the SU(2) gauge fields has also been uplifted to massive type IIA theory in [25] in which the solution is interpreted as a twodimensional conformal defect in N = (1, 0) SCFT. In the present work, we are interested in similar solutions in N = 2 gauged supergravity coupled to three vector multiplets with SO(4) ∼ SO(3) × SO(3) gauge group. In this case, the maximally supersymmetric AdS 7 vacuum is dual to an N = (1, 0) SCFT with flavor symmetry SO (3). The solutions presented here should give an extension of the results in [19] and [23] and represent more general solutions of N = 2 seven-dimensional gauged supergravity.
The paper is organized as follow. In section 2, we give a short review of the matter coupled N = 2 gauged supergravity in seven dimensions. The aim of this section is to give relevant formulae which will be used throughout the paper. Supersymmetric solutions interpolating between two asymptotically locally AdS 7 geometries and solutions flowing from an asymptotically locally AdS 7 with SO(4) symmetry to a singular geometry are given in section 3. These solutions are obtained by using the AdS 3 × S 3 -sliced domain wall ansatz. We also discuss the embedding of the latter type of solutions in eleven-dimensional supergravity. In section 4, we study similar solutions with non-vanishing SO(3) diag gauge fields and argue that this does not give rise to supersymmetric solutions. Some conclusions and comments on the results are given in section 5. All bosonic field equations of N = 2 gauged supergravity coupled to vector multiplets are given in appendix A. A consistent reduction ansatz for special values of the gauge coupling constants is reviewed in appendix B.

N = gauged supergravity in seven dimensions
We first give a brief review of N = 2 gauged supergravity in seven dimensions with topological mass term. All of the conventions and notations are essentially the same as those in [18] to which the reader is referred for more detail. The half-maximal N = 2 supergravity in seven dimensions can couple to an arbitrary number n of vector multiplets, the only matter multiplets in N = 2 supersymmetry. The field contents are given respectively by Supergravity multiplet : Curved and flat space-time indices are denoted by µ, ν, . . . andμ,ν, . . ., respectively. B µν and σ are the two-form and the dilaton fields. The two-form field will be dualized to a three-form field C µνρ which admits a topological mass term leading to a massive deformation of the N = 2 supergravity. Indices i, j = 1, 2, 3 label triplets of SU(2) R ∼ SO(3) R symmetry. The dilaton σ can be described by a coset space SO(1, 1) ∼ R + . Each of the vector multiplets, labelled by indices r, s = 1, 2, . . . , n, consists of a vector field A µ , two gaugini λ A and 3 scalars φ i . Indices A, B, . . . = 1, 2 label a doublet of the SU(2) R symmetry and will be generally suppressed throughout this paper. There are 3n scalar fields φ ir parametrizing SO(3, n)/SO(3) × SO(n) coset manifold. These can be efficiently described by a coset representative of the form L = (L i I , L r I ), I = 1, . . . , 3 + n .
The inverse of L will be denoted by Since L is an element of SO(3, n), we have the following relations It should be noted that indices i, j and r, s are raised and lowered by δ ij and δ rs , respectively while the full SO(3, n) indices I, J are raised and lowered by the SO(3, n) invariant tensor η IJ = diag(− − − + . . . +). With these conventions, relations involving components of L can be written as Gaugings of N = 2 supergravity can be obtained by promoting a subgroup G 0 of the global symmetry group R + × SO(3, n) to be a local symmetry. If the gauging does not invove the R + factor, the embedding of G 0 in SO(3, n) is described by the SO(3, n) tensor f IJ K identified with the structure constants of the gauge group G 0 via the gauge algebra where T I denote the gauge generators. In the embedding tensor formalism, f IJ K is a component of the full embedding tensor, see [26] for more detail.
