Supersymmetric domain walls in maximal 6D gauged supergravity I

We find a large class of supersymmetric domain wall solutions from six-dimensional N=(2,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=(2,2)$$\end{document} gauged supergravity with various gauge groups. In general, the embedding tensor lives in 144c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {144}}}_c$$\end{document} representation of the global symmetry SO(5, 5). We explicitly construct the embedding tensors in 15-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {15}}}^{-1}$$\end{document} and 40¯-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\mathbf {40}}}^{-1}$$\end{document} representations of GL(5)∼R+×SL(5)⊂SO(5,5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$GL(5)\sim {\mathbb {R}}^+\times SL(5)\subset SO(5,5)$$\end{document} leading to CSO(p,q,5-p-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,5-p-q)$$\end{document} and CSO(p,q,4-p-q)⋉Rs4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,4-p-q)\ltimes {\mathbb {R}}^4_{{\varvec{s}}}$$\end{document} gauge groups, respectively. These gaugings can be obtained from S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^1$$\end{document} reductions of seven-dimensional gauged supergravity with CSO(p,q,5-p-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,5-p-q)$$\end{document} and CSO(p,q,4-p-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,4-p-q)$$\end{document} gauge groups. As in seven dimensions, we find half-supersymmetric domain walls for purely magnetic or purely electric gaugings with the embedding tensors in 15-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {15}}}^{-1}$$\end{document} or 40¯-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\mathbf {40}}}^{-1}$$\end{document} representations, respectively. In addition, for dyonic gauge groups with the embedding tensors in both 15-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {15}}}^{-1}$$\end{document} and 40¯-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\mathbf {40}}}^{-1}$$\end{document} representations, the domain walls turn out to be 14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{4}$$\end{document}-supersymmetric as in the seven-dimensional analogue. By the DW/QFT duality, these solutions are dual to maximal and half-maximal super Yang–Mills theories in five dimensions. All of the solutions can be uplifted to seven dimensions and further embedded in type IIB or M-theories by the well-known consistent truncation of the seven-dimensional N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} gauged supergravity.


