Supersymmetric domain walls in 7D maximal gauged supergravity

We give a large class of supersymmetric domain walls in maximal seven-dimensional gauged supergravity with various types of gauge groups. Gaugings are described by components of the embedding tensor transforming in representations $\mathbf{15}$ and $\overline{\mathbf{40}}$ of the global symmetry $SL(5)$. The embedding tensor in $\mathbf{15}$ representation leads to $CSO(p,q,5-p-q)$ gauge groups while gaugings in $\overline{\mathbf{40}}$ representation describes $CSO(p,q,4-p-q)$ gauge groups. These gaugings adimits half-supersymmetric domain walls as vacuum solutions. On the other hand, gaugings involving both $\mathbf{15}$ and $\overline{\mathbf{40}}$ components lead to $\frac{1}{4}$-supersymmetric domain walls. In this case, the gauge groups under consideration are $SO(2,1)\ltimes \mathbf{R}^4$ and $CSO(2,0,2)\sim SO(2)\ltimes \mathbf{R}^4$. All of the domain wall solutions are analytically obtained. For $SO(5)$ gauge group, the gauged supergravity admits an $N=4$ supersymmetric $AdS_7$ vacuum dual to $N=(2,0)$ SCFT in six dimensions. The corresponding domain walls can be interpreted as holographic RG flows from the $N=(2,0)$ SCFT to non-conformal $N=(2,0)$ field theories in the IR. The solutions can be uplifted to eleven dimensions by using a truncation ansatz on $S^4$. Furthermore, the gauged supergravity with $CSO(4,0,1)\sim SO(4)\ltimes \mathbf{R}^4$ gauge group can be embedded in type IIA theory via a truncation on $S^3$. The uplifted domain walls, describing NS5-branes of type IIA theory, are also given.


Introduction
Supersymmetric p-branes have played an important role throughout the development of string/M-theory. These extended objects can be effectively described by using (p + 2)-dimensional gauged supergravity (possibly including massive deformations in higher dimensions) in which they become domain walls. The latter are of particular interest in the DW/QFT correspondence [1,2,3], a generalization of the AdS/CFT correspondence [4], and in cosmology, see for example [5,6,7]. In addition, classifications of supersymmetric domain walls can give some insight to the underlying structure of M-theory [8] through the algebraic structure E 11 [9].
In ten dimensions, there is only one massive type IIA supergravity and hence only one possible domain wall [10]. In nine and eight dimensions, halfsupersymmetric domain walls have been studied in [11] and [12,13] using maximal gauged supergravities. In this paper, we will consider supersymmetric domain walls within maximal gauged supergravity in seven dimensions. General discussions about this type of solutions and examples of domain walls in N = 4 gauged supergravity with SO(5) gauge group have already been given in previous works [14,15,16,17]. However, as pointed out in [17], a systematic study of these domain walls and explicit solutions in other gauge groups have not appeared so far. Similar solutions in lower-dimensional gauged supergravities can also be found in [18,19,20,21,22,23].
We will give a large number of supersymmetric domain wall solutions in maximal N = 4 gauged supergravity with various gauge groups. The first N = 4 gauged supergravity with SO(5) gauge group has been constructed for a long time in [24,25]. It can be obtained from a consistent truncation of elevendimensional supergravity on a four-sphere S 4 [26,27,28]. The most general deformations of the N = 4 supergravity are obtained by using the embedding tensor formalism. These gaugings have been constructed in [29]. There are two components of the embedding tensor transforming in 15 and 40 representations of the global SL(5) symmetry. As shown in [17], each of these components leads to half-supersymmetric domain walls. In addition, the 15 and 40 parts give rise to domain walls supporting respectively tensor and vector multiplets on their world-volumes. Unlike higher-dimensional analogues, when both representations of the embedding tensor are present simultaneously, the domain walls are only 1 4 -supersymmetric. In this paper, we will analytically give solutions for domain walls of all these types.
For gaugings in 15 representation, we will consider CSO(p, q, 5 − p − q) ∼ SO(p, q)⋉R (p+q)(5−p−q) gauge groups. For SO(5) gauge group with known elevendimensional origin, solutions to N = 4 gauged supergravity can be embedded in M-theory. Furthermore, this gauged supergravity also admits a maximally supersymmetric AdS 7 vacuum which is, according to the AdS/CFT correspondence, dual to N = (2, 0) superconformal field theory (SCFT) in six dimensions. The domain walls with an AdS 7 asymptotic can be interpreted as holographic RG flows from the N = (2, 0) SCFT to non-conformal field theories in the IR. We consider this type of domain walls in the context of the AdS/CFT correspondence and carry out their uplift to eleven dimensions. The gauging from 15 representation with gauge group CSO(4, 0, 1) can be obtained from a truncation of type IIA supergravity on S 3 [30]. We also give uplifted solutions of the domain walls from this gauge group in type IIA theory.
For gaugings in 40 representation, the gauge groups under consideration are CSO(p, q, 4 − p − q) ⊂ SL(4) ⊂ SL (5). The existence of a higher-dimensional origin of the SO(4) gauge group from a truncation of type IIB theory on S 3 has been pointed out in [1]. However, the full non-linear truncation ansatz in this case currently has not been constructed. Finally, for gaugings with the embedding tensor from both 15 and 40 representations, we consider non-semisimple SO(2, 1) ⋉ R 4 and SO(2) ⋉ R 4 gauge groups which give rise to 1 4 -supersymmetric domain walls.
The paper is organized as follow. In section 2, we give a review of the maximal gauged supergravity in seven dimensions using the embedding tensor formalism. Half-supersymmetric domain walls for gauge groups CSO(p, q, 5 − p − q) are given in sections 3. For SO(5) gauge group, admitting a supersymmetric AdS 7 vacuum, we consider holographic RG flows from N = (2, 0) six-dimensional SCFT to non-conformal field theories in the IR and study an uplift to eleven dimensions of these solutions. Uplifted solutions to type IIA theory of domain walls in CSO(4, 0, 1) gauge group are also given. We then perform a similar analysis for CSO(p, q, 4 − p − q), SO(2, 1) ⋉ R 4 and SO(2) ⋉ R 4 gauge groups in sections 4 and 5. Conclusions and comments on the results are given in section 6. Consistent reduction ansatze for M-theory on S 4 and type IIA theory on S 3 which are useful to the discussion in the main text are reviewed in the appendix.

