Supersymmetric domain walls in maximal 6D gauged supergravity III

We continue our study of gaugings the maximal $N=(2,2)$ supergravity in six dimensions with gauge groups obtained from decomposing the embedding tensor under $\mathbb{R}^+\times SO(4,4)$ subgroup of the global symmetry $SO(5,5)$. Supersymmetry requires the embedding tensor to transform in $\mathbf{144}_c$ representation of $SO(5,5)$. Under $\mathbb{R}^+\times SO(4,4)$ subgroup, this leads to the embedding tensor in $(\mathbf{8}^{\pm 3}$, $\mathbf{8}^{\pm 1},\mathbf{56}^{\pm 1})$ representations. Gaugings in $\mathbf{8}^{\pm 3}$ representations lead to a translational gauge group $\mathbb{R}^8$ while gaugings in $\mathbf{8}^{\pm 1}$ representations give rise to gauge groups related to the scaling symmetry $\mathbb{R}^+$. On the other hand, the embedding tensor in $\mathbf{56}^{\pm 1}$ representations gives $CSO(4-p,p,1)\sim SO(4-p,p)\ltimes \mathbb{R}^4\subset SO(4,4)$ gauge groups with $p=0,1,2$. More interesting gauge groups can be obtained by turning on more than one representation of the embedding tensor subject to the quadratic constraints. In particular, we consider gaugings in both $\mathbf{56}^{-1}$ and $\mathbf{8}^{+3}$ representations giving rise to larger $SO(5-p,p)$ and $SO(4-p,p+1)$ gauge groups for $p=0,1,2$. In this case, we also give a number of half-supersymmetric domain wall solutions preserving different residual symmetries. The solutions for gaugings obtained only from $\mathbf{56}^{-1}$ representation are also included in these results when the $\mathbf{8}^{+3}$ part is accordingly turned off.