For the gauging to be a consistent one, preserving all of the original supersymmetry, f IJ K must satisfy the conditions Apart from the gauging, there is also a massive deformation given by adding a topological mass term to the three-form field C µνρ . This additional deformation is crucial for the gauged supergravity to admit AdS 7 vacua. The bosonic Lagrangian including both the gauging and the massive deformation can be written as where we have used the form language for convenience in dealing with the field equations. The constant h describes the topological mass term for the three-form C (3) with H (4) = dC (3) . The scalar potential is given by where C and C ir are defined in term of the coset representative as The scalar kinetic term is written in term of the vielbein P ir µ on the SO(3, n)/SO(3) × SO(n) coset defined by The scalar matrix a IJ appearing in the kinetic term of vector fields is given by Finally, the Chern-Simons three-form satisfying with the gauge field strength tensors F I 1) . The associated bosonic field equations are collected in appendix A. The gauge coupling constants are included in f IJ K . Other ingredients which are relevant for finding supersymmetric solutions are supersymmetry transformations of fermions. With all fermionic fields vanishing, these are given by where σ i are the usual Pauli matrices. The covariant derivative of ǫ is given by where 3 AdS 3 × S 3 -sliced domain walls with the threeform field In this section, we will study supersymmetric solutions involving the sevendimensional metric, scalars and the three-form field. We will consider the case of n = 3 vector multiplets and SO(4 The corresponding structure constants are given by For a particular case of g 2 = g 1 , the resulting gauged supergravity can be embedded in eleven-dimensional supergravity [27]. An explicit parametrization of SO(3, 3)/SO(3) × SO(3) coset can be achieved by defining thirty-six 6 × 6 matrices of the form Non-compact generators of SO(3, 3) are accordingly given by We first truncate all of the nine scalars in SO . There is only one singlet scalar corresponding to the non-compact generator, see [19] for more detail, The coset representative can be written as The scalar potential is straightforwardly computed to be There are two supersymmetric AdS 7 critical points for this potential.
• AdS 7 with SO(4) symmetry: • AdS 7 with SO(3) diag symmetry: We have set g 1 = −16h in order to make the dilaton σ vanish at the SO(4) critical point. This is equivalent to a redefinition of σ by an appropriate shift. V 0 is the value of the scalar potential at the critical point. Holographic RG flow solutions interpolating between these two critical points and flows to non-conformal field theories have already been given in [19].

Solutions flowing between AdS 7 vacua
In this paper, we generalize the solutions studied in [19] by including a nonvanishing three-form field in the solutions. Following [23], we take the metric ansatz to be an AdS 3 × S 3 -sliced domain wall with the metrics on AdS 3 and S 3 given by The seven-dimensional coordinates are taken to be x µ = (x a , r, x m ) with a = 0, 1, 2 and m = 4, 5, 6. We will also use x 0 = t and x 3 = r in the following analysis. The corresponding flat indices will be denoted byμ = (â,3,m). The S 3 part is described by Hopf coordinates x m = (θ 1 , θ 2 , θ 3 ). In the limit τ → 0 and κ → 0, the AdS 3 and S 3 become flat Minkowski space and flat space R 3 , respectively.
With the vielbeins on AdS 3 and S 3 of the form and the spin connections take a simple form with ǫ012 = ǫ456 = 1. We will use ′ to denote the r-derivative throughout the paper.
As in the usual domain wall solutions, the scalar fields σ and φ are functions of only r while the ansatz for the three-form field is taken to be in which Vol AdS 3 and Vol S 3 are volume forms on AdS 3 and S 3 , respectively. We will also set A I (1) = 0 since, in this section, we are interested only in solutions with vanishing vector fields.