Maximal gauged supergravity in seven dimensions
In this section, we give a brief review of N = 4 gauged supergravity in seven dimensions in the embedding tensor formalism. This section closely follows the original construction given in [29] to which the reader is referred for more detail. The maximal N = 4 supersymmetry consists of only the supergravity multiplet with the field content given by and Ω ab = Ω [ab] is USp(4) symplectic form with the inverse Ω ab = (Ω ab ) * satisfying Ω ab Ω cb = δ c a . Raising and lowering of USp(4) indices a, b, . . . by Ω ab and Ω ab are related to complex conjugation for example All fermions are symplectic Majorana spinors subject to the conditions where C denotes the charge conjugation matrix obeying The Dirac conjugate on a spinor Ψ is defined by Ψ = Ψ † γ 0 . We will denote space-time gamma matrices by γ µ as opposed to Γ µ in the convention of [29]. The SL(5)/SO(5) coset representative V M A transform under the global SL(5) and local SO(5) ∼ USp(4) by left and right multiplications, respectively. Accordingly, the index A can be described by an anti-symmetric pair of USp(4) fundamental indices, and V M A can be written as V M ab subject to the condition The inverse of V M ab will be denoted by V ab M . We then have the following relations It should be noted that the SL(5)/SO(5) coset can also be described by a unimodular symmetric matrix M M N defined by with its inverse given by M M N = V ab M V cd N Ω ac Ω bd . The most general gaugings of N = 4 supergravity can be efficiently described by using the embedding tensor Θ M N,P Q . This tensor describes an embedding of a gauge group G 0 in the global symmetry group SL(5) via the covariant derivative with ∇ µ being the space-time covariant derivative including (possibly) composite SO(5) connections. t P Q are SL(5) generators and g is the gauge coupling constant.
The covariant derivative implies that the embedding tensor lives in the product representation between the conjugate representation of the vector fields and the adjoint representation of SL (5) 10 ⊗ 24 = 10 + 15 + 40 + 175.
Among the resulting irreducible representations, supersymmetry allows only the embedding tensor in 15 and 40. These representations will be denoted respectively by Y M N and In terms of Y M N and Z M N,P , the embedding tensor can be written as In order to define a viable gauging, the embedding tensor needs to satsisfy the so-called quadratic constraint to ensure that the gauge generators X M N = Θ M N,P Q t P Q form a closed subalgebra of SL(5) In the fundamental representation 5 of SL(5), gauge generators (X M N ) P Q can be written as while in the 10 representation, these generators are given by In terms of Y M N and Z M N,P , the quadratic constraint (12) reads It should also be noted that this constraint implies that the embedding tensor is gauge invariant The introduction of the minimal coupling (9) usually breaks the original supersymmetry. To restore supersymmetry, modifications to the Lagrangian and supersymmetry transformations are needed. In addition to the introduction of fermionic mass-like terms and scalar potential, gaugings also lead to hierarchies of non-abelian vector and tensor fields of different ranks. In this paper, we are interested only in domain wall solutions with only the metric and scalars nonvanishing. We will set all vector and tensor fields to zero from now on. It is straightforward to verify that for all solutions under consideration here, this is indeed a consistent truncation.
The bosonic Lagrangian with only the metric and scalar fields reads and the supersymmetry transformations of ψ a µ and χ abc are given by δχ abc = 2Ω cd P µde ab γ µ ǫ e + gA d,abc The covariant derivative of ǫ a is defined as The vielbein on the SL(5)/SO(5) coset P µab cd and the SO(5) ∼ USp(4) composite connection Q µa b are obtained from the relation A 1 and A 2 tensors are given in terms of scalar fields and the embedding tensor A d,abc in which B and C tensors are defined by with Finally, the scalar potential is given by It should also be noted that the Lagrangian (17) can be written in a USp(4) invariant form as with the scalar potential given by

