Supersymmetric solutions from N=6 gauged supergravity

We study supersymmetric solutions in four-dimensional $N=6$ gauged supergravity with $SO(6)$ gauge group. There is a unique $N=6$ supersymmetric $AdS_4$ vacuum with $SO(6)$ symmetry dual to an $N=6$ SCFT in three dimensions. We find a number of domain walls interpolating between this $AdS_4$ vacuum and singular geometries in the IR with $SO(2)\times SO(4)$, $U(3)$, $SO(3)$ and $SO(2)\times SO(2)\times SO(2)$ symmetries. The $SO(3)$ case admits $N=6$ or $N=2$ solutions depending on whether the pseudoscalars are present or not. On the other hand, all the remaining solutions preserve $N=6$ supersymmetry. These solutions describe RG flows from the $N=6$ SCFT to non-conformal field theories driven by mass deformations. In particular, the $SO(2)\times SO(4)$ solution is in agreement with the previously known mass deformations of the dual $N=6$ SCFT. We also give a supersymmetric Janus solution with $SO(2)\times SO(4)$ symmetry, describing two-dimensional conformal defects in the $N=6$ SCFT with unbroken $N=(4,2)$ supersymmetry. Finally, we find an $N=2$ supersymmetric $AdS_2\times H^2$ solution with $SO(2)\times SO(4)$ symmetry and the corresponding domain wall interpolating between this fixed point and the $AdS_4$ vacuum. The solution describes an $AdS_4$ black hole with a magnetic charge and is dual to a twisted compactification of the $N=6$ SCFT on a hyperbolic space $H^2$. We also give a domain wall interpolating between a locally supersymmetric $AdS_4$ and a curved domain wall with $SO(2)\times SO(2)\times SO(2)$ symmetry dual to an RG flow across dimensions from the $N=6$ SCFT to a supersymmetric quantum mechanics.


Introduction
Supersymmetric solutions of gauged supergravities in various space-time dimensions play an important role in string/M-theory. In the AdS/CFT correspondence [1,2,3], these solutions provide holographic descriptions of strongly coupled systems such as (non) conformal field theories, conformal defects, AdS-black holes and condensed matter physics. In many cases, solutions of lower-dimensional gauged supergravities can be uplifted to ten or eleven dimensions via consistent truncations resulting in complete AdS/CFT dualities in the context of string/Mtheory.
In this paper, we are interested in supersymmetric solutions of fourdimensional N = 6 gauged supergravity with SO(6) gauge group. This has been constructed in [4] in the embedding tensor formalism obtained from truncating the maximal N = 8 gauged supergravity [5], see also [6,7,8]. The N = 6 gauged supergravity has been shown to admit a unique N = 6 supersymmetric AdS 4 vacuum, with the full SO(6) symmetry unbroken, dual to an N = 6 superconformal field theory (SCFT) in three dimensions. The uniqueness of the N = 6 AdS 4 vacua has also been shown in a recent result on supersymmetric AdS vacua [9]. It has been pointed out in [4] that this AdS 4 fixed point describes a truncation of type IIA theory on CP 3 , so the AdS 4 vacuum can be uplifted to AdS 4 × CP 3 geometry in type IIA theory. This truncation has been studied long ago in [10] in which the full mass spectrum has also been given, for more recent studies see for example [11,12,13,14].
The very first example of the dual N = 6 SCFT from type IIA theory has been given in [15]. In general, SCFTs in three dimensions take the form of Chern-Simons-Matter (CSM) theories since the usual gauge theories with Yang-Mills gauge kinetic terms are not conformal. A number of these SCFTs with different numbers of supersymmetries have already been constructed, see [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] for an incomplete list. These SCFTs arise as world-volume theories of M2-branes on various transverse spaces and play an important role in understanding the dynamics of M2-branes. Supersymmetric solutions of four-dimensional gauged supergravities are expected to be very useful for their holographic descriptions at least in the large-N limit.
Various types of supersymmetric solutions from gauged supergravities have been considered and given interpretations in terms of the corresponding dual field theories. We will study these solutions in N = 6 gauged supergravity beginning with supersymmetric domain walls interpolating between the N = 6 supersymmetric AdS 4 vacuum and singular geometries. The solutions describe holographic RG flows from the dual N = 6 SCFT in the UV to non-conformal phases in the IR arising from mass deformations of the UV N = 6 SCFT. Similar solutions have been extensively studied in N = 8 and N = 2 gauged supergravities, see for example [36,37,38,39,40,41,42,43,44]. Solutions in gauged supergravities with N = 3, 4, 5 supersymmetries have also been considered recently in [45,46,47,48,49,50]. This work could hopefully fill the existing gap by providing a number of supersymmetric solutions in N = 6 gauged supergravity.