Introduction
Supersymmetric domain walls in gauged supergravities in various space-time dimensions have provided a useful tool for studying various aspects of the AdS/CFT correspondence since the original proposal in [1], see also [2,3]. In particular, these solutions play an important role in the so-called DW/QFT correspondence [4,5,6], a generalization of the AdS/CFT correspondence to non-conformal field theories. They are also useful in studying some aspects of cosmology, see for example [7,8,9]. Due to their importance in many areas of applications, many domain wall solutions in gauged supergravities have been found in different spacetime dimensions [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. A systematic classification of supersymmetric domain walls from maximal gauged supergravity in various space-time dimensions can also be found in [26].
In this paper, we are interested in maximal N = (2, 2) six-dimensional gauged supergravity with SO(5, 5) global symmetry. Compared to other dimensions, supersymmetric solutions to this six-dimensional gauged supergravity have not been systematically studied since the original construction of the ungauged N = (2, 2) supergravity long ago in [27]. The first N = (2, 2) six-dimensional gauged supergravity with SO(5) gauge group has been constructed in [28] by performing an S 1 reduction of the SO(5) maximal gauged supergravity in seven dimensions [29]. More recently, the most general gaugings have been constructed and classified in [30] using the embedding tensor formalism. From the results of [30], there are two particularly interesting classes of gaugings under GL (5) and SO(4, 4) subgroups of SO (5,5). The former contains gaugings obtained from an S 1 reduction of seven-dimensional maximal gauged supergravity while the latter can be truncated to half-maximal N = (1, 1) gauged supergravity.
We will consider only gaugings in the first class with the embedding tensor in 15 −1 and 40 −1 representations of GL (5). These gaugings have known sevendimensional origins via an S 1 reduction and can also be embedded in string/Mtheory using the truncations to maximal gauged supergravity in seven dimensions. The fact that there does not exist an N = 4 superconformal symmetry in five dimensions [31] is in agreement with the recent classification of maximally supersymmetric AdS vacua given in [32]. This implies that there is no AdS 6 /CFT 5 duality in the case of 32 supercharges. Therefore, maximally supersymmetric vacuum solutions of the N = (2, 2) gauged supergravity are expected to be halfsupersymmetric domain walls. In this work, we will systematically study this type of solutions and give a large number of them including 1 4 -supersymmetric solutions.
It has been shown recently that maximally supersymmetric Yang-Mills theory in five dimensions plays an important role in the dynamics of (conformal) field theories in both higher and lower dimensions via a number of dualities, see for example [33,34,35,36,37,38]. In particular, this theory could even be used to define the less known N = (2, 0) superconformal field theory in six dimensions compactified on S 1 . The latter is well-known to describe the dynamics of strongly coupled theory on M5-branes. Accordingly, we expect that supersymmetric domain walls of the maximal gauged supergravity in six dimensions could be useful in studying various aspects of the maximal super Yang-Mills theory in five dimensions via the DW/QFT correspondence. A simple domain wall solution with SO(5) symmetry has already been given in [28] for SO (5) gauging, see [39] and [40] for the holographic interpretation of this solution. In this paper, we extend this study by including a large class of supersymmetric domain walls with different unbroken symmetries in N = (2, 2) gauged supergravity with various gauge groups.
The paper is organized as follows. In section 2, the construction of sixdimensional maximal gauged supergravity in the embedding tensor formalism is 2 N = (2, 2) gauged supergravity in six dimensions We begin by giving a brief review of six-dimensional N = (2, 2) gauged supergravity in the embedding tensor formalism constructed in [30]. We will mainly collect relevant formulae for constructing the embedding tensor and finding supersymmetric domain wall solutions. For more details, we refer the reader to the original construction in [30]. As in other dimensions, N = (2, 2) maximal supersymmetry in six dimensions allows only a unique graviton supermultiplet with the following field content eμ µ , B µνm , A A µ , V A αα , ψ +µα , ψ −µα , χ +aα , χ −ȧα . (2.1) Most of the conventions are the same as in [30].  5. We use ± to indicate space-time chiralities of the spinors. Under the local SO(5) × SO (5) symmetry, the two sets of gravitini ψ +µα and ψ −µα transform as (4,1) and (1,4) while the spin- 1 2 fields χ +aα and χ −ȧα transform as (5,4) and (4,5). In ungauged supergravity, only the electric two-forms B µνm appear in the Lagrangian while the magnetic duals B µν m transforming in 5 representation of GL(5) are introduced on-shell. The electric and magnetic two-forms are combined into 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. On the other hand, the full SO (5,5) duality symmetry is the on-shell symmetry interchanging field equations and Bianchi identities of the two-form potentials. However, the most general gaugings of the ungauged supergravity can involve a symmetry that is not a subgroup of the off-shell GL(5) symmetry. Moreover, the magnetic two-forms can also appear in the gauged Lagrangian via topological terms.
In N = (2, 2) supergravity, there are 25 scalar fields parametrizing the coset space SO(5, 5)/ (SO(5) × SO(5)). In chiral spinor representation, we can describe the coset manifold by a coset representative V A αβ transforming under the global SO(5, 5) and local SO(5) × SO(5) by left and right multiplications, respectively. The inverse elements (V −1 ) αβ A will be denoted by V A αβ satisfying the relations In vector representation, the coset representative is given by a 10 × 10 matrix V M A = (V M a , V Mȧ ) with A = (a,ȧ). This is related to the coset representative in chiral spinor representation by the following relations In these equations, (Γ M ) AB and (Γ A ) αα ββ = ((γ a ) αα ββ , (γȧ) αα ββ ) are respectively SO(5, 5) gamma matrices in non-diagonal η M N and diagonal η AB bases, see appendix A.3 for more detail.
The inverse will be denoted by V M A satisfying 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 can be efficiently described by using the embedding tensor Θ A M N . This tensor introduces the minimal coupling of various fields via the covariant derivative where g is a gauge coupling constant. 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 . Supersymmetry requires the embedding tensor to transform as 144 c representation of SO (5,5). Accordingly, Θ A M N can be parametrized in term of a vector-spinor θ AM of SO(5, 5) as With the SO(5, 5) generators in vector and spinor representations given by in which η M N is the off-diagonal SO(5, 5) invariant tensor given in (A.1), the corresponding gauge generators take the forms For consistency, the gauge generators must form a closed subalgebra of SO (5,5), so the embedding tensor needs to satisfy the quadratic constraint In terms of θ AM , the quadratic constraint reduces to the following two conditions It follows that any θ AM ∈ 144 c satisfying this quadratic constraint defines a consistent gauging of the theory. To identify possible gaugings, we first decompose θ AM under a given subgroup of SO(5, 5). As pointed out before, the GL(5) subgroup of SO(5, 5) is of particular interest since this is the symmetry of the ungauged Lagrangian. As given in [30], θ AM ∈ 144 c decomposes under GL(5) ⊂ SO (5,5) as (2.14) The explicit form of all the seven irreducible components can be found in appendix A.4. In this case, determining consistent gaugings is to find the irreducible components satisfying the quadratic constraint (2.13). By decomposing the SO(5, 5) vector index under GL(5), we can write θ AM = (θ Am , θ A m ) with θ Am and θ A m containing the following irreducible components It is easily seen that the first equation in (2.13) is automatically satisfied for purely electric or purely magnetic gaugings that involve only θ Am or θ A m components. We note that as pointed out in [30], gaugings triggered by θ Am are electric in the sense that only electric two-forms participate in the resulting gauged theory while magnetic gaugings triggered by θ A m involve magnetic two-forms together with additional three-form tensor fields. Comparing (2.15) and (2.16) to (2.14), we immediately see that gaugings in 24 −5 ⊕ 40 −1 and 5 +7 ⊕ 15 −1 ⊕ 45 +3 representations are respectively purely electric and purely magnetic whereas those in 5 +3 ⊕ 10 −1 representation correspond to dyonic gaugings involving both electric and magnetic two-forms. Other dyonic gaugings can also arise from combinations of various electric and magnetic components leading to many possible gauge groups. Apart from the minimal coupling implemented by the covariant derivative (2.7), gaugings also lead to hierarchies of non-abelian vector and tensor fields of various ranks. However, since we are only interested in domain wall solutions which only involve the metric and scalar fields, we will, from now on, set all vector and tensor fields to zero. It is straightforward to verify that this is indeed a consistent truncation. With only the metric and scalars non-vanishing, the bosonic Lagrangian of the maximal N = (2, 2) gauged supergravity takes the form and supersymmetry transformations of fermionic fields are given by The covariant derivatives of supersymmetry parameters, +α and −α , are defined by withγ µ = eμ µγμ . Matricesγμ are space-time gamma matrices, see the convention in appendix B. For simplicity, we will suppress all space-time spinor indices. The scalar vielbein P aȧ µ and SO(5) × SO(5) composite connections, Q ab µ and Q˙a˙b µ , are given by in which Ω αβ and Ωαβ are the two U Sp(4) symplectic forms whose explicit forms can be found in (A.23). These definitions can be derived from the following relation The scalar potential is given by where we have introduced the T-tensors defined by 3 Supersymmetric domain walls from gaugings in 15 −1 representation In this section, we consider gauge groups arising from the embedding tensor in 15 −1 representation. These are purely magnetic gaugings with the corresponding embedding tensor given by The matrix T An is the inverse of the transformation matrix T An given in (A.59) and Y mn is a symmetric 5 × 5 matrix. As previously mentioned, for θ Am = 0, the embedding tensor θ AM = ( 0 , T An Y nm ) automatically satisfies the quadratic constraint (2.13). Therefore, every symmetric tensor Y mn defines a viable gauging in 15 −1 representation. As in [41], we can use SL(5) ⊂ GL(5) symmetry to bring Y mn to the form where p + q + r = 5. Under GL (5), the gauge generators transforming as a spinor 16 s of SO(5, 5) decompose as follows For the embedding tensor in 15 −1 representation, the only non-vanishing gauge generators are given by with t m n being GL(5) generators. In vector representation, the explicit form of X mn is given by These generators satisfy the commutation relations [X mn , X pq ] = (X mn ) pq rs X rs (3.6) in which (X mn ) pq . Therefore, the corresponding gauge group is determined to be These gaugings arise from an S 1 reduction of seven-dimensional maximal gauged supergravity with the same gauge groups. In the case of SO(5) gauge group (p = 5 and q = r = 0), the complete reduction ansatz has already been constructed in [28].