Supersymmetric domain walls from gaugings in 15 representation
In this section, we consider gauge groups arising from the embedding tensor in 15 representation. It is readily seen from (15) that setting Z M N,P = 0 trivially satisfies the quadratic constraint. Therefore, any symmetric tensor Y M N leads to an admissible gauge group. The SL(5) symmetry can be used to fix the form of Y M N to be The corresponding gauge generators are given by which give rise to the gauge group In order to find supersymmetric solutions, we restrict ourselves to a subset of scalars invariant under a certain symmetry group H 0 ⊂ G 0 following the approach introduced in [31]. The metric takes the form of standard domain wall ansatz where α, β = 0, 1, .., 5 and A(r) is a warp factor depending only on the radial coordinate r. Non-compact generators of SL(5) are given by 5 × 5 symmetric traceless matrices. To obtain an explicit parametrization of the coset representative V M A , it is useful to introduce GL(5) matrices To convert the SO(5) vector indices A, B, . . . = 1, 2, . . . , 5 to a pair of antisymmetric USp(4) indices a, b, . . . , = 1, 2, 3, 4, we use a convenient choice of SO(5) gamma matrices given by where σ i are the usual Pauli matrices. Γ A satisfy the following relations The symplectic form of USp(4) is taken to be The coset representative of the form V M ab and the inverse V ab M are then obtained by using the relations We are now in a position to set up BPS equations and look for domain wall solutions with different unbroken symmetries.
The coset representative can be written as The scalar potential for this SO(4) invariant scalar is given by It can be verified that, for κ = 1, this potential admits two AdS 7 critical points at φ = 0 and φ = 1 10 ln 2. These critical points have already been studied in [25]. The first critical point has SO(5) symmetry and preserves all supersymmetry. Upon uplifting, this vacuum corresponds to AdS 7 × S 4 solutions of eleven-dimensional supergravity. The cosmological constant and AdS 7 radius are given by The second critical point is SO(4) symmetric and breaks all supersymmetry. This non-supersymmetric AdS 7 vacuum is unstable [25].
In order to setup the corresponding BPS equations, we impose a projector and obtain the following BPS equations from δψ a α = 0 and δχ abc = 0 conditions The condition δψ a r = 0 gives the usual solution for the Killing spinors with the constant spinors ǫ a 0 satisfying γ r ǫ a 0 = ǫ a 0 . The solution is then halfsupersymmetric.
The above BPS equations can be readily solved to obtain the solution with the new radial coordinate ρ defined by dρ dr = e 3φ . The integration constant C can be removed by shifting the coordinate ρ. We have neglected an additive integration constant for A since it can be absorbed by rescaling the coordinates x α .
Note also that for κ = −1, the solution for φ can be written as For κ = 0, we find

SO(3) × SO(2) symmetric domain walls
We now consider another residual symmetry SO(3) × SO(2) which is possible only for SO(5) and SO(3, 2) gauge groups. In this case, we write the embedding tensor as with σ = 1 and σ = −1 corresponding to SO(5) and SO(3, 2), respectively. The only one SO(3) × SO(2) singlet scalar corresponds to the noncompact generatorỸ With the coset representative we find the scalar potential which admits an AdS 7 critical point at φ = 0 for σ = 1.
The BPS equations are given by By defining a new radial coordinate ρ by the relation dρ dr = e φ , we obtain the solution This solution is very similar to the SO(4) symmetric solution.

