Supersymmetric solutions of 7D maximal gauged supergravity

We study a number of supersymmetric solutions in the form of $Mkw_3\times S^3$- and $AdS_3\times S^3$-sliced domain walls in the maximal gauged supergravity in seven dimensions. These solutions require non-vanishing three-form fluxes to support the $AdS_3$ and $S^3$ subspaces. We consider solutions with $SO(4)$, $SO(3)$, $SO(2)\times SO(2)$ and $SO(2)$ symmetries in $CSO(p,q,5-p-q)$, $CSO(p,q,4-p-q)$ and $SO(2,1)\ltimes \mathbf{R}^4$ gauge groups. All of these solutions can be analytically obtained. For $SO(5)$ and $CSO(4,0,1)$ gauge groups, the complete truncation ansatze in terms of eleven-dimensional supergravity on $S^4$ and type IIA theory on $S^3$ are known. We give the full uplifted solutions to eleven and ten dimensions in this case. The solutions with an $AdS_3\times S^3$ slice are interpreted as two-dimensional surface defects in six-dimensional $N=(2,0)$ superconformal field theory in the case of $SO(5)$ gauge group or $N=(2,0)$ nonconformal field theories for other gauge groups. For $SO(4)$ symmetric solutions, it is possible to find solutions with both the three-form fluxes and $SO(3)$ gauge fields turned on. However, in this case, the solutions can be found only numerically. For $SO(3)$ symmetric solutions, the three-form fluxes and $SO(3)$ gauge fields cannot be non-vanishing simultaneously.


I. INTRODUCTION
Gauged supergravities in various space-time dimensions have become a useful tool for studying different aspects of the AdS/CFT correspondence [1][2][3] and the DW/QFT correspondence [4][5][6]. Solutions to gauged supergravities provide some insight to the dynamics of stongly-coupled conformal and non-conformal field theories via holographic descriptions, see for example [7][8][9][10][11]. The study along this line is particularly fruitful in the presence of supersymmetry. In this case, many aspects of both the gravity and field theory sides are more controllable even at strong coupling. This makes finding various types of supersymmetric solutions in gauged supergravities worth considering.
In this paper, we are interested in supersymmetric solutions in the maximal gauged supergravity in seven dimensions. The solutions under consideration here take the form of Mkw 3 × S 3 and AdS 3 × S 3 -sliced domain walls. This type of solutions has originally been considered in the minimal N = 2 gauged supergravity in [12], see also [13] for similar solutions in the matter-coupled N = 2 gauged supergravity. Some of these solutions have been interpreted as surface defects within N = (1, 0) superconformal field theory (SCFT) in six dimensions in [14], see [15,16] for similar solutions in six dimensions and [17][18][19][20][21][22] for examples of another holographic description of conformal defects in terms of Janus solutions.
We will find these Mkw 3 × S 3 and AdS 3 × S 3 -sliced domain walls in the maximal N = 4 gauged supergravity with various types of gauge groups. The most general gaugings of the N = 4 supergravity can be constructed by using the embedding tensor formalism [23], for an earlier construction see [24] and [25]. The embedding tensor describes the embedding of an admissible gauge group G 0 in the global symmetry group SL (5) and encodes all information about the resulting gauged supergravity. Supersymmetry allows for two components of the embedding tensor transforming in 15 and 40 representations of SL (5). We will consider CSO(p, q, 5 − p − q) and CSO(p, q, 4 − p − q) gauge groups obtained from the embedding tensor in 15 and 40 representations, respectively. We will also study similar solutions in SO(2, 1) ⋉ R 4 gauge group from the embedding tensor in both 15 and 40 representations.
Vacuum solutions in terms of half-supersymmetric domain walls for all these gauge groups have already been studied in [26]. In this paper, we will extend these solutions, which involve only the metric and scalars, by including non-vanishing two-and three-form fields. In some cases, in addition to two-and three-form fields, it is also possible to couple SO(3) gauge fields to the solutions.
As shown in [27] using the framework of exceptional field theory, seven-dimensional gauged supergravity in 15 representation with CSO(p, q, 5 − p − q) gauge group can be obtained from a consistent truncation of eleven-dimensional supergravity on H p,q • T 5−p−q .
On the other hand, a consistent truncation of type IIB theory on H p,q • T 4−p−q gives rise to CSO(p, q, 4 − p − q) gauging from 40 representation. This has been shown in [28] along with a partial result on the corresponding truncation ansatze. In particular, internal components of all the ten-dimensional fields have been given.
For SO (5) and CSO(4, 0, 1) gauge groups, the complete truncation ansatze have already been constructed long ago in [29,30] and [31]. In this work, we will mainly consider uplifted solutions from these two gauge groups using the truncation ansatze given in [29][30][31] which are more useful for solutions involving two-and three-form fields in seven dimensions. We leave uplifting solutions from other gauge groups for future work.
The paper is organized as follows. In section II, we give a brief review of the maximal gauged supergravity in seven dimensions. Supersymmetric Mkw 3 ×S 3 -and AdS 3 ×S 3 -sliced domain walls in CSO(p, q, 5 − p − q) gauge group together with the uplifted solutions to eleven and ten dimensions in the case of SO (5) and CSO(4, 0, 1) gauge groups are presented in section III. Similar solutions for CSO(p, q, 4 − p − q) and SO(2, 1) ⋉ R 4 gauge groups obtained from gaugings in 40 and (15,40) representations are given in sections IV and V, respectively. Conclusions and comments are given in section VI. In the two appendices, all bosonic field equations of the maximal gauged supergravity and consistent truncation ansatze for eleven-dimensional supergravity on S 4 and type IIA theory on S 3 are given.