Supersymmetric domain walls
In order to find supersymmetric domain wall solutions, we take the space-time metric to be the standard domain wall ansatz ds 2 6 = e 2A(r) ημνdxμdxν + dr 2 (3.8) whereμ,ν = 0, 1, . . . , 4, and A(r) is a warped factor depending only on the radial coordinate r. To parametrize the coset representative of SO(5, 5)/(SO(5)× SO(5)), we first identify the corresponding non-compact generators of SO (5,5) in the basis with diagonal SO(5, 5) metric η AB . These are given bŷ is the inverse of the transformation matrix M given in (A.50).
We then split these generators into two parts that are symmetric and antisymmetric in a andḃ indices as followŝ It is now straightforward to check that symmetric generatorst (3.12) We can also separate the trace part oft + aḃ , corresponding to the dilaton scalar field ϕ in GL(5)/SO(5) ∼ R + × SL(5)/SO(5) scalar coset. This generator is the R + ∼ SO(1, 1) generator defined in (A.4). In terms oft + aḃ , this is given by d =t (3.13) The remaining generators can be identified as the fourteen non-compact generators corresponding to scalar fields {φ 1 , ..., φ 14 } in the SL(5)/SO(5) coset. These generators are given by the symmetric traceless part satisfying δ aḃt aḃ = 0. The other ten scalars denoted by {ς 1 , ..., ς 10 } correspond to the shift generators s mn . These will be called the axions or shift scalars in this work. The decomposition in equation (3.12) is in agreement with that in [28] in which the consistent circle reduction of seven-dimensional SO(5) gauged supergravity giving rise to SO(5) gauged theory in six dimensions is performed. From a higherdimensional perspective, the fourteen scalars are the seven-dimensional scalars parameterizing the SL(5)/SO(5) coset in seven dimensions while the dilaton and shift scalars descend from the reduction of seven-dimensional metric and vector fields, respectively, see appendix C for more detail.
In order to find supersymmetric solutions, we consider first-order BPS equations derived from the supersymmetry transformations of fermionic fields in the background with vanishing fermionic fields. In this section, we only discuss a general structure of the procedure leaving a more detailed analysis and explicit results in subsequent sections. We begin with the variations of the gravitini which are given by In these equations, we have used the notation A = dA dr . We will use a prime to denote an r-derivative throughout the paper.
Multiply the first equation by A γ r and use the second equation or viceversa, we find the following consistency conditions

18)
in which we have introduced the "superpotential" W. We then obtain the BPS equations for the warped factor Using this result in equations (3.16) and (3.17) leads to the following projectors on the Killing spinorŝ satisfying P αα Pα β = δ α β and Pα α P αβ = δαβ. The conditions δψ +rα = 0 and δψ −rα = 0 determine the Killing spinors as functions of the radial coordinate r as usual.
Using these projectors in δχ +aα = 0 and δχ −ȧα = 0 equations, we eventually obtain the BPS equations for scalars. These equations are of the form in which G IJ is the inverse of the scalar metric G IJ defined in (3.15).
In addition, the scalar potential can also be written in term of W as It is well-known that the BPS equations of the form (3.20) and (3.23) satisfy the second-order field equations derived from the bosonic Lagrangian (3.15) with the scalar potential given by (3.24), see [42,43,44,45,46,47] for more detail.
As in other dimensions, we will follow the approach introduced in [48] to explicitly find supersymmetric domain wall solutions involving only a subset of the 25 scalars that is invariant under a particular subgroup H 0 ⊂ G 0 to make the analysis more traceable.

SO(5) symmetric domain walls
We first consider supersymmetric domain walls with the maximal unbroken symmetry SO(5) ⊂ CSO(p, q, 5 − p − q). The only gauge group containing SO(5) as a subgroup is SO(5) with Y mn = δ mn . In this case, only the dilaton ϕ corresponding to the non-compact generator (3.13) is invariant under SO(5). Thus, the coset representative can be written as (3.25) We recall that this coset representative is a 16×16 matrix with an index structure V A B . To compute the T-tensor, we need to write the SO(5) × SO(5) index as a pair of SO(5) spinor indices resulting in the coset representative of the form V A αα . To achieve this, we use the transformation matrices p introduced in (A.35) so that V A αα and its inverse V A αα are given by With all these, it is now straightforward to find the T-tensor from which the superpotential is given by The resulting scalar potential reads which does not admit any stationary points. The general analysis given above leads to the BPS equation for the warped factor 3.30) and the following projectorγ For definiteness, we have chosen a particular sign choice in the A equation and theγ r projector. The condition δψ ±r = 0 gives the standard solution for the Killing spinors with the constant spinors 0 ± satisfyingγ r 0 ± = 0 ∓ . Accordingly, the solution is half-supersymmetric.
The BPS equation for the dilaton can be found from the condition δχ ± = 0 with the projector (3.31). This results in a simple equation All of these equations can be readily solved to obtain the solution The integration constant C can be removed by shifting the radial coordinate r. We have also neglected an additive integration constant for A since it can be absorbed by rescaling the coordinates xμ. This is the SO(5) domain wall originally found in [28]. In order to recover the same form of the solution, we redefine the radial coordinate as r → 4 and set ϕ = 1 2 √ 10 σ.

SO(4) symmetric domain walls
We now look for more complicated solutions with SO(4) symmetry. The gauge groups that contain SO(4) as a subgroup are SO(5), SO(4, 1), and CSO(4, 0, 1). To incorporate all of these gauge groups within a single framework, we write the embedding tensor in the form with κ = 1, 0, −1 corresponding to SO(5), CSO(4, 0, 1), and SO(4, 1) gauge groups, respectively. There are two SO(4) singlet scalars. The first one is the dilaton corresponding to the non-compact generator (3.13), and the other one comes from the SL(5)/SO(5) coset corresponding to the non-compact generator (3.36) Using the coset representative we find that the T-tensor is given by This leads to the superpotential and the scalar potential of the form Using the projector (3.31), we find the BPS equations The resulting solutions for the dilation ϕ and the warped factor A as functions of φ are given by To obtain the solution for φ, we change r to a new radial coordinate ρ defined by dρ dr = e ϕ+6φ . The solution for φ is then given by for an integration constant C 1 . It is useful to note that for κ = −1, the solution for φ can be written as e 10φ = tan √ 2gρ + C 1 . For κ = 0, the solution is simply given by