We will also find Janus solutions in the form of AdS 3 -sliced domain walls interpolating between asymptotically AdS 4 spaces. These are holographically dual to two-dimensional conformal defects within the N = 6 SCFT and break the superconformal symmetry in the three-dimensional bulk to a smaller one on the two-dimensional surfaces. Solutions of this type in other four-dimensional gauged supergravities have previously been studied in [47,48,50,51,52,53,54]. Finally, we will look for AdS 2 × Σ 2 geometries with Σ 2 being a Riemann surface together with solutions interpolating between these backgrounds and the supersymmetric AdS 4 vacuum. The solutions describe supersymmetric black holes in asymptotically AdS 4 space, and a number of these solutions have already been studied in other gauged supergravities in [50,55,56,57,58,59,60,61,62,63,64,65]. In the dual field theory, the solutions describe RG flows from the N = 6 SCFT to superconformal quantum mechanics in the IR which play a prominent role in microscopic computation of black hole entropy in asymptotically AdS 4 spaces, see for example [66,67,68]. In this context, the superconformal quantum mechanics, or one-dimensional SCFT, is obtained from the N = 6 SCFT via twisted compactifications on Σ 2 .
Four-dimensional N = 6 gauged supergravity has SO * (12) global symmetry with the maximal compact subgroup U(6) ∼ SU(6) × U(1). There are thirty scalars encoded in SO * (12)/U(6) coset manifold. The SO(6) gauging of this supergravity can be obtained from a consistent truncation of the maximal N = 8 gauged supergravity with SO(8) gauge group. The latter is in turn a consistent truncation of eleven-dimensional supergravity on S 7 [69,70,71,72,73,74]. The N = 6 gauged supergravity with SO(6) gauge group can accordingly be uplifted to eleven dimensions via a series of consistent truncations. On the other hand, the SO(6), N = 6 gauged supergravity is a consistent truncation of type IIA theory on CP 3 . Therefore, all the solutions given here have known higher-dimensional origins and can be embedded in ten-or eleven-dimensional supergravities. The scalar potential of the N = 6 gauged supergravity has been analyzed for a long time in [75]. More recently, this gauged supergravity has been rewritten in a more general setting of the embedding tensor formalism in [4] in which the fermionshift matrices and the scalar potential have been given by truncating the N = 8 theory. In this paper, we first complete the task by extending the truncation to all terms in the bosonic Lagrangian and fermionic supersymmetry transformations. Both of these are of course a relevant part in the present analysis.
The paper is organized as follows. In section 2, we review four-dimensional N = 6 gauged supergravity with SO(6) gauge group in the embedding tensor formalism. In section 3, we study supersymmetric domain wall solutions describing RG flows in the dual N = 6 SCFT to non-conformal phases in the IR. We then turn to supersymmetric Janus solutions in section 4 and finally look for possible supersymmetric AdS 2 × Σ 2 solutions together with flow solutions interpolating between the AdS 4 vacuum and these geometries in section 5. Conclusions and comments are given in section 6.
2 N = 6 gauged supergravity with SO(6) gauge group We first give a review of N = 6 gauged supergravity in the embedding tensor formalism as described in [4]. We will follow most of the convention in [4] but with a mostly plus signature for the space-time metric. The only supermultiplet in N = 6 supersymmetry is the gravity multiplet with the field content The component fields correspond to the graviton eμ µ , six gravitini ψ µA , sixteen vectors A AB µ = −A BA µ and A 0 µ , twenty-six spin-1 2 fields χ ABC = χ [ABC] and χ A together with fifteen complex scalars φ AB = −φ BA . Real and imaginary parts of φ AB are usually called scalars and pseudoscalars, respectively.