Introduction
Supersymmetric domain walls are solutions to gauged supergravities that play many important roles in various aspects of string/M-theory.These solutions have provided a useful tool for studying different aspects of the AdS/CFT correspondence since the beginning of the original proposal in [1], see also [2,3].They are also vital in the so-called DW/QFT correspondence [4,5,6], a generalization of the AdS/CFT correspondence to non-conformal field theories.In particular, these solutions give holographic descriptions to RG flows in strongly coupled dual field theories in various space-time dimensions.Domain walls also appear in the study of cosmology via the domain wall/cosmology correspondence, see for example [7,8,9].A systematic classification of supersymmetric domain walls from maximal gauged supergravity in various space-time dimensions has been performed in [10], and many domain wall solutions in gauged supergravities have been found in different space-time dimensions, see [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] for an incomplete list.
In this paper, we are interested in domain wall solutions from maximal N = (2, 2) six-dimensional gauged supergravity.The ungauged N = (2, 2) supergravity has been constructed in [29], and the first N = (2, 2) six-dimensional gauged supergravity with SO(5) gauge group has been found in [30] by performing an S 1 reduction of the maximal SO (5) gauged supergravity in seven dimensions [31].The most general N = (2, 2) gauged supergravity has been constructed in [32] using the embedding tensor formalism.The embedding tensor transforms in 144 c representation of SO (5,5) global symmetry as required by supersymmetry, and some possible gaugings classified under GL (5) and SO(4, 4) subgroups of SO (5,5) have also been identified in [32].Many gaugings arising from GL(5) decomposition together with a large number of supersymmetric domain wall solutions have been constructed recently in [17] and [18].
In this work, we will continue the study of the maximal N = (2, 2) gauged supergravity and the corresponding supersymmetric domain walls by considering gaugings arising from decomposing the embedding tensor under R + × SO(4, 4) ⊂ SO (5,5).Under R + × SO (4,4), the embedding tensor in 144 c representation of SO (5,5) decomposes into 8 ±1 , 8 ±3 , and 56 ±1 representations of R + × SO (4,4).We will determine explicit solutions of the embedding tensor giving rise to consistent gauge groups of the N = (2, 2) gauged supergravity and look for possible supersymmetric domain wall solutions.
According to the DW/QFT correspondence, the aforementioned domain wall solutions should be dual to maximally supersymmetric Yang-Mills theory in five dimensions.The latter plays an important role in defining the N = (2, 0) superconformal field theory in six dimensions compactified on S 1 and also describing nonperturbative dynamics of N = 1, 2, class S, theories in four dimensions, see for example [33,34,35,36,37,38,39].We expect domain wall solutions studied here could be useful in this context as well.
The paper is organized as follows.In section 2, we briefly review the construction of six-dimensional maximal gauged supergravity in the embedding tensor formalism.Possible gauge groups arising from decomposing the embedding tensor under SO (4,4) are determined in section 3.In section 4, we find supersymmetric domain wall solutions from gaugings in 56 −1 and 8 +3 representations.Conclusions and discussions are given in section 5. Relevant branching rules for SO (5,5) representations under SO (4,4) are given in the appendix.
2 N = (2, 2) gauged supergravity in six dimensions We first give a brief review of six-dimensional N = (2, 2) gauged supergravity in the embedding tensor formalism constructed in [32].We will only collect relevant formulae for determining possible gauge groups and finding supersymmetric domain wall solutions.For more details, we refer the reader to the original construction in [32].
The supergravity multiplet of the maximal N = (2, 2) supersymmetry in six dimensions consists of the following component fields The electric two-form potentials B µνm , appearing in the ungauged Lagrangian, transform as 5 under GL(5) while the vector fields A A µ transform as 16 c under SO (5,5).Together with the magnetic duals B µν m transforming in 5 representation of GL(5), the electric two-forms B µνm transform in a vector representation 10 of the full global symmetry group SO(5, 5) denoted by B µνM = (B µνm , B µν m ).Therefore, only the subgroup GL(5) ⊂ SO(5, 5) is a manifest off-shell symmetry of the theory.Indices M, N, . . .denote fundamental or vector representation of SO (5,5).Finally, there are 25 scalar fields parametrizing the coset space SO(5, 5)/SO(5) × SO (5).