II. MAXIMAL GAUGED SUPERGRAVITY IN SEVEN DIMENSIONS
In this section, we briefly review N = 4 gauged supergravity in seven dimensions in the embedding tensor formalism. We mainly focus on the bosonic Lagrangian and fermionic supersymmetry transformations which are relevant for finding supersymmetric solutions. The reader is referred to [23] for the detailed construction of the maximal gauged supergravity.
As in other dimensions, the maximal N = 4 supersymmetry in seven dimensions allows only the supergravity multiplet with the field content This multiplet consists of the graviton eμ µ , four gravitini ψ a µ , ten vectors five two-form fields B µνM , sixteen spin-1 2 fermions χ abc = χ [ab]c , and fourteen scalar fields described by the SL(5)/SO(5) coset representative V M A .
Throughout the paper, we will use the following convention on various types of indices. The gravitini then transform as 4 while the spin-1 2 fields χ abc transform as 16 of USp(4). The latter satisfy the following conditions and Ω ab χ abc = 0 (2) with Ω ab = Ω [ab] being the USp(4) symplectic form satisfying the properties (Ω ab ) * = Ω ab and Ω ac Ω bc = δ b a .
It should also be noted that raising and lowering of USp(4) indices by Ω ab and Ω ab correspond to complex conjugation. Furthermore, all fermions are symplectic Majorana spinors subject to the conditions ψ T µa = Ω ab Cψ b µ andχ T abc = Ω ad Ω be Ω cf Cχ def (4) where C denotes the charge conjugation matrix obeying With the space-time gamma matrices denoted by γ µ , the Dirac conjugate on a spinor Ψ is defined by Ψ = Ψ † γ 0 .
The fourteen scalars parametrizing SL(5)/SO (5) Similarly, the inverse of V M A denoted by V A M will be written as V ab M . We then have the following relations Gaugings are deformations of the N = 4 supergravity by promoting a subgroup G 0 ⊂ SL(5) to be a local symmetry. The most general gaugings of a supergravity theory can be efficiently described by using the embedding tensor formalism. The embedding of G 0 within SL (5) is achieved by using a constant SL(5) tensor Θ M N,P Q = Θ [M N ],P Q living in the product representation [23] 10 ⊗ 24 = 10 + 15 + 40 + 175 .
In term of the embedding tensor, gauge generators are given by in which t M N , satisfying t M M = 0, are SL(5) generators. In particular, the gauge generators in the fundamental 5 and 10 representations are given by and with ǫ M N P QR being the invariant tensor of SL (5). To ensure that the gauge generators form a closed subalgebra of SL(5) the embedding tensor needs to satisfy the quadratic constraint Gaugings introduce minimal coupling between the gauge fields and other fields via the covariant derivative where ∇ µ is the spacetime covariant derivative including (possibly) composite SO(5) connections. To restore supersymmetry of the original N = 4 supergravity, fermionic mass-like terms and the scalar potential at first and second orders in the gauge coupling constant are needed. In addition, to ensure gauge covariance, the field strength tensors of vector and two-form fields need to be modified as where the non-abelian gauge field strength tensor is defined by Note that the three-form fields S M µνρ in H µνρ only appear under the projection of Y M N . In ungauged supergravity, all of the three-form fields can be dualized to two-form fields.
However, this is not the case in the gauged supergravity. Therefore, different gaugings lead to different field contents in the resulting gauged supergravity.
Following [23], we first define s ≡ rank Z and t ≡ rank Y . In a given gauging, t two-forms can be set to zero by tensor gauge transformations of the three-form fields. This results in t self-dual massive three-forms. Similarly, s gauge fields can be set to zero by tensor gauge transformations of the two-forms giving rise to s massive two-form fields. It should also be pointed out that there can be massive vector fields arising from broken gauge symmetry via the usual Higgs mechanism. We can see that the numbers of two-and three-form tensor fields depend on the gauging under consideration. However, the quadratic constraint ensures that t + s ≤ 5, so the degrees of freedom from the ten vector and five two-form fields in the ungauged supergravity are redistributed into two-and three-form fields in the gauged theory. This fact will affect our ansatz for finding supersymmetric solutions in subsequent where the covariant field strengths of the three-form fields are given by It should be emphasized that the three-forms S M µνρ and its field strength tensors always appear under the projection by Y M N .
With all these ingredients, the bosonic Lagrangian of the seven-dimensional maximal gauged supergravity can be written as In this equation, the scalar fields are described by a unimodular symmetric matrix Its inverse is given by We will not give the explicit form of the vector-tensor topological term L V T here due to its complexity but refer the reader to [23]. Finally, the scalar potential is given by The supersymmetry transformations of fermionic fields which are essential for finding supersymmetric solutions read δχ abc = 2Ω cd P µde ab γ µ ǫ e + gA d,abc The covariant derivative of the supersymmetry parameters is defined by The composite connection Q µa b and the vielbein on the SL(5)/SO(5) coset P µab cd are obtained from the following relation The fermion shift matrices A 1 and A 2 are given by A d,abc with various components of B and C tensors defined by In the above equations, we have introduced "dressed" components of the embedding tensor defined by and Finally, we note that the scalar potential can also be written in terms of the fermion-shift matrices A 1 and A 2 as In the following sections, we will find supersymmetric solutions in a number of possible gauge groups.
This corresponds to the gauge group To give an explicit parametrization of the SL(5)/SO(5) coset, we first introduce GL(5) We will use the following choice of SO(5) gamma matrices to convert an SO(5) vector index to a pair of antisymmetric spinor indices where σ i are the usual Pauli matrices. Γ A satisfy the following relations The symplectic form of USp(4) is chosen to be The coset representative of the form V M ab and the inverse V ab M are then obtained from the following relations We will use the metric ansatz in the form of an AdS 3 × S 3 -sliced domain wall The seven-dimensional coordinates are taken to be x µ = (x m , r, x i ) with m = 0, 1, 2 and i = 4, 5, 6. Note that V (r) is an arbitrary non-dynamical function that can be set to zero with a suitable gauge choice. The explicit forms for the metrics on AdS 3 and S 3 are given in Hopf coordinates by in which τ and κ are constants. In the limit τ → 0 and κ → 0, the AdS 3 and S 3 parts become flat Minkowski space Mkw 3 and flat space R 3 , respectively.
With the following choice of vielbeins we find the following non-vanishing components of the spin connection with the convention that ε012 = −ε012 = ε456 = ε456 = 1. Throughout this paper, we will use a prime to denote the r-derivative.
Following [12], we take the ansatz for the Killing spinors to be with ǫ a 0 being constant spinors. In addition, we will use the following ansatz for the threeform field strength tensors In subsequent analysis, we will call the solutions with non-vanishing H M "charged" domain walls.