SO(3) symmetric domain walls
When the residual symmetry of the solutions is smaller, we find more interesting solutions. We now consider domain wall solutions with SO(3) symmetry. There are many gauge groups containing SO(3) subgroup with the embedding tensor given by There are three scalar singlets under SO(3) symmetry generated by gauge generators X M N , M, N = 1, 2, 3. These singlets correspond to the following noncompact generators of SL(5) Using the parametrization of the coset representative we obtain the scalar potential This potential admits two AdS 7 critical point for κ = σ = 1. The first one is at φ 1 = φ 2 = φ 3 = 0 corresponding to the N = 4 supersymmetric AdS 7 with SO(5) symmetry. Another critical point is non-supersymmetric and given by It should also be noted that for φ 2 = 5φ 1 , the residual symmetry is enhanced to SO(4). As a check, we can compute all scalar masses at this critical point. The result is which contains the value m 2 L 2 that violates the BF bound m 2 L 2 = −9. Therefore, this critical point is unstable. The four Goldstone bosons corresponding to the broken generators X a4 − X a5 , a = 1, 2, 3 and X 45 . Using the same procedure as in the previous cases, we find the following BPS equation To find explicit solutions, it is useful to separately discuss various possible values of κ and σ.

Domain walls in SO(4, 1) gauge group
In this case, σ = −κ = 1, we find that φ ′ 2 = 0. It can also be checked that φ 2 can be consistently truncated out. Note also that the corresponding noncompact generatorŶ 2 is one of the non-compact generators of SO(4, 1), namely X 45 . φ 2 is then identified with a Goldstone boson of the symmetry breaking SO(4, 1) → SO(4) → SO(3) at the vacuum.
Taking φ 2 = 0 and redefining the radial coordinate r to ρ via dρ dr = e 6φ 1 , we obtain a domain wall solution 3.3.4 Domain walls in SO(5) and CSO(3, 2) gauge groups We now look at the last possibility κ = σ = ±1 corresponding to SO(5) and SO(3, 2) gauge groups. In term of the new radial coordinate ρ as defined in the previous cases, we find a domain wall solution

SO(2) × SO(2) symmetric domain walls
We consider another truncation to SO(2)×SO(2) invariant scalars corresponding to SL(5) non-compact generators In this case, the embedding tensor takes the form of which encodes various possible gauge groups depending on the values of σ and κ. These gauge groups are SO (5) With the parametrization of the coset representative we find the scalar potential For SO(5) gauge group, there are two AdS 7 critical points at φ 1 = φ 2 = 0 and φ 1 = φ 2 = 1 10 ln 2. The former is as in other cases the N = 4 supersymmetric one while the latter is a non-supersymmetric critical point. Note also that this nonsupersymmetric AdS 7 has SO(4) symmetry since the SO(2) × SO(2) symmetry is enhanced to SO(4) when φ 1 = φ 2 . This critical point is unstable as previously mentioned.
The BPS equations in this case read Defining a new radial coordinate ρ by dρ dr = e −2φ 1 , we find a domain wall solution

Uplift to eleven dimensions and holographic RG flows
For SO(5) gauge group, the seven-dimensional gauged supergravity can be obtained from a consistent truncation of eleven-dimensional supergravity on S 4 . Therefore, the domain wall solutions obtained previously can be uplifted to solutions of eleven-dimensional supergravity. Furthermore, these solutions are asymptotic to the N = 4 supersymmetric AdS 7 vacuum corresponding to N = (2, 0) SCFT in six dimensions. According to the AdS/CFT correspondence, the domain walls can then be interpreted as holographic RG flows from six-dimensional N = (2, 0) SCFT to non-conformal field theories in the IR, see for example [32,33]. We will consider this type of solutions including the uplift to eleven dimensions.