The ansatz for Killing spinors corresponding to the unbroken supersymmetry takes the form of with the constant spinor ǫ 0 satisfying the projection condition Y (r) and θ(r) are functions of r to be determined. To find supersymmetric solutions, we consider BPS equations obtained from supersymmetry transformations of fermionic fields (ψ µ , χ, λ r ). Using the Killing spinor (38) and the projection (39), we obtain two equations from δλ r = 0 conditions For the coset representative (23), we can readily compute P ir µ and C ir . The result is given by Note also that the three-form field does not enter the δλ r equations. Compatibility between equations (40) and (41) implies cos(2θ) = ±1 leading to sin θ = 0 or cos θ = 0. Up to a redefinition of ǫ 0 toǫ 0 = γ012ǫ 0 and a sign change in the projection condition (39), the two choices give equivalent BPS equations. For definiteness, we will choose sin θ = 0 in the following analysis. This leads to the BPS equation for φ The Killing spinor then takes a simpler form We now consider δχ = 0 equation. This condition involves a contribution from the three-form field of the form H µνρσ γ µνρσ ǫ. We will use the same convention for spinors and gamma matrices as in [23]. Using the relation γ0γ1γ2γ3γ4γ5γ6 = 1 8 or more compactly ǫâbĉγâbĉγr = −ǫmnpγmnp, we find Since there is no other term contributing γ012 matrix in the δχ variation, this term must vanish by itself. This can be achieved by setting which leads to the BPS equation for σ We then move on to the BPS equations from δψ µ conditions. After using the γr projection (39) and the three-form ansatz (37) in the conditions δψ a = 0 and δψ m = 0, we find two types of terms one with γ012 and the other with 1 8 . The former gives rise to the BPS equations for k and l while the latter gives the corresponding equations for U and W The last equation implies that U = W + C for a constant C. In order to find solutions interpolating between AdS 7 vacua, we require that the solutions be asymptotically locally AdS 7 at which U = W . This implies that C = 0 or U = W .
Using this relation in equation (46), we find that k ′ = l ′ or k = l +C for some constantC. This constant can be set to zero by a suitable redefinition of k and l. We will accordingly set k = l. With all these, equation (48) gives In summary, we end up with the BPS equations for the warped factor U and k in the form of It should be noted that the contribution from C (3) is cancelled by the spin connections on AdS 3 and S 3 . Therefore, for non-vanishing C (3) and k = l, there can be no background with Mink 3 and R 3 . This is perfectly in agreement with a similar solution considered in [23] but without the scalar from vector multiplets. It can also be easily checked that any solutions to the above BPS equations solve all the field equations.
We finally consider the equation from δψ 3 condition. This gives the BPS equation for Y (r) which can be solved by a solution Y ∼ e U 2 . We are now in a position to solve all of the BPS equations. To find an analytic solution, we first choose a function V (r) = σ 2 . This is equivalent to changing to a new radial coordinater defined by the relation dr dr = e − σ 2 in [19]. The procedure is very similar to that used in [19], so we will not repeat all the details here. After choosing V (r) = σ 2 , we obtain the solution for (43) (54) where an irrelevant additive integration constant has been neglected. By treating U, σ and k as functions of φ, we find the solution of equations (47), (51) and (52) in which irrelevant integration constants in U and k have been removed. The integration constant in σ is however important and has been chosen such that the solution for σ interpolates between the two supersymmetric AdS 7 critical points, see [19] for more detail.
As r → ±∞, the solution is asymptotic to the AdS 7 critical points with for r → ∞, and for r → −∞. In these equations, we have set g 1 = −16h.
It should be noted that the four-form field strength does not actually vanish in the limit r → ±∞ as can be seen from the BPS equation for k ′ . Moreover, the existence of C (3) is needed to support the AdS 3 and S 3 factors as mentioned above. However, its effect in the limit r → ±∞ is highly suppressed compared to the scalar potential. The solution is then asymptotically locally AdS 7 as r → ±∞.

Solutions with known higher dimensional origin
For a particular case of g 2 = g 1 , solutions of the N = 2 gauged supergravity can be uplifted to eleven dimensions. The corresponding reduction ansatz has been constructed in [27]. Setting g 2 = g 1 , we obtain the BPS equations It can be clearly seen from the φ ′ equation that there is only one supersymmetric AdS 7 background at φ = 0. The solutions interpolating between this AdS 7 and physically acceptable, singular geometries dual to non-conformal field theories in the case of k = 0 have already been studied in [27]. In this paper, we will give the solution with non-vanishing three-form field. This solution can be found by the analysis similar to the previous case. The resulting solution is given by It can be seen that φ diverges at a finite value of r. Therefore, the solution is singular at this point. Without loss of generality, we can shift the coordinate r such that the singularity occurs at r = 0. The integration constant C 1 controls the behavior near the singularity, see [27] for more detail.