A. SO(4) symmetric charged domain walls
We first consider charged domain wall solutions with SO(4) symmetry. As in [26], we will find supersymmetric solutions with a given unbroken symmetry from many gauge groups within a single framework. Gauge groups that can give rise to SO(4) symmetric solutions are SO(5), SO(4, 1) and CSO(4, 0, 1). We will accordingly write Y M N in the following form where ρ = +1, −1, 0 corresponding to SO(5), SO(4, 1), and CSO(4, 0, 1) gauge groups, respectively. With this embedding tensor, the SO(4) residual symmetry is generated by With the coset representative the scalar potential is given by For ρ = 1, this potential admits two AdS 7 critical points with SO(5) and SO(4) unbroken symmetries. The former preserves all supersymmetry while the latter is non-supersymmetric.
These vacua are given respectively by and φ = 1 10 ln 2 and The cosmological constant is denoted by V 0 , the value of the scalar potential at the vacuum.
To preserve SO(4) symmetry, we will keep only the following components of H At this point, it is useful to consider the H Since the four-form field strengths do not enter the supersymmetry transformations of fermionic fields, the functions k M (r) and l M (r) will appear, in this case, algebraically in the resulting BPS equations. This is in contrast to the pure N = 2 gauged supergravity considered in [12] in which the four-form field strength of the massive three-form field appears in the supersymmetry transformations. Therefore, in that case, the BPS conditions result in differential equations for k(r) and l(r).
For CSO(4, 0, 1) gauge group with Y 55 = 0, S 5 µνρ does not contribute to H M , but, in this case with s = 0 and t = 4, there is 5 − t = 1 massless two-form field B µν5 with the field To satisfy the Bianchi's identity DH (3) = 0, we need k ′ = l ′ = 0 or constant three-form fluxes. We will see that this is indeed the case for our BPS solutions. Taking this condition into account, we can write the ansatz for the two-form field as with vol AdS 3 = dω 2 and vol S 3 = dω 2 . With the metrics given in (47) and (48), the explicit form of ω 2 andω 2 is given by After imposing two projection conditions we find the following BPS equations from the conditions δψ a µ = 0 and δχ abc = 0 together with an algebraic constraint We note here that the appearance of the SO(5) gamma matrix Γ 5 in the projection conditions is due to the non-vanishing H µνρ5 . Note also that the solutions are 1 4 -BPS since the Killing spinors ǫ a 0 are subject to two projectors. We now consider various possible solutions to these BPS equations.