SO(3) × SO(2) symmetric domain walls
We now consider SO(3) × SO(2) residual symmetry, which is possible only for SO(5) and SO(3, 2) gauge groups. In this case, we write the embedding tensor as with κ = 1 and κ = −1 corresponding to SO(5) and SO(3, 2), respectively. The SO(3) × SO(2) symmetry is generated by X ij , i, j = 1, 2, 3, and X 45 . There are three singlet scalars corresponding to the dilaton and the following non-compact generators With the coset representative we find the scalar potential The superpotential reads which can be found from the T-tensor given by In this case, it turns out that consistency of the supersymmetry conditions from δχ ± requires ς = 0. Therefore, in order to find a consistent set of BPS equations, we need to truncate the axion out. With ς = 0, the superpotential is given by With the projector (3.31), we find the following BPS equations It can be verified that all these equations satisfy the corresponding field equations as expected.
With a new radial coordinate ρ given by dρ dr = e ϕ+2φ , we obtain the domain wall solution

SO(3) symmetric domain walls
We now move to domain wall solutions with SO(3) symmetry. Many gauge groups contain SO(3) as a subgroup with the embedding tensor parameterized by for κ, λ = 0, ±1. With this embedding tensor, the SO(3) symmetry is generated by X mn , m, n = 1, 2, 3. In addition to the dilaton, there are four singlet scalars corresponding to the following non-compact generators With the only exception for κ = λ = 0 corresponding to CSO(3, 0, 2) gauge group, we need to truncate out the scalar corresponding to s 45 generator in order to find a consistent set of BPS equations as in the previous case. For the moment, we will set this shift scalar to zero and consider the particular case of κ = λ = 0 afterward. For vanishing shift scalars, the coset representative is given by giving rise to the superpotential and the scalar potential of the form We also note the matrix G IJ in this case In this case, the Killing spinors are different from the ansatz given in (3.32) due to the non-vanishing composite connections Q 45 r and Q˙4˙5 r appearing in δψ ±r = 0 conditions. In more detail, there are additional terms involving (γ 45 ) α β +β and (γ˙4˙5)αβ −β in the covariant derivative of the supersymmetry parameters, see equations (2.22) and (2.23). According to this, we modify the ansatz for the Killing spinors to where B(r) is an r-dependent function, and 0 ± are constant symplectic-Majorana-Weyl spinors satisfyingγ r 0 ± = 0 ∓ . Using this ansatz for the Killing spinors satisfying the projector (3.31), we find the following set of BPS equations from the supersymmetry transformations of fermions To find explicit solutions, we will separately discuss various possible values of κ and λ.

83)
. For κ = λ = ±1 corresponding to SO(5) and SO(3, 2) gauge groups. we find the following domain wall solution in terms of the new radial coordinate ρ defined previously. The function B(r) appearing in the Killing spinors is given in term of φ 2 as 90) in which the integration constant has been set to zero.

Domain walls in CSO(3, 0, 2) gauge group
In the case of CSO(3, 0, 2) gauge group with κ = λ = 0, supersymmetry allows a non-vanishing axion corresponding to Y 4 generator. We write the coset representative as and find a simple scalar potential We also note that this potential does not depend on ς and can be obtained from (3.66) by setting κ = λ = 0. This potential can also be written in the form (3.24) using the superpotential and the symmetric matrix With all these and the usual ansatz for the Killing spinors (3.32) together with the projector (3.31), we find the BPS equations Except for an additional equation for ς, these are the BPS equations obtained from (3.69) to (3.73) by setting κ = λ = 0. Furthermore, φ 2 and φ 3 can be consistently truncated out since the scalar potential (3.92) is independent of φ 2 and φ 3 . With all these, we find a domain wall solution in which ρ is a new radial coordinate defined by dρ dr = e ϕ−4φ 1 . It should also be noted that the axion ς can also be truncated out.

SO(2) × SO(2) symmetric domain walls
As a final example of domain wall solutions in 15 −1 representation, we consider an SO(2) × SO(2) unbroken symmetry. In this case, the embedding tensor for all possible gauge groups takes the form There are five scalars invariant under SO(2) × SO(2) generated by X 12 and X 34 . As usual, one of these is the dilaton and the other four are associated with the following non-compact generators (3.101) As in many previous cases, we need to truncate out the axions corresponding to the shift generators s 12 and s 34 in order to find a consistent set of BPS equations that are compatible with the field equations. We then take the coset representative of the form The resulting scalar potential reads which can be written in terms of the superpotential Using the projector (3.31) together with the Killing spinors (3.32), we find the following BPS equations Solving these BPS equations gives a domain wall solution

112)
in which ρ is the new radial coordinate defined by the relation dρ dr = e ϕ−4φ 1 . For domain walls preserving smaller residual symmetries such as SO(2) diag ⊂ SO(2)×SO(2) and SO (2), there are many more scalars, and the analysis is much more involved without any possibility for complete analytic solutions. We will not consider these cases in this work.

Domain walls from gaugings in 40
−1 representation In this section, we consider gaugings in which the irreducible part of the embedding tensor transforms in 40 −1 representation. These gauged theories are obtained from a consistent circle reduction of the maximal seven-dimensional CSO(p, q, 4 − p − q) gauged supergravity constructed in [41].
In six dimensions, gaugings in 40 −1 representation are purely electric and triggered by , the second condition from the quadratic constraint (2.13) reduces to U mn,r U pq,s ε mnpqt = 0 . This condition can be solved by setting in which v m is a GL(5) vector and w mn is a symmetric tensor, w mn = w (mn) . To classify possible gauge groups, we follow [41] by using the SL(5) symmetry to further fix v m = δ m 5 and split the index m = (i, 5), i = 1, .., 4. For simplicity, we also restrict to cases with w i5 = w 55 = 0. The remaining SL(4) residual symmetry can be used to diagonalize the 4 × 4 block corresponding to w ij as with p + q + r = 4. From the decomposition in (3.3), we find that in this case, only X ij and X i gauge generators are non-vanishing. The generators X ij are given in terms of the GL(5) generators while X i only involve the shift generators. Explicitly, these generators are given by It is now straightforward to show that the gauge generators satisfy the following commutation relations l] . This implies that the corresponding gauge group is of the form The CSO(p, q, 4 − p − q) factor and the four-dimensional translation group from the shift symmetries R 4 s are respectively generated by X ij and X i . We should note here that the corresponding gauge group in seven dimensions is just CSO(p, q, 4 − p − q). After an S 1 reduction, this gauge group is accompanied by a translation group R 4 s . As pointed out in [30], the complete off-shell symmetry group of the maximal six-dimensional gauged supergravity is GL(5) 10 −4 , with 10 −4 being shift symmetries of scalar fields. The gauge group given in (4.7) is embedded in GL(5) 10 −4 as CSO(p, q, 4 − p − q) ⊂ GL(5) and R 4 s ⊂ 10 −4 . We also note that in vector representation of SO(5, 5), the gauge generators are given by Since the SL(5) generatorst aḃ are traceless, the generatort 55 is related to the trace part oft ij according tot 11 +t 22 +t 33 +t 44 = −t 55 . It is then convenience to define new non-compact generators t ij as t ij =t ij + 1 4t 55 δ ij (4.10) which are symmetric traceless. The nine scalar fields corresponding to these generators then parametrize an SL(4)/SO(4) coset. The other four scalars associated witht i5 =t + i5 are nilpotent scalars and will be denoted by b i as in seven dimensions. In addition, there are also ten axions corresponding to the antisymmetric shift generators as in the previous section.
As in the previous section, we will systematically find supersymmetric domain walls invariant under some residual symmetries of the CSO(p, q, 4−p−q) factor in the gauge group.