RG flow preserving SO(4) symmetry
We first consider a simple solution with SO(4) symmetry. For SO(5) gauge group, the domain wall solution reads with dρ dr = e 3φ . As ρ → ∞, we find φ → 0 and ρ ∼ r with an asymptotic behavior which indicates that the solution approaches the supersymmetric N = 4 AdS 7 critical point. The scalar φ is dual to an operator of dimension ∆ = 4. Indeed, all scalars of the N = 4 gauged supergravity are dual to operators of dimension four.
As gρ → C, the solution is singular We can now check that the scalar potential is bounded above with V → −∞ as φ → −∞. This implies that the singularity is physically acceptable according to the criterion of [34]. In addition, we can use the truncation ansatz, reviewed in the appendix, to uplift this solution to eleven dimensions.
With the parametrization of the SL(5)/SO (5) coset and the coordinates on S 4 withμ i being coordinates on S 3 satisfyingμ iμi = 1, we find the eleven-dimensional metric and four-form field strength tensor with dΩ 2 (3) being the metric on a unit S 3 and ∆ = e 8φ cos 2 ξ + e −2φ sin 2 ξ, We see that the internal S 4 is deformed in such a way that an S 3 inside the S 4 is unchanged. The isometry of this S 3 is the SO(4) residual symmetry of the seven-dimensional solution.
In addition, we can look at the behavior of the metric componentĝ 00 = e 2A ∆ which means the singularity is physical according to the criterion given in [35]. This solution then describes an RG flow from six-dimensional N = (2, 0) SCFT to a non-conformal field theory in the IR. The flow is driven by a vacuum expectation value of an operator of dimension ∆ = 4 that breaks conformal symmetry and preserves only SO(4) ⊂ SO(5) R-symmetry.

RG flow preserving SO(3) × SO(2) symmetry
In this case, the flow solution reads as in the previous case. As gρ → C, the solution becomes Near the singularity, we find that the scalar potential is bounded above V → −∞.
The uplifted solution can be obtained by using the S 4 coordinates µ M = (sin ξμ a , cos ξ cos α, cos ξ sin α), a = 1, 2, 3 We find the eleven-dimensional solution g 2 e −6φ cos 2 ξdα 2 + (e 4φ cos 2 ξ + e −6φ sin 2 ξ)dξ 2 +e 4φ sin 2 ξdμ a dμ a , where We can see that the unbroken SO(3) × SO(2) symmetry corresponds to the isometry of the S 2 inside the S 4 and the isometry of the S 1 parametrized by the coordinate α.
From the eleven-dimensional metric, we find The singularity is accordingly physical.
The four-form field strength is much more complicated than the previous cases. We refrain from giving its explicit form here. The unbroken symmetry SO(2) × SO(2) corresponds to the isometry of S 1 × S 1 parametrized by coordinates α and β.
As r → ∞, the solution becomes which again implies that φ 1 and φ 2 are dual to operators of dimension ∆ = 4 in the dual N = (2, 0) SCFT. There are two possibilities for the IR behaviors.
As gρ → 2C 1 , we have Near the singularity, the scalar potential is unbounded above V → ∞. The eleven-dimensional metric givesĝ This singularity is unphysical. As gρ → 2C 2 , we have Near the singularity, we find V → −∞ and In this case, the singularity is physical, and the solution describes an RG flow from N = (2, 0) SCFT to a non-conformal field theory in the IR.

RG flow preserving SO(3) symmetry
In this case, the solution is more complicated. We will consider only a truncation of the full solution here. Making a consistent truncation by setting φ 3 = 0, we obtain a simple solution to the truncated BPS equations with dρ dr = e 6φ 1 . Near the AdS 7 critical point in the UV as r → ∞, we find, as in the previous cases, and, near the IR singularity as gρ → C, the solution becomes In this case, the scalar potential diverges near the singularity V → ∞, and the component of the eleven-dimensional metric giveŝ The singularity is then unphysical. We will not give the corresponding elevendimensional solution. It can be verified that a truncation with φ 3 = 0 also gives similar result.

Uplifted solutions to type IIA supergravity
We now consider the uplift of the domain wall solutions in CSO(4, 0, 1) gauge group to type IIA theory. Relevant parts of the truncation ansatz are reviewed in the appendix. We first decompose the SL(5)/SO(5) coset in term of the SL(4)/SO(4) where V is the coset representative of SL (4) with M = V V T . Relations between seven-dimensional fields and eleven-dimensional ones are given in the appendix. In all of the solutions considered here, we have b i = χ i = 0, so only the ten-dimensional metric, dilaton and three-form field strength are non-vanishing. The resulting solutions then, as expected for domain walls in seven dimensions, describe NS5-branes in the transverse space with different symmetries.