M kw 3 × R 3 -sliced domain walls
We begin with a simple case of Mkw 3 × R 3 -sliced domain walls with vanishing τ and κ.
Imposing τ = κ = 0 into the constraint (72) gives Setting g = 0 corresponds to ungauged N = 4 supergravity and gives rise to a supersym- Another possibility to satisfy the condition (73) is to set tan 2θ = 0 which implies θ = nπ 2 , n = 0, 1, 2, 3, . . .. For even n, we have sin θ = 0 and, from (51), the Killing spinors take the form ǫ a = e U (r)/2 ǫ a 0 (74) with ǫ a 0 satisfying the projection conditions given in (65). For odd n with cos θ = 0, the Killing spinors become We can redefine ǫ a 0 toǫ a 0 = γ012ǫ a 0 satisfying the projection conditions This differs from the projectors in (65) only by a minus sign in the γ3 projector. Therefore, the two possibilities obtained from the condition tan 2θ = 0 are equivalent by flipping the sign of γ3 projector. We can accordingly choose θ = 0 without losing any generality.
With θ = 0, the BPS equations (66) to (71) become By choosing V = −3φ, we find the following solution with an integration constant C. Since k = l = θ = 0, the Γ 5 projection in (65) is not needed.
This is then a half-supersymmetric solution with vanishing three-form fluxes and is exactly the SO(4) symmetric domain wall studied in [26]. Therefore, the Mkw 3 × R 3 -sliced solution is just the standard flat domain wall.

M kw 3 × S 3 -sliced domain walls
In this case, we look for domain wall solutions with Mkw 3 × S 3 slice. Following [12], we choose the follwing gauge choice By setting τ = 0, we can solve the BPS equations (66) -(71) and obtain the following solution, for ρ = ±1, with κ = −g/2. C is an integration constant in the solution for φ.
For SO(5) gauge group with ρ = 1, the solution is locally asymptotic to the N = 4 supersymmetric AdS 7 in the limit r → ∞ with It should be noted that in this limit, the main contribution to the solution is obtained from the scalar. The contribution from the three-form field strength is highly suppressed as can be seen from its components in flat basis given in (60). In the limit r → 0, the solution is singular similar to the solution studied in [12].
For SO(4, 1) gauge group with ρ = −1, there is no AdS 7 asymptotic since this gauge group does not admit a supersymmetric AdS 7 vacuum. In this case, the solution is the SO(4) symmetric domain wall studied in [26] with a dyonic profile of the three-form flux.
Furthermore, a non-vanishing θ gives a non-trivial three-form flux according to (93) to support the S 3 part. For constant θ = 0, we can find the following solution, after choosing V = 0 gauge choice, with an integration constant C. The constant θ is given by As in the SO(4, 1) gauge group, it can be verified that for a given constant θ, this solution is the SO(4) symmetric domain wall of CSO(4, 0, 1) gauge group given in [26] with a magnetic profile of a constant three-form flux.

AdS 3 × S 3 -sliced domain walls
We now consider more complicated solutions with an AdS 3 × S 3 slice. As in [12], we begin with a simpler solution with a single warp factor U = W . From the BPS equations Setting θ = 0, we find that the BPS equations become By choosing V = −3φ, we obtain the following solution with an integration constant C. This solution is the SO(4) symmetric domain wall coupled to a dyonic profile of the three-form flux.
For SO(5) gauge group, the solution is locally asymptotic to the supersymmetric AdS 7 dual to N = (2, 0) SCFT in six dimensions. This solution is then expected to describe a surface defect, corresponding to the AdS 3 part, within the six-dimensional N = (2, 0) SCFT.
Similarly, according to the DW/QFT correspondence, the usual Mkw 6 -sliced domain wall without the three-form flux is dual to an N = (2, 0) non-conformal field theory in six dimensions. We then interpret the solutions for SO(4, 1) and CSO(4, 0, 1) gauge groups as describing a surface defect within a non-conformal N = (2, 0) field theory in six dimensions.
We now consider more general solutions with the AdS 3 × S 3 slice. We will find the solutions for the cases of ρ = ±1 and ρ = 0, separately. With the same gauge choice given in (82), the BPS equations (66) -(71) for ρ = 0 are solved by together with the following relation obtained from the constraint (72) As in the previous case, for SO(5) gauge group, the solution is locally asymptotically AdS 7 given in (89) as r → ∞. For SO(4, 1) gauge group, the solution is a charged domain wall with a non-vanishing three-form flux. In general, these solutions describe respectively holographic RG flows from an N = (2, 0) SCFT and N = (2, 0) non-conformal field theory to a singularity at r = 0 except for a special case with τ = g(ρC + 1)/4. This is very similar to the solutions of pure N = 2 gauged supergravity studied in [12] For the particular value of τ = g(ρC + 1)/4, the scalar potential is constant as r → 0, and the solution turns out to be described by a locally AdS 3 × T 4 geometry with the following leading profile To obtain real solutions, we choose the integration constant C < 1 4 and C < − 1 4 for SO(5) and SO(4, 1) gauge groups, respectively.
For CSO(4, 0, 1) gauge group with ρ = 0, we find the following solution, after setting where the constant κ is given by Note also that, in this case, θ is constant since the corresponding BPS equation gives θ ′ = 0 as can be seen from equation (69).