In chiral spinor representation, the SO(5, 5)/SO(5) × SO(5) coset is described by a coset representative V A α β transforming under the global SO(5, 5) and local SO(5) × SO(5) symmetries by left and right multiplications, respectively.The inverse elements (V −1 ) α β A will be denoted by V A α β satisfying the relations On the other hand, in vector representation, the coset representative is given by a 10 × 10 matrix with A = (a, ȧ) and related to the coset representative in chiral spinor representation by the following relations In these equations, we have explicitly raised the SO(5) × SO(5) vector index A = (a, ȧ) resulting in a minus sign in equation (2.6).
The most general gaugings of six-dimensional N = (2, 2) supergravity are described by the embedding tensor in 144 c representation of SO (5,5).This can be written in terms of a vector-spinor of SO (5,5), θ AM , subject to the linear constraint (LC) The gauge covariant derivative is then given by with g being a gauge coupling constant and Θ A M N defined by (2.9) As usual, the embedding tensor identifies generators X A = Θ A M N t M N of the gauge group G 0 ⊂ SO(5, 5) with particular linear combinations of the SO(5, 5) generators t M N .Consistency also requires the gauge generators to form a closed subalgebra of SO (5,5) implying the quadratic constraint (QC) In terms of θ AM , this constraint reduces to the following two conditions (2.12) Any θ AM ∈ 144 c satisfying these quadratic constraints defines a consistent gauging.
In this work, we are only interested in classifying possible gauge groups and finding supersymmetric domain wall solutions involving only the metric and scalar fields.We have explicitly checked that the truncation of vector and tensor fields is consistent in all the domain wall solutions given in section 4.This follows from the fact that the corresponding Yang-Mills currents vanish for all of the solutions considered here.With all vector and tensor fields set to zero, the bosonic Lagrangian of the maximal N = (2, 2) gauged supergravity is given by while the supersymmetry transformations of fermionic fields read ) ) (2.17) The covariant derivatives of supersymmetry parameters are defined as with γµ = e μ µ γμ .γμ are space-time gamma matrices, and for simplicity, we will suppress all space-time spinor indices.
The scalar vielbein P a In these equations, Ω αβ and Ω α β are the U Sp(4) ∼ SO(5) symplectic forms that satisfy the following relations and similarly for Ω α β .The scalar potential is given by with the T-tensors defined by and We also note useful identities involving various components of the T-tensors 3 Gaugings of six-dimensional N = (2, 2) supergravity under SO (4,4) In this section, we will determine explicit forms of the embedding tensor for a number of possible gauge groups leading to consistent N = (2, 2) gauged supergravities in six dimensions.To find gauge groups by decomposing the embedding tensor under R + × SO(4, 4) ⊂ SO(5, 5), we decompose the SO(5, 5) vector index as M = (−, I, +) with I = 1, 2, . . ., 8 being the SO(4, 4) vector index.The SO(5, 5) generators t M N are decomposed accordingly as t M N = (t −+ = d, t +I = p I , t −I = k I , t IJ = τ IJ ) with d and τ IJ being R + and SO(4, 4) generators, respectively.
Similarly, the SO(5, 5) spinor index A will also be split as A = (m, ṁ) with m = 1, 2, . . ., 8 and ṁ = 1, 2, . . ., 8 being SO(4, 4) spinor indices, see more detail in the appendix.As given in the appendix, the embedding tensor transforming in 144 c representation of SO(5, 5) will split into the following representations under R + × SO(4, 4) For convenience, we also recall the identification of various components of the embedding tensor of the form The components θ m− 1 , θ ṁI 4 , and θ m+ ) and (θ n− 1 , θ ṁI 4 , ), as required by the LC, are defined in (A.43).For later convenience, we also repeat these relations here and In this section, we will determine explicit forms of the embedding tensor by imposing the quadratic constraint on the embedding tensor.With the above decomposition, the first condition of QC given in (2.11) reduces to with η IJ being the SO(4, 4) invariant tensor defined in (A.3).On the other hand, the second condition (2.12) splits into In these equations, c mn and c ṁ ṅ are elements of the SO(4, 4) charge conjugation matrix defined in (A.32), and (γ I ) m ṅ = (γ I ) ṅm are chirally decomposed SO(4, 4) gamma matrices given in (A.34)Some possible gauge groups under SO(4, 4) have also been discussed in [40], and it has been pointed out that turning on only θ m− need to be zero, respectively.Accordingly, we conclude that gaugings with only 8 +1 or 8 −1 components non-vanishing are not consistent.