Solution with SO(4) symmetry
In this case, we simply have M ij = δ ij and The solutions for φ 0 and A are given by φ 0 = 1 2 ln gr 10 + C and A = ln gr 10 + C These are obtained from solving the BPS equations in (48) and (49) by renaming φ to φ 0 and setting κ = 0. We identify the resulting ten-dimensional solution with NS5-branes in the transverse space R 4 .
Solutions for scalars φ 0 and φ can be obtained from the SO(3) symmetric domain wall given in section 3.3 by setting φ 2 = 0 and using the relations With κ = 0 and σ = 1, the domain wall solution is given by In this solution, 2 F 1 is the hypergeometric function. We now choose a specific form of the S 3 coordinates 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. With all these, we find the ten-dimensional fields e −6φ sin 2 ξ + e 2φ cos 2 ξ dξ 2 in which ∆ = e 6φ cos 2 ξ + e −2φ sin 2 ξ, The unbroken SO(3) symmetry corresponds to the isometry of S 2 ⊂ S 3 .
In this case, the solutions for φ 0 and φ can be obtained from the BPS equations given in section 3.4 by setting σ = 1, κ = 0 and using the relations The resulting seven-dimensional domain wall is given by Choosing the coordinates on S 3 to be µ i = (cos ξ cos α, cos ξ sin α, sin ξ cos β, sin ξ sin β), we find In this case, the SO(2) × SO(2) symmetry corresponds to the isometry of S 1 × S 1 parametrized by coordinates α and β.

Supersymmetric domain walls from gaugings in 40 representation
In this section, we consider gaugings with the embedding tensor in 40 representation. Setting Y M N = 0, the quadratic constraint reads This condition can be solved by the following tensor with w M N = w (M N ) . The SL(5) symmetry can be used to fix v M = δ M 5 . It is useful to split the index M = (i, 5).
If, in addition, we set w 55 = w i5 = 0, the remaining SL(4) symmetry, under which the vector v M = δ M 5 is invariant, can be used to diagonalize w ij . Accordingly, w ij can be written as The resulting gauge generators take the form of which gives rise to CSO(p, q, r) gauge group with p + q + r = 4. In these gaugings, following [29], it is convenient to parametrize the SL(5)/SO(5) coset representative in term of SL(4)/SO(4) submanifold as given in (139). After setting Y M N = 0, the scalar potential is given by It should be noted that the nilpotent scalars b i appear quadratically in the potential, so setting them to zero is a manifestly consistent truncation.

SO(4) symmetric domain walls
We firstly consider domain walls with the largest possible unbroken symmetry, SO(4) ⊂ CSO(p, q, 4 − p − q). The only gauge group containing SO(4) as a subgroup is SO(4). The embedding tensor is simply w ij = δ ij , and there are no SO(4) singlet scalars from SL(4)/SO(4). We then take the coset representative to be V = I 4 . The scalar potential takes a simple form The Killing spinor still takes the fom (50), but unlike the previous cases, the appropriate projector for this type of gaugings is given by The appearance of Γ 5 rather than other Γ A with A = 1, 2, 3, 4 is due to the specific form of v M = δ M 5 in the embedding tensor Z M N,P . It is now straightforward to derive the corresponding BPS equations We can readily find the solution

SO(3) symmetric domain walls
We now look for more complicated solutions with SO(3) symmetry. Gauge groups with an SO(3) subgroup are SO(4), SO(3, 1) and CSO(3, 0, 1). We descibe them all at once by taking the symmetric matrix w ij in the form with κ = 1, −1, 0, respectively. For simplicity, we truncate scalars b i out and consider only φ 0 and SL(4)/SO(4) scalars. With an explicit form of the SL(4)/SO(4) coset representative we obtain the scalar potential Using the projector in (166), we can derive the following set of BPS equations The solutions for A and φ 0 are given by The solution for φ(r) is given by for κ = 0 and 4grσ(e 8φ − σ) for κ = ±1.