Coupling to SO(3) gauge fields
In this section, we extend the analysis by coupling the previously obtained solutions to SO(3) vectors describing a Hopf fibration of the three-sphere. With the projector (Γ 5 ) a b ǫ b 0 = ǫ a 0 and the identity Γ 1 . . . Γ 5 = I 4 , we turn on the gauge fields corresponding to the anti-selfdual SO(3) ⊂ SO(4). The ansatz for these gauge fields is chosen to be The function p(r) is the magnetic charge with the dependence on the radial coordinate. The corresponding two-form field strengths can be computed to be To preserve some amount of supersymmetry, we need to impose additional projectors on the constant spinors ǫ a 0 as follow It should be noted that the last projector is not independent of the first two. Therefore, together with the projectors given in (65), there are four independent projectors on ǫ a 0 , and the residual supersymmetry consists of two supercharges.
With all these, the resulting BPS equations for the AdS 3 × S 3 -sliced domain wall are given by In contrast to the previous case, it can also be verified that these equations satisfy the second-order field equations without imposing any constraint. By setting τ = 0, we can The BPS equations in this case are much more complicated, and we are not able to find analytic flow solutions. We then look for numerical solutions with some appropriate boundary conditions. We first consider the solutions in SO(5) gauge group with an AdS 7 asymptotic at large r. With ρ = 1, we find that the following locally AdS 7 configuration solves the BPS equations at the leading order as r → ∞ This is rather different from the solutions in [12] in which the BPS equations for k and l are differential.
From the numerical solution in figure 2, the solutions for k and l appear to be diverging as k ∼ e 2U and l ∼ e 2W for r → ∞. However, the contribution from the three-form flux is sufficiently suppressed for r → ∞ since the terms involving H

B. SO(3) symmetric charged domain walls
In this section, we consider charged domain walls preserving SO(3) residual symmetry.
There are three singlet scalars corresponding to the following noncompact generatorŝ the scalar potential reads For SO(5) gauge group, this potential admits a supersymmetric AdS 7 vacuum given in (58) at φ 1 = φ 2 = φ 3 = 0 and a non-supersymmetric AdS 7 given in (59) at φ 1 = 1 20 ln 2, φ 2 = ± 1 4 ln 2 and φ 3 = 0. We now repeat the same procedure as in the previous section to set up the BPS equations.
The SO(3) residual symmetry allows for two three-form field strengths, H µνρM with M = 4, 5. We will choose the following ansatz iĵk4 = l 4 (r)e −3W (r) εˆiĵk, With H µνρ4 non-vanishing, the SO(5) gamma matrix Γ 4 will appear in the BPS conditions. To avoid an additional projector, which will break more supersymmetry, we impose the following condition k 4 (r) = tanh φ 2 k 5 (r) and l 4 (r) = tanh φ 2 l 5 (r) .
This simply makes the coefficient of Γ 4 vanish. It would also be interesting to consider a more general projector.
With the projection conditions in (65), we can find a consistent set of BPS equations for The latter forbids the possibility of setting either τ = 0 or κ = 0 without ending up with κ = τ = 0. Therefore, the solutions in this case can only be AdS 3 × S 3 -sliced domain walls.
The resulting BPS equations take the form However, the compatibility between these BPS equations and the corresponding field equations requires either φ 2 = 0 or φ 3 = 0. It should be noted that setting φ 3 = 0 is consistent with equation (147), namely φ ′ 3 = 0, only for σ = ρ, so solutions with vanishing φ 3 can only be obtained in SO(5), SO(3, 2) and CSO(3, 0, 2) gauge groups. To find explicit solutions, we separately consider various possible values of ρ and σ. and with an integration constant C.