In this work, space-time and tangent space indices are denoted by µ, ν, . . . = 0, 1, 2, 3 andμ,ν, . . . = 0, 1, 2, 3, respectively. Indices A, B, . . . = 1, 2, . . . , 6 correspond to the fundamental representation of SU(6) which is in turn a subgroup of the R-symmetry U(6) ∼ SU(6) × U(1). The 30 real scalars within φ AB are coordinates of the scalar manifold SO * (12)/U(6) and can be described by the coset representative in representation 32 of SO * (12) of the form with the Claley matrix and We also note thatφ AB = (φ AB ) * . In subsequent analysis, it is useful to define 16 × 16 submatrices of V M M by the following identification The inverse of V M M can accordingly be written in terms of f and h as The sixteen electric gauge fields A AB and A 0 combine into A Λ = (A 0 , A AB ). Together with the magnetic dual A Λ , the gauge fields transform as 32 representation of SO * (12) Gaugings are efficiently described by the embedding tensor formalism in which the corresponding gauge generators are defined as with t n being the SO * (12) generators. θ M n is called the embedding tensor in term of which the covariant derivative implementing the minimal coupling of various fields can be written as ∇ µ is the usual space-time covariant derivative including (possibly) the local U(6) composite connection. The parameter g is the gauge coupling constant which can be absorbed in the definition of θ M m . In 32 representation, with SO * (12) generators (t n ) M N , the embedding tensor can be described by the generalized structure constants To define a proper gauging and preserve the full supersymmetry of the ungauged theory, the embedding tensor needs to satisfy the so-called linear and quadratic constraints given respectively by with f mn p being the SO * (12) structure constants. The former implies that the embedding tensor θ M m is in the representation 351 of SO * (12) while the latter gives rise to [X M , X N ] = −X M N P X P .
The gauge generators then form a closed subalgebra for which X M N P act as the corresponding structure constants.
As usual in gauging a supergravity theory, supersymmetry requires some modifications to the ungauged Lagrangian and supersymmetry transformations. These modifications are of first and second order in the gauge coupling constant and can be written in term of the so-called T-tensor The bosonic Lagrangian of the N = 6 gauged supergravity can be written as The scalar kinetic term is given in term of the vielbein P ABCD µ = (P µABCD ) * on the SO * (12)/U(6) coset which is defined by The scalar matrix appearing in the gauge kinetic terms is given by with N ΛΣ being its complex conjugate. The complex self-dual and anti-self-dual gauge field strengths are defined by with F Λ µν given by The scalar potential is obtained from the fermion-shift matrices as follows We note that upper and lower SU(6) indices are related by complex conjugation. In terms of the various components of the T-tensor with the spliting of indices Λ, Σ, . . . as (0, [AB]), we have The fermionic supersymmetry transformations, with all fermionic fields vanishing, are given by We note here the chiralities of the fermionic fields with ψ A µ , χ ABC and χ A having opposite chiralities. The tensorsF with and The connection Q µA B is given by with Q M A B obtained from by the relation Q M AB . In general, both electric and magnetic gauge fields can participate in the gaugings leading to many possibilities of viable gauge groups. However, in this work, we will only consider SO(6) gauge group embedded electrically in U(6) ⊂ SO * (12). This gauging only involves electric gauge fields A AB . In this case, we have and X I 1 J 1 with all remaining components vanishing. In particular, there are no X Λ M N components which couple to magnetic gauge fields.
With the splitting of indices Λ, Σ, . . . = (0, [IJ]), we find from the definition (14) that From these, it is straightforward to obtain all the fermion-shift matrices and the scalar potential.
In subsequent sections, we will look for various types of supersymmetric solutions to this N = 6 gauged supergravity with SO(6) gauge group. It has been shown in [4] that this gauged supergravity admits a supersymmetric N = 6 AdS 4 vacuum with the cosmological constant V 0 = −48g 2 when all scalars vanish. According to the AdS/CFT correspondence, this is dual to an N = 6 SCFT in three dimensions. We will find solutions that are asymptotic to this AdS 4 geometry and can be interpreted as different types of deformations of the dual N = 6 SCFT.

Holographic RG flows
We first consider holographic RG flow solutions in the form of domain walls interpolating between the supersymmetric AdS 4 vacuum and another AdS 4 vacuum (if exists) or a singular geometry. These solutions correspond respectively to RG flows of the dual UV N = 6 SCFT to another conformal fixed point or to a non-conformal phase in the IR.
The metric ansatz is taken to be with dx 2 1,2 being the flat metric on three-dimensional Minkowski space. Scalar fields are allowed to depend only on the radial coordinate r with all the other fields set to zero.

Solutions with SO(2) × SO(4) symmetry
We first consider a simple case of solutions with SO(2) × SO(4) symmetry. The embedding of SO(6) implies that the scalars φ AB transform as an adjoint representation of SO(6). There is one singlet scalar under SO(2) × SO(4) ⊂ SO(6) given explicitly by We will also write φ = ϕe iζ (36) for real scalars ϕ and ζ depending only on r. By a straightforward computation, we find the tensor S AB of the form We have introduced the "superpotential" W for convenience. In general, the function W is related to the eigenvalue of S AB corresponding to the unbroken supersymmetry. In the present case, S AB is proportional to the identity matrix indicating that the solutions will preserve either N = 6 supersymmetry with all ǫ A non-vanishing or no supersymmetry at all. Note also that W has a critical point at ϕ = 0 which is the supersymmetric N = 6 AdS 4 vacuum mentioned above.