Gaugings in 8 +3 representation
We begin with gauge groups arising from the embedding tensor in 8 +3 representation.In this case, we set all θ's components to be zero except for a spinor v ṁ.The θ AM matrix of the form makes the embedding tensor satisfy all the LC and QC.We have used the notation that all vanishing elements are left as blank spaces.
For A = (m, ṁ), the corresponding gauge generators split into X A = (X m , X ṁ).With the above embedding tensor, the last eight generators vanish, X ṁ = 0, while the first eight generators are given in terms of k I as They are all linearly independent and commute with each other [X m , X n ] = 0. Thus, the resulting gauge group is an eight-dimensional translational group R 8 associated with the k I generators.

Gaugings in 8 −3 representation
As in the previous case, we set all θ's components to be zero except for any spinor w m .All the LC and QC are satisfied by this embedding tensor.In this case, there are also eight non-vanishing gauge generators, but given in terms of the p I generators, i.e.
As in the previous case, they are all linearly independent and commute with each other, [X ṁ, X ṅ] = 0.This implies again that the resulting gauge group is an eight-dimensional translational group R 8 associated with the p I generators.

Gaugings in 56 −1 representation
We now move to gaugings in 56 −1 representation by choosing only θ mI 3 to be non-vanishing.The embedding tensor takes the form which is the same as the second condition in (A.45).For vanishing θ ṁ+ 6 component from 8 −1 representation, the embedding tensor is simply given by θ mI 3 .As in [40], all 56 components in θ mI 3 can be parametrized by an antisymmetric tensor The LC given in (3.15) is now identically satisfied, and the corresponding gauge generators are split into the following two sets The first set contains eight nilpotent generators that commute with each other, [X m , X n ] = 0, so they generate a translational subgroup associated with p I generators.The other set gives another subgroup embedded in the SO(4, 4) factor.According to [40], the QCs in terms of the antisymmetric tensor f ṁ ṅ ṗ can be written as We will discuss some possible solutions to these conditions.
We first consider a simple solution of the form for i, j, . . .= 1, 2, 3 and r, s, . . .= 5, 6, 7. To solve the first condition in (3.19), we need to impose the relation κ 1 = ±κ 2 .We will choose With this form of the embedding tensor, we find that the gauge generators X 4 and X 8 vanish.Commutation relations between X i and X r lead directly to The remaining eight generators correspond to translational generators, but in this case, there are four constraints among them Therefore, there are only four linearly independent translational generators.
To make the form of the resulting gauge group explicit, we redefine the gauge generators as follows.We first introduce the SO(4) ∼ SO(3) × SO (3) generators M μν = −M ν μ for μ, ν = 1, 2, 3, 4.These satisfy the standard SO(4) algebra of the form In terms of X i and X r generators, we find, for μ = (i, 4), with Furthermore, we redefine the four independent translational generators as and obtain the following commutation relations This implies that the gauge group takes the form of SO(4) ⋉ R 4 ∼ CSO(4, 0, 1). (3.28) There is another solution to the conditions (3.19) with the antisymmetric tensor f ṁ ṅ ṗ of the form for i, j, . . .= 1, 2, 3 and r, s, . . .= 5, 6, 7.As in the previous case, there are eight nilpotent generators X m subject to four constraints given in (3.22) together with six non-vanishing gauge generators X i and X r .The latter satisfy the following commutation relations ).These relations correspond to an SO(3, 1) algebra which can be explicitly seen by defining These generators satisfy the SO(3, 1) algebra Redefining the four linearly independent translational generators as we find the commutation relations with the SO(3, 1) generators Accordingly, the resulting gauge group is given by with ī, j, ... = 1, 6, 7 and r, s, ... = 2, 3, 5 being two sets of indices that can be raised and lowered by η īj = ηīj = diag(−1, 1, 1) and η rs = η rs = diag(1, 1, −1), respectively.There are again eight nilpotent generators X m subject to four constraints given in (3.22).We will choose the independent generators to be .37)Commutation relations between the remaining six non-vanishing gauge generators Xī and X r are given by These lead to SO(2, 1) × SO(2, 1) ∼ SO(2, 2) algebra.As in the previous cases, we can also redefine the generators as for i, j, . . .= 1, 2, 3 and These generators satisfy the algebra of the form given in (3.32) but with η μν = diag(1, 1, −1, −1).
Finally, the commutation relations between these SO(2, 2) generators and the four nilpotent generators given in (3.37) are the same as in (3.34) for η μν = diag(1, 1, −1, −1).Consequently, the resulting gauge group is given by  We can repeat the same analysis as in the case of gaugings from 56 −1 representation by solving these QCs and find the same CSO(4 − p, p, 1) ∼ SO(4 − p, p) ⋉ R 4 gauge group for p = 0, 1, 2. However, the gauge generators for the two sets of nilpotent and SO(4 − p, p) generators are interchanged as with the nilpotent generators given in terms of the k I instead of p I .

Gaugings in 56 −1 and 8 +3 representations
As an example for gaugings from an embedding tensor with more than one representation, we consider gaugings with both 56 −1 and 8 +3 representations.The embedding tensor takes the form Only θ mI 3 is constrained by the LC given in equation (3.15).By a similar analysis as in the previous cases, we can solve this condition by parametrizing θ mI 3 as in equation (3.16).Denoting θ ṁ− 2 = v ṁ, we find that the QCs lead to the following four conditions, in accordance with the analysis of [40], in which | SD means the self-dual part of a four-form.
The first two conditions are independent of v ṁ, and can be solved by the antisymmetric tensor f ṁ ṅ ṗ given in the case of gaugings from 56 −1 representation.We then consider the following three possibilities: for a real constant λ.With all these, the resulting gauge generators are given by It turns out that by a suitable redefinition of X ṁ generators, these generators can be shown to satisfy SO(4 − p, p) algebra 49), and η μν = diag(1, 1, −1, −1) for f ṁ ṅ ṗ in (3.50).In addition, there are four constraints among X m generators, given by (3.22), implying that only four generators are linearly independent.Choosing these four generators as in section 3.3, we find the following commutation relations and defining the generators we obtain the algebra, with µ = (0, μ) = 0, 1, 2, 3, 4, for η µν = diag(±1, η μν ).The two sign choices correspond to SO(5 − p, p) or SO(4 − p, p + 1) gauge groups.In particular, the SO(4) group with η μν = δ μν is enlarged to SO(5) or SO(4, 1) I .Similarly, the SO(3, 1) group with η μν = diag(1, 1, 1, −1) is enlarged to SO(4, 1) II or SO(3, 2) I while the SO(2, 2) group, with η μν = diag(1, 1, −1, −1), becomes SO(3, 2) II .We have used subscripts I and II to distinguish the gauge groups arising from different SO(4 − p, p) groups obtained from the embedding tensor in 56 −1 representation.We finally note that the analysis for gaugings from 56 +1 and 8 −3 representations can be carried out in the same way leading to the same gauge groups with the role of X m and X ṁ interchanged.