SO(2) × SO(2) symmetric domain walls
Possible domain wall solutions with SO(2) × SO(2) symmetry can be obtained from SO(4) and SO(2, 2) gauge groups. These gauge groups are described by the component of the embedding tensor in the form of With the parametrization for the SL(4)/SO(4) coset representative the scalar potential and the BPS equations are given by and The domain wall solution can be straightforwardly obtained 6grσ(e 4φ − σ)

SO(2) symmetric domain walls
As a final example for domain wall solutions from gaugings in 40 representation, we consider SO(2) symmetric solutions. We again truncate out scalar fields b i and parametrize the SL(4)/SO(4) coset representative as in which Y i , i = 1, 2, 3 are non-compact generators commuting with the SO(2) symmetry generated by X 12 . The explicit form of these generators is given by There are many gauge groups admitting an SO(2) subgroup. They are uniformly characterized by the following component of the embedding tensor 1, σ, κ).
The scalar potential is computed to be It should be noted that the scalar potential for CSO(2, 0, 2) gauge group with σ = κ = 0 vanish identically. This leads to a Minkowski vacuum. In this case, the BPS equations are much more complicated than those obtiained in the previous cases We are not able to completely solve these equations for arbitrary values of the parameters κ and σ. However, the solutions can be separately found for various values of κ and σ.

Domain walls from CSO(2, 0, 2) gauge group
The simplest case is CSO(2, 0, 2) gauge group corresponding to σ = κ = 0. In this case, φ ′ 2 = φ ′ 3 = 0 and the remaining BPS equations simplify considerably Scalars φ 2 and φ 3 can be consistently truncated out, and the solution for the remaining fields can be readily found In this case, σ = −κ = 1, and the BPS equations give φ ′ 2 = 0. Similar to the previous case, φ 2 does not appear in any BPS equations. After truncating out φ 2 , we find the domain wall solution with ρ defined by dρ dr = e −2φ 0 −2φ 1 .

Domain walls from SO
In this case, we set κ = σ = ±1 corresponding to SO(4) and SO(2, 2) gauge groups. The domain wall solution can be found as in the previous case with dρ dr = e −2φ 0 −2φ 1 .

Supersymmetric domain walls from gaugings in 1and 40 representations
We now consider gaugings with both components of the embedding tensor in 15 and 40 representations non-vanishing. Following [29], we will choose a particular basis such that nonvanishing components of the embedding tensor are given by in whih the ranges of indices are given by x = 1, ..., t and α = t + 1, ..., 5 for t ≡ rankY M N . We will also choose Y xy in the form with p + q = t. Tensors Y xy , Z xα,β and Z αβ,γ need to satisfy the quadratic constaint which is explicitly given by We will look for domain wall solutions in SO(2, 1) ⋉ R 4 and SO(2) ⋉ R 4 gauge groups. The corresponding embedding tensors for these gauge groups have already been given in [29]. We also emphasize that in the case of gaugings in 15 and 40 representations, domain walls are 1 4 -BPS, preserving only eight supercharges. All gaugings in this case can be obtained from Scherk-Schwarz reduction of the maximal gauged supergravity in eight dimensions.

1 4 -BPS domain wall from SO(2, 1) ⋉ R 4 gauge group
We begin with the t = 3 case in which Y xy can be chosen to be diag (1, 1, ±1).
The component Z αβ,γ of the embedding tensor is not constrained by the quadratic constraint and can be parametrized by an arbitrary two-component vector v α as Z αβ,γ = ǫ αβ v γ . Accordingly, Z αβ,γ does not affect the form of the gauge algebra. For simplicity, we will set v α = 0. On the other hand, the quadratic constraint imposes the following condition on Z xα,β which implies that the 2 × 2 matrices (Σ x ) α β = −16ǫ αγ Z xγ,β satisfy the algebra In terms of Σ x , Z xα,β component of the embedding tensor takes the form As pointed out in [29], a real, nonvanishing solution for Z xα,β is possible only for Y xy generating a non-compact SO(2, 1) group. In this case, we take Y xy = diag(1, 1, −1) and choose the explicit form for Σ x as follow The corresponding gauge generators are given by with λ z ∈ R. It should be noted that the SO(2, 1) subgroup is embedded diagonally. The nilpotent generators Q (4) xα transform as 4 under SO(2, 1) and are obtained from projecting the tensor product 3 ⊗ 2 = 2 + 4 to representation 4. The resulting gauge group is then given by SO(2, 1) ⋉ R 4 .
With the SL(5)/SO(5) coset representative we obtain the scalar potential which does not admit any critical points. Contrary to the previous cases, finding the BPS equations in this case requires two projection conditions on the Killing spinors. In more detail, A 1 and A 2 tensors have two parts, one from Y M N and the other from Z M N,P . The latter comes with an extra SO(5) gamma matrices Γ A while the former does not. To obtain a consistent set of BPS equations, we impose the following projectors which reduce the number of supersymmetry to 1 4 of the original amount or eight supercharges.
Following the same procedure as in the previous cases, we obtain the BPS equations g 48 e −2(φ 0 +φ 1 ) 3sech2φ 2 sech2φ 3 + 3 cosh 2φ 2 cosh 2φ 3 + 4e 6φ 1 , (232) Introducing a new radial coordinate ρ via dρ dr = e −4φ 0 −2φ 1 , we can find a domain wall solution to these equations The corresponding gauge generators read with λ ∈ R. t x y = iσ 2 generates the compact SO(2) subgroup, and Q x α ∈ R in general generate six translations R 6 resulting in SO(2) ⋉ R 6 gauge group. As pointed out in [29], the number of independent translations is reduced if there exist non-trivial solutions for Q satisfying We will consider the compact case with TrZ 2 = −2. In this case, the gauged supergravity admits a half-supersymmetric (N = 2) Minkowski vacuum, and the gauge group is reduced to SO(2) ⋉ R 4 ∼ CSO(2, 0, 2). For definiteness, we take an explicit form of Z α β to be There are four SO(2) singlet scalars corresponding to the following SL(5) noncompact generators Using the parametrization of the SL(5)/SO(5) coset representative in the form we find that the scalar potential vanishes identically. This is in agreement with CSO(2, 0, 2) gauge group considered in the previous section.