Charged domain walls in SO(4, 1) gauge group
In this case, the gauge group is a non-compact SO(4, 1) with σ = −ρ = 1. As in the previous case, it is not possible to set φ 3 = 0, so we only consider solutions with φ 2 = 0.
Using the same gauge choice V = −6φ 1 , we find the following solution e 2φ 3 =tan

Charged domain walls in SO(5) and SO(3, 2) gauge groups
We now look at the last possibility with ρ = σ = ±1 corresponding to SO(5) and SO (3,2) gauge groups. In this case, it is possible to set φ 2 = 0 or φ 3 = 0. With φ 2 = 0 and V = −6φ 1 , we find the following solution together with For φ 3 = 0, we find the same solution as in (162) -(164) with φ 3 replaced by φ 2 , but the solution for k 4,5 and l 4,5 are now given by Unlike the previous cases, this solution has two non-vanishing three-form fluxes.
It turns out that, in this case, the compatibility between the resulting BPS equations and the corresponding field equations requires that For τ = 0, we can have a constant magnetic charge p as required by the conditions in (167), but in this case, the three-form flux vanishes unless e W τ = e U κ as required by (142). This case corresponds to performing a topological twist along the S 3 part. Since this type of solutions is not the main aim of this paper, we will not consider them here. On the other hand, setting e W τ = e U κ does lead to non-vanishing three-form fluxes, but equation (167) gives vanishing gauge fields. This corresponds to the charged domain walls given above.
Therefore, there does not seem to be solutions with both SO(3) gauge fields and three-form fluxes non-vanishing at least for the ansatz considered here. This is very similar to the result of [13] in the matter-coupled N = 2 gauged supergravity.
Using the coset representative (170), we obtain the scalar potential As in the previous case, a consistent set of BPS equations can be found only for θ = 0 and τ e W = κe U . With the three-form flux (60), which is manifestly invariant under SO(2) × SO (2), and the projectors given in (65), the resulting BPS equations read By choosing V = 2φ 1 , we obtain the solution with the integration constants C 1 and C 2 . This solution is just the SO(2)×SO(2) symmetric domain wall found in [26] with a dyonic profile for the three-form flux. In this case, coupling to SO(3) gauge fields is not possible due to the absence of any unbroken SO(3) gauge symmetry.

D. Uplifted solutions in ten and eleven dimensions
We now give the uplifted solutions in the case of SO(5) and CSO(4, 0, 1) which can be obtained from consistent truncations of eleven-dimensional supergravity on S 4 and type IIA theory on S 3 , respectively. As shown in [27], other gauge groups of the form CSO(p, q, 5 − p − q) with the embedding tensor in 15 representation can also be obtained from truncations of eleven-dimensional supergravity on H p,q • T 5−p−q . However, in this paper, we will not consider uplifted solutions for these gauge groups since the complete truncation ansatze have not been constructed so far. Furthermore, we will not consider uplifting solutions with non-vanishing vector fields since, in this case, the uplifted solutions are not very useful due to the lack of analytic solutions.

Uplift to eleven dimensions
We first consider uplifting the seven-dimensional solutions in SO (5) withμ i being coordinates on S 3 satisfyingμ iμi = 1. With the formulae given in appendix B, the eleven-dimensional metric and the four-form field strength are given by with dΩ 2 (3) = dμ i dμ i being the metric on a unit S 3 and ∆ = e 8φ cos 2 ξ + e −2φ sin 2 ξ, The SO (4) We now separately discuss the uplifted solutions for the two cases with φ 2 = 0 and φ 3 = 0.
We will also denote k 5 and l 5 simply by k and l with k 4 = tanh φ 2 k and l 4 = tanh φ 2 l. Recall also that for SO(3) symmetric solutions, we only have M 3 = AdS 3 .
All of these solutions should describe bound states of M2-and M5-branes with different transverse spaces and are expected to be holographically dual to conformal surface defects in N = (2, 0) SCFT in six dimensions. Solutions with SO(2) × SO(2) symmetry can similarly be uplifted, but we will not give them here due to their complexity.

Uplift to type IIA theory
We now carry out a similar analysis for solutions in CSO(4, 0, 1) gauge group to find uplifted solutions in ten-dimensional type IIA theory. Relevant formulae are reviewed in appendix B. In the solutions we will consider, gauge fields, massive three-forms and axions b i = χ i vanish. The ten-dimensional fields are then given only by the metric, the dilaton and the NS-NS two-form field. Therefore, in this case, we expect the solutions to describe bound states of NS5-branes and the fundamental strings.
We begin with a simpler SO(4) symmetric solution in which the SL(4)/SO(4) scalar matrix is given by M ij = δ ij . The ten-dimensional metric, NS-NS three-form flux and the dilaton are given by It should be noted that, in this case, we have a constant NS-NS flux.
For SO(3) symmetric solutions, we parametrize the SL(4)/SO(4) scalar matrix as and choose the S 3 coordinates to be µ i = (sin ξμ a , cos ξ), a = 1, 2, 3 withμ a being the coordinates on S 2 subject to the conditionμ aμa = 1. We again recall that only solutions with φ 2 = 0 are possible in this case.
The solutions for φ 0 and φ are obtained from φ 1 and φ 3 in section III B by the following These are obtained by comparing the scalar matrices obtained from (137) and (B10).