For C 1 = 0, the solution near r = 0 becomes in which we have set g 1 = −16h. For C 1 = 0, we find φ ∼ − ln(4hr), σ ∼ 6 5 ln(4hr), k ∼ constant, As pointed out in [27], all of these singularities are physically acceptable since the scalar potential is bounded from above, in this case V → −∞, as required by the criterion in [28]. In this case, the solution can be embedded in eleven dimensions by using the reduction ansatz in [27]. For convenience, we give a brief review of this result in appendix B. The nine scalars from vector multiplets can be equivalently described by SL(4, R)/SO(4) coset due to the isomorphism SL(4, R) ∼ SO (3,3). For the SO(3) diag singlet scalar, we find the SL(4, R)/SO(4) coset representative which gives a symmetric 4 × 4 matrix with unit determinant T αβ = diag(e φ , e φ , e φ , e −3φ ) = (δ ab e φ , e −3φ ).
In the remaining parts of this section, we will use indices a, b = 1, 2, 3 to denote coordinatesμ a on the internal S 2 withμ aμa = 1. We will also use the S 3 coordinates µ α = (cos ψμ a , sin ψ) satisfying µ α µ α = 1.
With all these and the seven-dimensional fields given previously, we obtain the eleven-dimensional metric with the warped factor given by and the metric on a unit two-sphere can be written as dΩ 2 2 = dμ a dμ a . It should be noted that the S 2 in the internal S 3 is unchanged. Its isometry corresponds to the unbroken SO(3) diag symmetry.
The four-form field strength of eleven-dimensional supergravity is given bŷ where ǫ (2) = 1 2 ǫ abcμ a dμ b dμ c is the volume form on S 2 . In this equation, we have also used ǫ abc4 = ǫ abc . The scalar function U is given by Similar to the discussion in [23], we expect the uplifted solution to describe eleven-dimensional configurations involving M2-M5-brane bound states due to the dyonic profile of C (3) . It is also interesting to consider the (00)-component of the eleven-dimensional metricĝ Near the singularity at r = 0, we find that g 00 ∼ (4hr) 26 15 → 0 andĝ 00 ∼ (4hr) 13 60 → 0 (77) for C 1 = 0 and C 1 = 0, respectively. According to the criterion of [29], the singularities are physical in agreement with the seven-dimensional results obtained from the criterion of [28]. We then expect that the solution holographically describes a two-dimensional conformal defect in six-dimensional N = (1, 0) SCFT with known M-theory origin.

Domain walls with the three-form and vector fields
In this section, we consider more general solutions with non-vanishing vector fields. We first choose an appropriate ansatz for the SO(4) ∼ SO(3) × SO(3) gauge fields. As in [23], we will take this ansatz in the form of in which the components A I i will be functions of only the radial coordinate r. Explicitly, these components are given by It is now straightforward to compute the field strength tensors F i (2) = L I i F I (2) and F r (2) = L I r F I (2) . Non-vanishing components of these tensors are given by To implement SO(3) diag , we will set We still use the ansatz for the Killing spinor as given in (38) and the projection (39). Due to the extra contributions from non-vanishing gauge fields, we need more projectors The second condition is just the Symplectic-Majorana condition. Therefore, the BPS solutions (if exist) will preserve only two supercharges or 1 8 -BPS after imposing the projection (39).