SO(4) symmetric domain walls
We first consider domain walls with the largest possible unbroken symmetry namely SO(4). The only gauge group containing SO(4) as a subgroup is SO(4) s with the embedding tensor parametrized by w ij = δ ij . The SO(4) symmetry is generated by X ij , i, j = 1, 2, 3, 4, generators.
There are two SO(4) singlet scalars given by the dilaton ϕ and another dilatonic scalar corresponding to the SO(1, 1) factor in SL(4)×SO(1, 1) ⊂ SL(5). The latter is given by the non-compact generator and will be denoted by φ 0 .
The coset representative can be written as leading to the T-tensor given by 14) The appearance of γ 5 rather than other SO(5) gamma matrices is due to the specific choice of v m = δ m 5 for the tensor U mn,p . The scalar potential can also be directly computed and is given by The Killing spinors are given by the same ansatz as in (3.32) but in this case subject to the following projector because of the appearance of γ 5 in the T-tensor. With this new projector, it is now straightforward to derive the following BPS equations

Domain walls with the symmetric axion
For κ = 0 corresponding to CSO(3, 0, 1) R 4 s gauge group, it is possible to find solutions with the symmetric axion b non-vanishing. With κ = 0, the scalar potential and the T-tensor are given by and By the general procedure given in section 3.1, we find the superpotential andγ r projectors on the Killing spinors It should be noted that these projectors are not independent. Therefore, the resulting solutions will preserve half of the supersymmetry. Moreover, we can easily see that these projectors reduce to that given in (4.16) for b = 0. With all these, we find the following set of BPS equations together with ς = 0. Since the scalar potential does not depend on ς, we can consistently truncate ς out by setting ς = 0. The domain wall solution to the above BPS equations is then given by in which the new radial coordinate ρ is defined by dρ dr = e ϕ−4φ 0 +4φ , and 2 F 1 is the hypergeometric function.

SO(2) × SO(2) symmetric domain walls
Domain walls preserving SO(2) × SO(2) symmetry can be found in SO(4) R 4 s and SO(2, 2) R 4 s gauge groups described by the embedding tensor with In addition to the two dilatons, there are three SO(2) × SO(2) singlet scalars corresponding to the following SO(5, 5) non-compact generators In this case, a consistent set of BPS equations can be found only when the scalars corresponding to Y 2 and Y 3 generators vanish.
Using a new radial coordinate ρ defined by dρ dr = e ϕ−2φ 0 , we find a domain wall solution

SO(2) symmetric domain walls
As a final example in this case, we consider SO(2) symmetric domain walls. There are many gauge groups admitting an SO(2) subgroup. They are collectively characterized by the following component of the embedding tensor w ij = diag(1, 1, κ, λ). Together with the two dilatons, there are additional nine SO(2) singlet scalars. Three of them are in the SL(4)/SO(4) coset corresponding to noncompact generators The remaining ones consist of two nilpotent scalars associated with and four shift scalars corresponding to However, dealing with all eleven scalars turns out to be highly complicated, so we perform a subtruncation by setting the shift scalar corresponding to s 12 and the two nilpotent scalars to zero. It is straightforward to verify that this is a consistent truncation and still gives interesting solutions. We now end up with eight singlet scalars with the coset representative In what follows, we will for the moment set ς 3 = 0 and separately consider the CSO(2, 0, 2) R 4 s gauge group with ς 3 = 0.
With ς 3 = 0, we can compute the scalar potential and the superpotential of the form  This scalar potential can be written in term of the superpotential according to (3.24) using , (4.73) and . . , 5 and x, y = 6, 7. Note also that the scalar potential for CSO(2, 0, 2) R 4 s gauge group with κ = λ = 0 vanishes identically leading to a family of Minkowski vacua.

Domain walls in SO(3, 1) R 4 s gauge group
In this case, we set κ = −λ = 1, and the BPS equations give B = φ 2 = 0. We can again truncate φ 2 out and set the constant B = 0. As a result, we find a domain wall solution

Domain walls in
(4.98) In this solution, we have defined the coordinate ρ by dρ dr = e −2φ 0 −2φ 1 and set the integration constant for φ 2 solution to be C 2 = 1 16(1+2e C 3 ) 2 in order to simplify the expression for the solution. We also note that the two gauge groups have exactly the same domain wall solution since the parameter κ does not appear anywhere in the solution. In more detail, κ 2 appears in φ 2 solution as g 2 κ 2 ρ 2 , but this term is simply given by g 2 ρ 2 for κ = ±1.
For the remaining scalars ς 1 and ς 2 , we are not able to analytically find their solutions. We can instead perform a numerical analysis to find these solutions, but we will not pursue any further along this direction. In any case, these scalars can be consistently truncated out since they do not appear in the scalar potential.

Domain walls in SO(4) R 4
s and SO(2, 2) R 4 s gauge groups In this case, we set κ = λ = ±1 corresponding to SO(4) R 4 s and SO(2, 2) R 4 s gauge groups. As in the previous case, the resulting BPS equations are very complicated to find explicit solutions. Therefore, we will set ς 1 = ς 2 = 0 and find the domain wall solution for the remaining fields as follows with dρ dr = e ϕ−4(φ 0 +φ 1 ) .