IV. SUPERSYMMETRIC SOLUTIONS FROM GAUGINGS IN 40 REPRESEN-TATION
In this section, we repeat the same analysis for gaugings from 40 representation. Setting Y M N = 0, we are left with the quadratic constraint Following [23], we can solve this constraint by taking with w M N = w (M N ) and v M being a five-dimensional vector.
The SL(5) symmetry can be used to fix the vector v M = δ M 5 . Therefore, it is useful to split the SL(5) index as M = (i, 5). Setting w 55 = w i5 = 0 for simplicity, we can use the remaining SL(4) ⊂ SL(5) symmetry to diagonalize w ij as The resulting gauge generators read corresponding to a CSO(p, q, r) gauge group with p + q + r = 4.
With the split of SL(5) index M = (i, 5) and the decomposition SL(5) → SL(4) × SO(1, 1), we can parametrize the SL(5)/SO(5) coset representative in term of the SL(4)/SO(4) one as V is the SL(4)/SO(4) coset representative, and t 0 , t i refer to SO(1, 1) and four nilpotent generators, respectively. The unimodular matrix M M N is then given by with M ij = ( V V T ) ij . Using (25), we can compute the scalar potential for these gaugings The presence of the dilaton prefactor e φ 0 shows that this potential does not admit any critical points. Note also that we can always consistently set the nilpotent scalars b i to zero for simplicity since they do not appear linearly in any terms in the Lagrangian.
We will use the same ansatz as in the case of gaugings in the 15 representation to find charged domain wall solutions. However, we note here that, for gaugings in the 40 representation, there are no massive three-form fields S M µνρ . The three-form fluxes given in (52) in this case correspond solely to the two-form fields B µνM . We now consider a number of possible solutions with different symmetries.

A. SO(4) symmetric charged domain walls
For SO(4) residual symmetry under which only the scalar field φ 0 is invariant, we have M ij = δ ij . The only gauge group that can accommodate the SO(4) unbroken symmetry is SO(4) with the embedding tensor component w ij = δ ij . The scalar potential as obtained from (213) takes a very simple form which does not admit any critical points. We will consider solutions with non-vanishing µνρ5 which is an SO(4) singlet.
In this SO(4) gauging, there are four massive two-form fields B µνi , i = 1, . . . , 4, and one massless two-form field B µν5 with the latter being an SO(4) singlet. We will take the ansatz for B µν5 as given in (60). With the following projection conditions the BPS equations are given by together with an algebraic constraint In this case, we find that θ is constant. Choosing V = 0, we find the following solution with an integration constant C. For a particular value of θ = 0, we find the solution

Coupling to SO(3) gauge fields
We now consider charged domain wall solutions with non-vanishing SO(3) ⊂ SO(4) gauge fields. In this case, the projector (Γ 5 ) a b ǫ b 0 = −ǫ a 0 implies that the non-vanishing gauge fields correspond to the self-dual SO(3) ⊂ SO(4) given by The two-form field strengths are straightforward to obtain Since the components of the embedding tensor Z ij,5 vanish, the two-form field B 5 does not contribute to the modified two-form field strengths. Imposing the projection conditions (125) and (215), we find the following BPS equations It can be verified that these BPS equations satisfy the second-order field equations without any additional constraint.
Since there is no an asymptotically locally AdS 7 configuration, we will consider flow solutions from a charged domain wall without vector fields given in (221)-(224) to a singular solution with non-vanishing gauge fields. To find numerical solutions, we will consider the charged domain wall with θ = 0 given in (225) for simplicity. As r → − 5C 2g , we impose the following boundary conditions with τ = κ. An example of the BPS flows is shown in figure 7. From this solution, it can be seen that k is constant along the flow since the above BPS equations give U ′ = 2φ ′ 0 which implies the constancy of U − 2φ 0 . It should also be noted that this solution is similar to that in CSO(4, 0, 1) gauge group given in figure 5. We also expect this solution to describe a surface defect within an N = (2, 0) nonconformal field theory.
We then obtain the scalar potential using (213) To find the BPS equations, we use the same ansatz for the modified three-form field strength (60) and impose the projection conditions (215). We note here that, in this case, there are two two-form fields, B in the BPS equations. We will accordingly restrict ourselves to the solutions with only B (2) 5 non-vanishing. Consistency with the field equations also leads to the conditions given in (142). With all these, the resulting BPS equations are given by Setting W = U and V = 0, we find the solutions for U, φ 0 , l and k as functions of φ in which C 0 is an integration constant.
The solution for φ(r) is given by for ρ = 0 and 4gρr(e 8φ − ρ) 1/5 = 5e 2C 1 + 32 for ρ = ±1. In the last equation, 2 F 1 is the hypergeometric function. This solution is again the domain wall found in [26] with a non-vanishing three-form flux.
As in the SO(3) symmetric solutions from the gaugings in the 15 representation, coupling to SO(3) vector fields does not lead to new solutions. Consistency with the field equations implies either vanishing two-form fields or vanishing gauge fields. We also note that repeating the same analysis for SO(2) × SO(2) and SO(2) symmetric solutions leads to the domain wall solutions given in [26] with a constant three-form flux We will not give further detail for these cases to avoid a repetition.