Supersymmetric domain wall solutions
In this section, we find supersymmetric domain walls which are half-supersymmetric vacuum solutions of the maximal gauged supergravities considered in the previous section.We take the space-time metric to be the standard domain wall ansatz where μ, ν, . . .are space-time indices of five-dimensional Minkowski space, and A(r) is a warp factor depending only on the radial coordinate r.
Following [17] and [18], the coset representative of SO(5, 5)/SO(5) × SO(5), parametrized by 25 scalar fields, can be obtained by the following noncompact generators of SO(5, 5) in diagonal basis where M A M = (M a M , M ȧM ) is the inverse of the transformation matrix M given in (A.24).These non-compact generators are symmetric such that ( ta ḃ) T = ta ḃ.
The split of indices a and ȧ given above implies the branching 5 → 1 + 4 of SO(5) → SO(4).Therefore, under SO(4) × SO(4) ⊂ SO(5) × SO(5), the scalars tranform as (5, 5) With this form of the scalar fields, we can rewrite the kinetic terms of the scalar fields in (2.13) and obtain the following form of the bosonic Lagrangian We will find supersymmetric domain wall solutions from first-order Bogomol'nyi-Prasad-Sommerfield (BPS) equations derived from the supersymmetry transformations of fermionic fields.The procedure is essentially the same as that given in [17] and [18], so we will mainly state the final results.The variations of the gravitini in (2.14) and (2.15), δψ +μα and δψ −μ α respectively gives Throughout the paper, we use the notation ′ to denote an r-derivative.Multiply equation (4.8) by A ′ γr and use equation (4.9) or vice-versa, we find the following consistency conditions in which we have introduced the "superpotential" W. We then obtain the BPS equations for the warp factor With this result, equations (4.8) and (4.9) lead to the following (not independent) projectors on the Killing spinors Using these projectors in the variations δχ +a α and δχ − ȧα together with some identities involving the T-tensors, in particular (2.29), we may rewrite the BPS equations for scalar fields in the form in which G IJ is the inverse of the scalar metric G IJ .The remaining variations δψ +rα and δψ −r α determine the r dependence of the Killing spinors.
In addition, we also note that the scalar potential can be written in terms of W as It is also straightforward to show that the BPS equations of the form (4.12) and (4.14) satisfy the second-order field equations derived from the bosonic Lagrangian (4.7) with the scalar potential given by (4.15), see [41,42,43,44,45,46] for more detail.Finally, since we have imposed only one independent projector on the Killing spinors, all solutions found in this work are half-supersymmetric.

SO(4) symmetric domain walls
To deal with the 25-dimensional scalar manifold, we will follow the approach introduced in [47] by considering domain wall solutions that are invariant under a particular subgroup of the gauge groups.These solutions involve only a subset of all 25 scalars.We also restrict ourselves to only gauge groups obtained from the embedding tensor in 56 −1 and 8 +3 representations.
Using the coset representative of the form we find the superpotential and the scalar potential of the form It can be checked that the scalar potential can be written in terms of the superpotential according to (4.15) using the scalar matrix G IJ = diag( 1 2 , 1 8 ) for Φ I = {φ, ϕ} and I = 1, 2. The general analysis given above leads to the BPS equation for the warp factor together with the BPS equations for the scalar fields For σ = 1, −1 corresponding to SO(5) and SO(4, 1) I gauge groups, the solutions for the warp factor A and dilaton φ can be given in terms of ϕ as in which C 1 and C 2 are integration constants.To obtain the solution for ϕ, we change the radial coordinate r to ρ defined by dρ dr = e −3φ−4ϕ .The solution of ϕ is then readily found to be in which ± directly corresponds to the upper/lower signs in the BPS equations.Thus, the two sign choices in the BPS equations can be absorbed by flipping the sign of the radial coordinate.We will neglect these sign choices by choosing the upper sign of the BPS equations from now on.Moreover, the integration constants C 1 and C 3 can also be removed by rescaling the coordinates x μ and shifting the radial coordinate ρ.
For σ = 0 corresponding to CSO(4, 0, 1) gauge group, the superpotential and scalar potential are independent of ϕ and the BPS equations reduce to All of these equations can be readily solved to obtain the solution In this case, we can consistently truncate out the SO(4) invariant scalar ϕ since the scalar potential is independent of ϕ.
For SO(4, 1) II gauge group, the SO(4) compact subgroup is not embedded in SO(4, 4) since it involves M 0 ĩ generators obtained from the gauge generators X m .However, a similar analysis can be carried out by using f ṁ ṅ ṗ given in (3.49) together with v 4 = v 8 = κ 4 √ 2 .In this case, there are again two SO(4) singlet scalars corresponding to the non-compact generators Using the coset representative we find the same form of the domain wall solution as given in (4.23)-(4.25)with σ = −3, φ = −ϕ 1 , and ϕ = −ϕ 2 .
The residual symmetry SO(2) × SO( 2) is embedded in SO(4) as 4 → (1, 2)+(2, 1) with 2 denoting the fundamental or vector representation of SO (2).As in the previous cases, decomposing the transformation of all 25 scalar fields under SO(2) × SO(2) gives (4.48) Accordingly, there are five singles corresponding to the R + generator d and the following four non-compact generators leading to the coset representative of the form In this case, consistency between the BPS equations and field equations requires vanishing ϕ 3 and ϕ 4 .With ϕ 3 = ϕ 4 = 0, the superpotential is given by and the scalar potential takes the form The parameters u = ±1 and σ = ±1, 0 correspond to different gauge groups; SO