4.4.4
Domain walls in CSO(2, 0, 2) R 4 s gauge group Finally, we consider the case of κ = λ = 0 corresponding to CSO(2, 0, 2) R 4 s gauge group. Using the coset representative (4.69), we find the T-tensor given by By the general procedure given in section 3.1, we find the superpotential W = g 4 e ϕ−4(φ 0 −φ 1 ) ς 2 3 + 1 (4.107) and the following projectorŝ As expected for half-supersymmetric solutions, these projectors are not independent. In addition, for ς 3 = 0, they reduce to a simpler projector given in (4.16). At this point, it is useful to note that for this gauge group, the scalar potential vanishes as previously mentioned, so there exists a six-dimensional Minkowski vacuum for this gauge group. However, the superpotential (4.107) does not have any stationary points, so this Minkowski vacuum is not supersymmetric. With the following ansatz for the Killing spinors we obtain the BPS equations With a new radial coordinate ρ defined by dρ dr = e ϕ−4φ 0 , the corresponding solution is given by (4.117)

Domain walls from gaugings in (15+40) −1 representation
We now consider gaugings with non-vanishing components of the embedding tensor in both 15 −1 and 40 −1 representations. These gaugings are dyonic with the embedding tensor containing both electric and magnetic parts. The full embedding tensor is given by for Y mn = Y (mn) and U mn,p = U [mn],p satisfying U [mn,p] = 0. However, for dyonic gaugings, the first condition in the quadratic constraint (2.13) is not automatically satisfied. For the embedding tensor given in (5.1), we find that this constraint imposes the following condition To solve this condition, we follow [41] and split the GL(5) index as m = (i, x). By choosing a suitable basis, we can take Y mn to be The constraint (5.2) then implies that only the components U xy,z and U ix,y = U i(x,y) are non-vanishing. As a result, the embedding tensor is parametrized by the following tensors Y ij , U i(x,y) , U xy,z .
In the following, we will study supersymmetric domain walls in the two non-trivial cases with rankY = 3 and rankY = 2. Gaugings in these cases are expected to arise from a circle reduction of seven-dimensional maximal gauged supergravity with the embedding tensor in both 15 and 40 representations of SL (5). Similar to the seven-dimensional solutions given in [15], we will find that in these gaugings, the domain walls are 1 4 -BPS preserving eight supercharges. For the case of rankY = 1, the second condition from the quadratic constraint (2.13) is much more complicated to find a non-trivial solution for U i(x,y) and U xy,z . We refrain from discussing this case here.

1 4 -BPS domain walls for rankY = 3
We first consider the case of rankY = 3 with i, j = 1, 2, 3. The second condition from the quadratic constraint (2.13) becomes which can be solved by U ix,y of the form where (Σ i ) x y are 2 × 2 matrices. In terms of these Σ i , the quadratic constraint (5.5) can be rewritten as As pointed out in [41], a real, non-vanishing solution for U ix,y is possible only for with the explicit form of Σ i given in terms of Pauli matrices as The constraint (5.7) is then the Lie algebra of a non-compact group SO(2, 1). It should also be noted that the tensor U xy,z is not constrained by this condition, so it can be parametrized by an arbitrary two-component vector u x as U xy,z = ε xy u z . (5.10) We now consider the corresponding gauge algebra spanned by the following gauge generators To determine the form of the corresponding gauge group, we explicitly evaluate these generators in vector representation and find the following commutation relations [X x , X y ] = 0, [X ij , X x ] = (X ij ) y x X y , [X ix , X y ] = 0, (5.14) [X ix , X jy ] = 0, [X ij , X kx ] = −2(X ij ) kx ly X ly , Redefining the X ij generators as with η ij = diag(+1, +1, −1), we find that X ij generate an SO(2, 1) subgroup with the Lie algebra The remaining generators X ix and X x , which transform non-trivially under SO(2, 1), generate two translation groups. Note also that there are only four independent X ix generators. With all these, the resulting gauge group is then given by in which R 2 s is the translation group from the shift symmetries generated by X x . As also pointed out in [41], we see that the vector u x does not change the gauge algebra, so we can set u x = 0 for simplicity.
We now look for supersymmetric domain wall solutions invariant under SO(2) ⊂ SO(2, 1) generated by X 12 . There are five SO (2)  Using the coset representative of the form we find the scalar potential Consistency of the BPS equations from δχ ± conditions requires ς 1 = 0. After truncating out ς 1 , we find the T-tensor It turns out that only W 1 gives rise to the superpotential in term of which the scalar potential can be written. With the superpotential given by W 1 , the unbroken supersymmetry corresponds to 1 ± and 3 ± . Therefore, we set 2 ± = 4 ± = 0 in the following analysis. Alternatively, we can implement this by imposing an additional projector of the form By the same procedure as in the previous cases together with the projector (3.31), we obtain the BPS equations, with ς 2 = ς, Introducing a new radial coordinate ρ via dρ dr = e ϕ−8φ 1 +2φ 2 , we find a domain wall solution e 6φ 2 = 3 2 tan( √ 3gρ + C 2 ), (5.35)

1 4 -BPS domain walls for rankY = 2
In this case, i, j = 1, 2, we have Y ij = diag(1, ±1). The second condition from the quadratic constraint (2.13) allows only the components U xy,z , x, y, . . . = 3, 4, 5, which can be parametrized by a 3 × 3 traceless matrix u x y as with u x x = 0. The non-vanishing gauge generators read with the commutation relations given by X x and X ix commute with each other and separately generate two translation groups R 3 s and R 6 which transform non-trivially under X 12 . The single X 12 generator in turn leads to a compact SO(2) or a non-compact SO(1, 1) group for Y ij = diag(1, 1) or Y ij = diag(1, −1), respectively. The corresponding gauge groups are then given by SO(2) (R 6 × R 3 s ) or SO(1, 1) (R 6 × R 3 s ).