Conclusions and discussions
We have studied supersymmetric domain walls in N = 4 gauged supergravity in seven dimensions with various gauge groups. There are both half-supersymmetric and 1 4 -supersymmetric solutions depending on which components of the embedding tensor in the 15 and 40 representations of the global symmetry SL(5) lead to the gauging.
For SO(5) gauge group, the gauged supergravity admits a supersymmetric AdS 7 vacuum and can be embedded in eleven-dimensional supergravity. Accordingly, there exist domain walls that are asymptotic to the AdS 7 vacuum and can be interpreted as RG flows from N = (2, 0) SCFT, dual to the AdS 7 , to non-conformal field theories in the IR. The resulting solutions can be uplifted to eleven dimensions. Furthermore, solutions from CSO(4, 0, 1) gauged supergravity can be embedded in type IIA theory via a consistent S 3 truncation. These solutions with clear higher-dimensional origins should be useful in the study of the AdS/CFT correspondence and various dynamical aspects of M5-branes and NS5-branes in different transverse spaces.
There are a number of future directions to pursue. First of all, it is interesting to look for domain walls from CSO(1, 0, 4) and CSO(1, 0, 3) gauge groups that would presumably involve many non-vanishing scalars. These are called elementary domain walls in [17]. Working out the full non-linear ansatz for the truncation of type IIB supergravity on S 3 would be useful and could be used to embed the solutions from SO(4) gauge group to type IIB string theory. Using the solutions given here to holographically study field theories on M5-branes and NS5-branes also deserves further investigation. Finally, finding supersymmetric domain walls with non-vanishing tensor fields as in half-maximal gauged supergravity studied in [36,37] is worth considering.
with the coordinates µ M , M = 1, 2, 3, 4, 5, on S 4 satsifying 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 given by The ansatz for the four-form field strength readŝ with the following definitions After multiplied by 1 2 , the seven-dimensional Lagrangian can be written as Comparing with (31) A.2 Type IIA supergravity on S 3 By taking a limit in which the four-sphere S 4 degenerates to R × S 3 followed by a standard Kaluza-Klein reduction on S 1 , a consistent truncation of type IIA supergravity on S 3 has been obtained in [30]. To present this ansatz, we will split the index M as M = (i, 5), i = 1, 2, 3, 4. The SL(5)/SO(5) coset is decomposed under the SL(4)/SO(4) submanifold as where M ij is a unimodular 4 × 4 symmetric matrix describing the SL(4)/SO(4) coset. The ten-dimensional metric, dilaton and various form field strength tensors are given by with Using the relation (262) and comparing the SL(5)/SO(5) coset given in (139) with (263), we find the relations Φ = e 8φ 0 , In this case, µ i is the coordinates on S 3 satisfying µ i µ i = 1. The gauge couplingĝ is related to g byĝ = 1 4 g as in the S 4 truncation of eleven-dimensional supergravity.