.56)
Defining a new radial coordinate ρ by dρ dr = e φ−2(ϕ 1 +ϕ 2 ) , we find a domain wall solution ) , (4.57) ) ) , (4.58) The coset representative can be written as It turns out that the resulting T-tensor, superpotential, and scalar potential are highly complicated.Accordingly, we will look for some subtruncations to simplify the analysis but still obtain interesting results.One possibility is to impose the conditions ϕ 3 = ϕ 4 = 0 together with ϕ 8 = ϕ 7 and ϕ 6 = −ϕ 5 .We have checked that these indeed lead to a consistent subtruncation.

Conclusions and discussions
We have constructed the embedding tensors of six-dimensional maximal N = (2, 2) gauged supergravity for various gauge groups arising from the decomposition of the embedding tensor under R + × SO(4, 4) ⊂ SO(5, 5) symmetry.Under this decomposition, viable gauge groups can be determined from the embedding tensor in 8 ±1 , 8 ±3 , and 56 ±1 representations.We have pointed out that gaugings in 8 ±1 representation without 56 ±1 is not consistent due to the linear constraint, and gaugings in 8 ±3 representation only lead to a translational gauge group R 8 .On the other hand, gaugings in 56 ±1 representation give rise to CSO(4 − p, p, 1) ∼ SO(4 − p, p) ⋉ R 4 gauge groups with p = 0, 1, 2. Including 8 ±3 representation to 56 ∓1 can enlarge the gauge groups to SO(4 − p, p + 1) or SO(5 − p, p).We have also found a number of half-supersymmetric domain wall solutions from gaugings in 56 −1 and 8 +3 representations with various residual symmetries.The corresponding solutions for gaugings solely from 56 −1 representation can be straightforwardly obtained from these results by turning off the 8 +3 part.
As pointed out in [32], some of the gaugings under SO(4, 4) decomposition could be truncated to gaugings of half-maximal N = (1, 1) supergravity coupled to four vector multiplets in which supersymmetric AdS 6 vacua are known to exist [48,49,50].It would be interesting to explicitly truncate the results given here to half-maximal gauged supergravity and obtain new gaugings as well as new supersymmetric AdS 6 vacua.Along this direction, a classification of gauge groups with known eleven-dimensional origins, arising from truncating the maximal theory to half-maximal one, has been given in [40].It could also be interesting to extend this analysis to many new gaugings identified in this paper.
On the other hand, uplifting the six-dimensional gauged supergravity and the corresponding domain wall solutions in this work and in [17,18] to higher dimensions could also be worth considering.This could be done by constructing truncation ansatze of string/M-theory to six dimensions using SO(5, 5) exceptional field theory given in [51,52] and would lead to interesting holographic descriptions of maximal super Yang-Mills theory in five dimensions.Finally, finding a holographic interpretation of the domain wall solutions given in this paper could also be of particular interest.This could be done along the line of [53] and [54] in which a holographic description from a simple domain wall found in [30] with SO(5) symmetry has been studied.

I
P a ȧJ being a symmetric scalar metric.The vielbein P a ȧ I on the scalar manifold is related to P a ȧ µ via P a ȧ µ = P a ȧ I ∂ µ Φ I .