Domain walls in SO
(2) (R 6 × R 2 s ) gauge group To find solutions with a non-trivial residual symmetry, we will consider SO(2) (R 6 × R 2 s ) gauge group with Y ij = δ ij . In vector representation, the X 12 generator is given by Accordingly, we choose the matrix u x y to be with λ ∈ R. The SO(2) subgroup is then embedded diagonally with only X 4 and X 5 non-vanishing. Thus, the corresponding gauge group, in this case, is given by SO(2) (R 6 × R 2 s ). There are five SO(2) singlets corresponding to the following non-compact generators commuting with X 12 Y d =t With the coset representative it turns out that the scalar potential vanishes identically. On the other hand, the T-tensor is given by or explicitly This leads to two superpotentials e ϕ−12φ 1 (λ + 2) 1 + 4ς 2 1 , (5.56) Unlike the previous rankY = 3 case, both of these give a valid superpotential in term of which the scalar potential can be written. As in the previous case, half of the supersymmetry is broken by choosing any one of these two possibilities which again corresponds to imposing an additional γ 3 projector of the form for W = W 1 or W = W 2 , respectively. Together with the usualγ r projectorŝ the resulting solutions will preserve only eight supercharges or 1 4 of the original supersymmetry.
With the following ansatz for the Killing spinors for 0 ± satisfying the projectors (5.58) and (5.59), we obtain the following BPS The choices of plus or minus signs in these equations are correlated with the plus or minus signs of the two projectors given in (5.58). We can consistently set φ 2 = 0 and find a domain wall solution where ρ is the new radial coordinate defined by dρ dr = e ϕ−2φ 1 .
In this case, there are nine scalars invariant under the residual SO(2) symmetry generated by X 12 . They are given by the five scalars associated with the non-compact generators given in (5.48) to (5.52) together with additional two symmetric and two shift scalars respectively corresponding to However, with this large number of scalar fields, the analysis is highly complicated. To make things more manageable, we will further truncate the nine scalars to the previous five singlets together with each of the two sets of axionic scalars separately.
Turning on two shift scalars, denoted by ς 3 and ς 4 , corresponding to Y 8 and Y 9 generators, we find the solution given in equations (5.65) to (5.69) together with the solutions for ς 3 and ς 4 of the form More interesting solutions are obtained by including the scalars corresponding to Y 6 and Y 7 generators. With the coset representative we find that the scalar potential vanishes as in the previous case. There are also two superpotentials. One of them vanishes identically while the non-trivial one is given by (5.74) Unlike the previous case, the Minkowski vacuum in this case is half-supersymmetric with the unbroken supersymmetry corresponding to the vanishing superpotential. This is very similar to CSO(2, 0, 2) gauged supergravity in seven dimensions [41].
Only the supersymmetry corresponding to the superpotential (5.74) is preserved by the domain wall. This again amounts to imposing a γ 3 projector of the form (5.58). Furthermore, consistency of the BPS equations from δχ ± requires ς 1 = ς 2 = ς. It is useful to note the explicit form of the T-tensor for ς 1 = ς 2 = ς which is given by (5.75) Using the Killing spinors (5.60) subject to the projectors in (5.59) and the first projector in (5.58), we can derive the following BPS equations Introducing a new radial coordinate ρ via dρ dr = e ϕ−12φ 1 √ 1+4ς 2 , we eventually find a domain wall solution ln(e 4φ 3 − 1) + 1 10 ln(e 4φ 3 + 1), (5.84) We end this section by noting that a domain wall solution with ς = 0 can similarly be obtained with the coordinate ρ defined by dρ dr = e ϕ−12φ 1 . In this case, the solutions for the dilaton and warped factor are given by while solutions for the remaining scalars are the same as given above.

Conclusions and discussions
We have constructed the embedding tensors of six-dimensional maximal N = (2, 2) gauged supergravity for various gauge groups with known seven-dimensional origins via an S 1 reduction. These gaugings are triggered by the embedding tensor in 15 −1 and 40 −1 representations of GL(5) ⊂ SO(5, 5) duality symmetry. In 15 −1 representation, the corresponding gauge group is CSO(p, q, 5 − p − q) which is the same as its seven-dimensional counterpart. On the other hand, for gaugings in 40 −1 representation, additional translation groups R n s associated with the shift symmetries on the scalar fields appear in the gaugings resulting in CSO(p, q, 4 − p − q) R 4 s gauge group. This is also the case for gaugings in (15 + 40) −1 representation with gauge groups SO(2, 1) (R 4 × R 2 s ), SO(2) (R 6 × R 2 s ), and CSO(2, 0, 2) R 2 s . We have also studied supersymmetric domain wall solutions and found a large number of half-supersymmetric domain walls from purely magnetic and purely electric gaugings in 15 −1 and 40 −1 representations, respectively. In addition, we have given 1 4 -supersymmetric domain walls for dyonic gaugings involving the embedding tensor in both 15 −1 and 40 −1 representations. These are similar to the seven-dimensional solutions and in agreement with the general classification of supersymmetric domain walls in [26] in which the existence of 1 4 -BPS domain walls has been pointed out.
Apart from solutions with seven-dimensional analogues, we have also found solutions that are not uplifted to seven-dimensional domain walls due to the presence of axionic scalars leading to non-vanishing vector fields in seven dimensions. This can be explicitly seen from the truncation ansatz collected in appendix C. Although this ansatz has originally been given only for SO(5) gauge group, a similar ansatz with possibly suitable modifications in the tensor field content is also applicable for other gauge groups. In particular, the fact that a truncation of seven-dimensional vectors leads to axionic scalars in six dimensions is still true. Therefore, domain wall solutions with non-vanishing axionic scalars obtained in this work cannot be obtained from an S 1 reduction of any domain wall solutions in seven dimensions. Accordingly, these solutions are genuine six-dimensional domain walls without seven-dimensional analogues. As a final comment, we note that there is no SO(5) symmetric domain wall in seven dimensions since there is no SO(5) singlet scalar in SL(5)/SO(5) coset. The six-dimensional SO(5) symmetric domain wall, on the other hand, arises form an S 1 reduction of the supersymmetric AdS 7 vacuum by the general result of [49].
The seven-dimensional origin of all the gaugings considered in this work can also be embedded in ten or eleven dimensions, so the six-dimensional domain wall solutions can be embedded in string/M-theory via the corresponding sevendimensional truncations. Accordingly, the solutions given here are hopefully useful in the study of DW 6 /QFT 5 duality for maximal supersymmetric Yang-Mills theory in five dimensions from both six-dimensional framework and string/Mtheory context. It is interesting to explicitly uplift the domain wall solutions to seven dimensions and subsequently to ten or eleven dimensions using the truncation ansatze given in [50,51,52,53,54].
Constructing truncation ansatze of string/M-theory to six dimensions using SO(5, 5) exceptional field theory given in [55] is also of particular interest. This would allow uplifting the six-dimensional solutions directly to ten or eleven dimensions. In this paper, we have considered only gaugings with the embedding tensor in 15 −1 and 40 −1 representations. It is natural to extend this study by performing a similar analysis for the embedding tensors in other GL (5) representations as well as finding supersymmetric domain walls. Unlike the solutions obtained in this paper, these solutions will not have seven-dimensional counterparts via an S 1 reduction. It is also interesting to construct the embedding tensors for various gaugings under SO(4, 4) ⊂ SO (5,5). These gaugings can be truncated to gaugings in half-maximal N = (1, 1) supergravity coupled to four vector multiplets in which supersymmetric AdS 6 vacua are known to exist in the presence of both conventional gaugings and massive deformations [56,57,58]. Finding supersymmetric solutions from these gauge groups could be useful in the study of AdS 6 /CFT 5 correspondence. Finally, finding supersymmetric curved domain walls with nonvanishing vector and tensor fields as in seven-dimensional maximal gauged supergravity in [59,60] is worth considering. This type of solutions can describe conformal defects or holographic RG flows from five-dimensional N = 4 super Yang-Mills theories to lower dimensions. Along this line, examples of solutions dual to surface defects from N = (1, 1) gauged supergravity have appeared recently in [61].