RESENTATIONS
In this section, we consider gaugings with both components of the embedding tensor in 15 and 40 representations non-vanishing. We first give a brief review of these gaugings as constructed in [23]. A particular basis can be chosen such that non-vanishing components of the embedding tensor are given by In terms of these components, the quadratic constraint (14) reads Y xy is chosen to be We will consider two gauge groups namely SO(2, 1) ⋉ R 4 and SO(2) ⋉ R 4 given in [23]. The latter can be obtained from Scherk-Schwarz reduction of the maximal gauged supergravity in eight dimensions.
We begin with the t = 3 case in which Y xy = diag(1, 1, −1) corresponding to SO(2, 1)⋉R 4 gauge group. The corresponding gauge generators are given by with λ z ∈ R and (t z ) x y = ǫ zyu Y ux being generators of SO(2, 1) in the adjoint representation.
The nilpotent generators Q The explicit form of ζ x can be given in terms of Pauli matrices as We now consider charged domain wall solutions with SO(2) ⊂ SO(2, 1) symmetry. As shown in [26], there are four SO(2) singlet scalars corresponding to the following non-compact generatorsȲ 1 = 2e 1,1 + 2e 2,2 + 2e 3,3 − 3e 4,4 − 3e 5,5 , The SL(5)/SO(5) coset representative can be written as The resulting scalar potential is given by which does not admit any critical points.
We now repeat the same analysis as in the previous sections. We first discuss the three- these two-forms transform as 1 + 2 + 2. Therefore, there is only one singlet two-form field under the SO(2) unbroken symmetry. In gauged supergravity, this two-form field will be gauged away by a three-form gauge transformation due to the non-vanishing component Y 33 of the embedding tensor. The SO(2) singlet is then described by a massive three-form field S 3 .
We will take the ansatz for the three-form field strength to be After imposing the following projection conditions we find the following BPS equations In these equations, we have imposed the conditions (142) for consistency.
By choosing V = 4φ 1 + 2φ 2 and taking W = U for convenience, we obtain a charged domain wall solution This is just the 1 4 -BPS domain wall obtained in [26] together with the running dyonic profile of the three-form flux. It is useful to emphasize here that this solution is 1 4 -supersymmetric. In general, domain wall solutions from gaugings in both 15 and 40 representations preserve only 1 4 of the original supersymmetry, see a general discussion in [33] and explicit solutions in [26]. From the above solution, we see that the solutions with a non-vanishing three-form flux do not break supersymmetry any further.
We end this section by giving a comment on the t = 2 case with SO(2) ⋉ R 4 gauge group.
Repeating the same procedure leads to a charged domain wall given by the solution found in [26] with a constant three-form flux given in (253). In contrast to the t = 3 case, the three-form flux H (3) 3 is due to the massless two-form field B (2) 3 since, in this case, we have Y 33 = 0. We will not give the full detail of this analysis here as it closely follows that of the previous cases. lifted to eleven-dimensional supergravity and type IIB theory as shown in [27] and [28].
We have performed only the uplift for solutions in SO(5) and CSO(4, 0, 1) gauge groups with SO(4) and SO(3) symmetries. In these cases, the complete truncation ansatze of eleven-dimensional supergravity on S 4 and type IIA theory on S 3 are known. Similar to the solutions in [12], the uplifted solutions in these two gauge groups should describe bound states of M2-and M5-branes and of F1-strings and NS5-branes, respectively. It is natural to extend this study by constructing the full truncation ansatze of eleven-dimensional supergravity on H p,q • T 5−p−q and type IIB theory on H p,q • T 4−p−q . These can be used to uplift the solutions in CSO(p, q, 5 − p − q) and CSO(p, q, 4 − p − q) gauge groups for any values of p and q leading to the full holographic interpretation of the seven-dimensional solutions found here.  The ansatz for the eleven-dimensional metric is given by with the coordinates µ M , M = 1, 2, 3, 4, 5, on S 4 satisfying µ M µ M = 1. T M N is a unimodular 5×5 symmetric matrix describing scalar fields in the SL(5)/SO(5) coset. The warped factor is defined by The ansatz for the four-form field strength readŝ In these equations, we have used the following definitions We have denoted the vector and massive three-form fields byÃ M N (1) andS M (3) to avoid confusion with those appearing in (22).
To find the identification between the seven-dimensional fields and parameters obtained from the S 4 truncation and those in seven-dimensional gauged supergravity of [23], we consider the kinetic terms of various fields and the scalar potential. After multiplied by 1 2 , the relevant terms in the seven-dimensional Lagrangian of [31] can be written as