To solve all the BPS conditions, we will, as in other previous works, impose the following projector for a real function Λ. Throughout the paper, we will use Majorana representation for gamma matrices in which all γ µ are real, but γ 5 is purely imaginary. This implies that ǫ A and ǫ A are related by complex conjugation. Note also that the projector (38) relates the two chiralities of ǫ A , so the full flow solutions will preserve only half of the original supersymmetry or twelve supercharges. Considering the conditions δψ µA = 0 for µ = 0, 1, 2, we find with ′ denoting r-derivatives. This equation gives In what follow, we will write W = |W| for convenience. We will also choose the upper signs in order to make the supersymmetric AdS 4 critical point correspond to r → ∞. Since, in this case, the superpotential is real, we simply have The condition δψ rA = 0 gives the standard Killing spinors of the domain walls for spinors ǫ A(0) satisfying (38).
Using the projection (38) in the variations δχ ABC and δχ A gives the following BPS equations We have now obtained the BPS equations that solve all the supersymmetry conditions. It can also be readily verified that these equations imply the second-order field equations. We can analytically solve the above BPS equations with the following solution We have neglected the integration constants in these equations since they can be removed by shifting the radial coordinate and scaling the x 0,1,2 coordinates, respectively. As r → ∞, we find that with L being the AdS 4 radius related to the cosmological constant by We have also taken g > 0 for convenience. The behavior of ϕ implies that ϕ is dual to a relevant operator of dimensions ∆ = 1, 2 in the dual SCFT. In addition, the solution is singular at r → 0 with ϕ ∼ ± ln(4gr) and A ∼ ln(4gr) .
We then find that ϕ → ±∞ near the singularity. From the explicit form of the scalar potential, we have By the criterion given in [76], we conclude that the singularity is physical. Therefore, the solution decribes an RG flow from the UV N = 6 SCFT to a nonconformal phase in the IR. The flow is driven by an operator of dimensions ∆ = 1, 2 corresponding to scalar or fermion mass terms in three dimensions. The flow breaks superconformal symmetry but preserves the full N = 6 Poincare supersymmetry. Moreover, the R-symmetry SO(6) is broken to SO(2) × SO (4) subgroup. This is precisely in agreement with the field theory result given in [24]. We then expect the solution to describe mass deformations of the threedimensional N = 6 SCFT.

Solutions with U (3) symmetry
We now consider another residual symmetry namely U(3) ∼ SU(3) × U(1) ⊂ SO (6). The U(3) generators in the fundamental representation of SO(6) can be written as in which A 3×3 and S 3×3 are anti-symmetric and symmetric 3 × 3 matrices, respectively. The matrices There is only one U(3) singlet scalar given by The matrix J AB is identified with the Kahler form of CP 3 on which the tendimensional type IIA theory compactifies [4]. By writting φ = ϕe iζ and repeating the same analysis as in the previous case, we find the scalar potential which is exactly the same as that given in [4]. As in the SO(2) × SO(4) case, this potential admits an N = 6 AdS 4 critical point at ϕ = 0 dual to an N = 6 SCFT in three dimensions. The matrix S AB is proportional to the identity with a complex superpotential The variations δχ A and δχ ABC lead to which implies ζ ′ = 0. It turns out that ζ ′ = 0 is also required by the field equations. For constant ζ = ζ 0 , we have verified that all the resulting BPS equations are compactible with the field equations. In the following analysis, we will set ζ 0 = 0 and end up with the BPS equations The solution can be readily obtained As in the previous case, the solution is asymptotic to the supersymmetric AdS 4 with ϕ dual to an operator of dimensions ∆ = 1, 2 while at r = 0, the solution is singular with ϕ ∼ ln(gr) and and ϕ ∼ − ln(gr) and A ∼ −ϕ ∼ ln(gr).
Both of these give so the two singularities are physical. We can accordingly interpret the solution as a holographic dual of RG flows from the N = 6 SCFT to non-conformal phases in the IR. The flow preserves N = 6 Poincare supersymmetry in three-dimensions as in the SO(2) × SO(4) case but breaks the SO(6) R-symmetry to U(3). It would be interesting to identify the corresponding mass deformations in the dual N = 6 SCFT similar to the SO(2) × SO(4) case.