Acknowledgement
This work is supported by the Second Century Fund (C2F), Chulalongkorn University. P. K. is supported by The Thailand Research Fund (TRF) under grant RSA6280022.

A GL(5) Branching rules
In this appendix, we collect all of the SO(5, 5) → GL(5) branching rules used throughout the paper. Relevant decompositions have already been given in [30], but in order to construct the embedding tensor, we need a concrete realization. Therefore, we will determine the decompositions for various representations of SO (5,5) in terms of GL(5) representations using explicit matrix forms. In vector representation, the SO(5, 5) algebra

A.1 Vector
we find the explicit form of the R + generator in vector representation given by With an SO(5, 5) vector decomposing as V M = (V m , V m ), we obtain the commutation relations These imply that we can assign the R + weights ±2 to the 5 and 5 representations of SL(5) ⊂ GL (5). Therefore, the branching rule for a vector representation reads

A.2 Adjoint
The decomposition of adjoint representation follows from the branching rule of vector representations. Using (A.7), we can decompose the SO(5, 5) generators as with τ m m = 0. We denote the shift and hidden generators by s mn = t mn and h mn = t mn , respectively. In vector representation, the SO(5, 5) generators can be written as In the second line of (A.17), we have used (A.9) to rewrite the commutation relation in terms of the GL(5) generators. Note also that (A.14) is the SL(5) algebra. It follows that the GL(5) branching rule for adjoint representation is given by

A.3 Spinor
Unlike the vector, decomposition of SO(5, 5) spinor representation under GL (5) is not straightforward. To describe this branching rule, we begin with the following two sets of U Sp(4) ∼ SO(5) gamma matrices satisfying  (5) vector indices raised and lowered by δ ab and δ˙a˙b, respectively. For both sets of SO(5) gamma matrices, we will use the following explicit representation where {σ 1 , σ 2 , σ 3 } are the usual Pauli matrices given by The matrices A, B, and C, which respectively realize Dirac, complex, and charge conjugation, have the following defining properties (A.28) In our explicit representation, the matrices A and B are given by The charge conjugation matrix C can be obtained from A and B through the relation The SO(5, 5) chirality matrix takes the following diagonal form We can then relate these two decompositions of SO(5, 5) spinor indices into A, A and a pair of U Sp(4) indices (αα) by using the following transformation matrices These matrices satisfy the relations We can now write chiral SO(5, 5) gamma matrices in terms of the SO(5) ones as Its elements can be explicitly expressed as Similarly, the matrix c AB satisfying the relations is also antisymmetric c A B = −c BA . By raising and lowering the SO(5, 5) spinor index, we can define gamma matrices with all upper or lower indices In terms of the U Sp (4)  With all these, we can eventually find the following relations It should be noted that the SO(5, 5) generators in spinor representation given in (A.54) also decompose according to (A.18) and satisfy the same algebra given in (A.11) to (A.17) for vector representation.
As pointed out in [30], the branching rules for spinor and conjugate spinor representations of SO(5, 5) are respectively given by In addition, we also note that a complex conjugation of the SO(5, 5) gamma matrices is related to raising the indices ((Γ M ) AB ) * = (Γ M ) AB . We can then similarly decompose a conjugate spinor of SO(5, 5) transforming in 16 c as follows The following commutation relations Here,γμ are 8 × 8 Dirac matrices and ημν = diag(−1, +1, +1, +1, +1, +1) witĥ µ,ν, ... = 0, 1, ..., 5 being six-dimensional flat space-time indices. We will use the following explicit representation of the gamma matriceŝ In this representation, the Dirac, complex, and charge conjugation matrices are respectively given byÂ The diagonal form ofγ * implies that a Dirac spinor Ψ can be chirally decomposed as Ψ = Ψ + + Ψ − with P ± Ψ ± = ±Ψ ± (B.7) where the projection operators are given by Therefore, we can define two irreducible Weyl spinors ψ + and χ − from the Dirac spinor Ψ by Although the second property in (B.5) implies that a reality condition cannot be imposed on the Dirac or Weyl spinors, we can define a symplectic-Majorana-Weyl spinor of the form Ψ +α = ψ +α 0 and Ψ −α = 0 χ −α (B.10) with Ψ +α and Ψ −α satisfying the pseudo-reality condition given by

C Truncation ansatze
In this appendix, we collect some useful formulae for a consistent truncation of seven-dimensional SO(5) gauged supergravity on a circle (S 1 ), giving rise to SO(5) gauged supergravity in six dimensions. This truncation has been constructed in [28]. The truncation ansatze for the seven-dimensional metric, scalar, vector, and tensor fields are respectively given by 10 , we find that (C.1) becomes e 2Â dx 2 1,5 + dr 2 = e 2A+2ϕ dx 2 1,4 + e −8ϕ dz 2 + e 2ϕ dr 2 = e 8A 5 (dx 2 1,4 + dz 2 ) + e 2ϕ dr 2 (C.5) In the second line, we have substituted ϕ = − A 5 from the domain wall solutions given here. This is also necessary for dx 2 1,4 and dz 2 to form a six-dimensional flat space-time matching dx 2 1,5 on the left hand side. We can also see the relations between the warped factorsÂ = 4A 5 and the radial coordinates dr = e ϕ dr. The ansatz (C.2) implies that the scalars parametrizing SL(5)/SO(5) c coset in seven-and six-dimensional supersymmetric domain walls are the same since they are independent of z and depend only on the corresponding radial coordinatesΠ Therefore, domain wall solutions with non-vanishing axionic scalars obtained in this work cannot be obtained from an S 1 reduction of any domain wall solutions in seven dimensions.