With all these, we can now set up the BPS equations. By the relation (87), the composite connection along S 3 takes a very simple form where ω i+3,j+3,k+3 is the spin connection given in (36). Using the same procedure as in the previous section, we find the following set of BPS equations where the quantities C and C are defined by In addition to these flow equations, there is an algebraic constraint arising from the fact that the supersymmetry transformations from the gravity multiplet (δψ µ and δχ) and those from the vector multiplets (δλ r ) lead to different BPS equations for A. Consistency between these two equations results in a constraint 0 = e σ 2 sin 2θ e −σ 3 This means supersymmetric solutions must satisfy the above BPS equations as well as the constraint (101) in order for the Killing spinors to exist. We have explicitly verified that the BPS equations (90) to (98) together with the constraint (101) imply all of the field equations. However, it turns out that there are no solutions with non-trivial threeform and vector fields satisfying the BPS equations (90) to (98) and the constraint (101). This can be readily checked by differentiating equation (101) and substituting the BPS equations (90) to (98). The result gives the following condition which can not be satisfied by the BPS equations (94) and (97). This implies that φ and k cannot flow independently but relate to each other by equation (102). Note that relation (102) is trivially satisfied for A = 0 which gives f = g = 0. This results in the BPS equations considered in the previous section. Another possibility is to set k ′ = 0 and g 2 2 = g 2 1 . However, with k ′ = 0 and the constraint (101), equation (94) implies θ = 0 and A = 1 g 1 for generic values of σ and φ. Although this result is compatible with the BPS equations given in (93) and (98), it also leads to l ′ = 0 as can be seen from equation (95) after using the constraint (101). The three-form field then has vanishing field strength. In this case, the gauge fields do not depend on the radial coordinate r. This type of solutions has already been studied in [21] in the context of twisted compactifications. Therefore, we conclude that there are no supersymmetric solutions with non-vanishing SO(3) diag gauge fields and non-trivial three-form field.

Conclusions and discussions
We have studied supersymmetric solutions of matter-coupled N = 2 gauged supergravity in seven dimensions with SO(4) ∼ SO(3) × SO(3) gauge group. The resulting solutions are generalizations of the previously known solutions of N = 2 gauged supergravity in the sense that all possible bosonic fields, from both gravity and vector multiplets, are considered. These solutions take the form of asymptotically locally AdS 7 solutions and should be useful in the holographic study of N = (1, 0) six-dimensional SCFTs.
For vanishing vector fields, we have found analytic solutions to the BPS equations for all the fields that are singlets of the residual SO(3) diag ⊂ SO(3) × SO(3) symmetry. For special values of SO(3) × SO(3) gauge couplings, namely g 2 = g 1 , the solutions can be uplifted to eleven dimensions. We have performed this uplift and given the explicit form of the eleven-dimensional metric and the four-form field strength tensor. Unlike the solutions found in [23], we have found that the solutions to the matter-coupled gauged supergravity are more restrictive. As a result, only solutions in the form of AdS 3 × S 3 sliced-domain walls are possible. From a general notion of the AdS/CFT correspondence and the recent result in [25], we expect these solutions to describe some supersymmetric, twodimensional, conformal defects in N = (1, 0) SCFT with flavor symmetry SO(3). It is interesting to find the dual descriptions of these solutions in the N = (1, 0) SCFT. A generalization of these solutions to include more scalars, such as those invariant under smaller residual symmetries, is straightforward since the threeform field does not couple directly to scalars from vector multiplets.
We have also derived a set of first-order flow equations together with an algebraic constraint for the case of non-vanishing SO(3) diag gauge fields. In this case, we have performed the analysis and argued that supersymmetric solutions do not exist at least within the truncation considered here. This is due to the fact that, in general, the constraint, arising from the supersymmetry variations of the gaugini, is violated by the solutions of the flow equations.
Solutions in N = 2 gauged supergravity with other gauge groups and the slicing different from AdS 3 × S 3 are worth considering. Similar solutions in the maximal N = 4 gauged supergravity in seven dimensions are also of particular interest. These would describe lower dimensional defects within N = (2, 0) SCFT dual to the AdS 7 ×S 4 solution of M-theory. We hope to come back to these issues in future works.
A Bosonic field equations of N = 2 gauged supergravity coupled to vector multiplets In this appendix, we give all of the bosonic field equations obtained from the Lagrangian given in (8). These equations read where C irs is defined by The Yang-Mills equation (105) can also be written in terms of C ir and C irs by using the relation In deriving the scalar field equation (106), it is useful to adopt the following projections, given in [13], in the form of δL i I = X i r L r I + X i j L j I , δL r I = X r s L s I + X r i L i I .
With these relations, variations with respect to the scalar fields lead to the following results