Solutions with SO(2) × SO(2) × SO(2) symmetry
To obtain more interesting and more complicated solutions, we consider solutions with a smaller symmetry namely There are three complex scalars which are singlets under this SO(2) × SO(2) × SO (2). The explicit parametrization of these singlets can be written as By setting we find the scalar potential It is clearly seen that this potential admits only one critical point at ϕ 1 = ϕ 2 = ϕ 3 = 0 which is the aforementioned N = 6 AdS 4 vacuum. The matrix S AB is given by with W 2 and W 3 take a similar form with the phase e i(ζ 1 −ζ 2 −ζ 3 ) replaced by e i(ζ 2 −ζ 1 −ζ 3 ) and e i(ζ 3 −ζ 1 −ζ 2 ) , respectively.
It turns out that none of these W α gives rise to the superpotential in term of which the scalar potential (64) can be written unless ζ 1 = ζ 2 = ζ 3 = 0. This is also implied by the consistency between the resulting BPS equations and the field equations. We now set ζ 1 = ζ 2 = ζ 3 = 0, resulting in e iΛ = ±1, and obtain the following BPS equations To find the solution to these equations, we first take a linear combination After changing to a new radial coordinate ρ defined by we find for a constant C 2 . Similarly, taking a linear combination ϕ ′ 1 + ϕ ′ 3 gives rise to Using these results in equation (67), we find Finally, with all these results, the solution for A is given by We now look at the behavior of the solution as ϕ α ∼ 0 which gives ρ ∼ r and This is the expected behavior of the solution asymptotic to the supersymmetric AdS 4 vacuum. As in the previous cases, the solution is singular as 4gρ → ln(−C α ). For C 1 = C 2 = C 3 , there are three possibilities: • C 1 < C 2,3 : In this case, the solution is singular when 4gρ → ln(−C 1 ) with • C 2 < C 1,3 or C 3 < C 1,2 : In this case, we find In the first case, we have ϕ 1 → −∞ and ϕ 2,3 → ∞ while in the second case, the solution gives ϕ 1,2,3 → ∞. It can be easily verified that all of these behaviors lead to V → −∞, so the singularities are physically acceptable. The solution then describes different types of mass deformations within the dual N = 6 SCFT to non-conformal phases with SO(2) × SO(2) × SO(2) symmetry. The solution also preserves N = 6 Poincare supersymmetry as in the previous cases.

Solutions with SO(3) symmetry
We further reduce the residual symmetry to SO(3) ⊂ SO(3) × SO(3) ⊂ SO(6) generated by the antisymmetric matrices A 3×3 in the upper-left block of (50). There are three singlet scalars parametrized by By writtingφ α = ϕ α e iζα with and we find the scalar potential In this case, the scalar potential depends on phases of the complex scalars, and the analysis is more complicated. To make the analysis more traceable, we will further truncate to two scalars by setting ϑ = 0 and ξ = −ζ. This is equivalent to settingφ 3 = 0. We now begin with the eigenvalues of the matrix S AB . These are of the form, after diagonalization, with W ± given by The corresponding eigenvectors arê ǫ ± = − 1 2 sec 2θ 2 cos η sin 2θ ∓ 3 + cos 2η + 2 cos 4θ sin 2 η ǫ 5 + ǫ 6 .
The scalar kinetic term is given by with φ α = (Φ, θ, ζ, η). It is useful to give an explicit form of the inverse of G αβ here The scalar potential can be written in term of the real superpotential W = |W + | = |W − | as After setting ǫ 1,2,3,4 = 0 and imposing the projection conditions we find the following BPS equations The flow equations for the scalars can be written in a compact form as It can straightforwardly be verified that all these equations satisfy the secondorder field equations. We also see that from these equations, there is only one supersymmetric critical point, with Φ ′ = θ ′ = ζ ′ = η ′ = 0, at φ α = 0. We note that although the superpotential and the scalar potential do not depend on ζ, ζ ′ is still non-vanishing due to the mixed terms between ζ and η in the matrix G αβ . Moreover, further truncations such as η = 0 or θ = 0 will lead to the BPS equations in the case of N = 6 supersymmetry with all the six eigenvalues of S AB leading to This is very similar to the N = 5 gauged supergravity studied in [50] in which the differences in the phases of the scalars are crucial for breaking the original supersymmetry to lower amount. We now look at the solution to the above equations. Combining η ′ and θ ′ equations gives dθ dη = 1 4 sin 4θ tan η with the solution given by cot 2θ = C 1 cos η .
Similarly, taking the combination between ζ ′ and η ′ equations leads to After using the above solution for θ, we find the solution for constant ζ 0 . Combining Φ ′ and η ′ equations by taking into account all the previous results together with a redefinitionΦ = sinh Φ, we find whose solution is given bỹ . (105) Using all these results in η ′ equation, we find the solution for η(r) implicitly from in which we have defined Ξ = cos 2η .
Finally, we can find the solution for A(Ξ) as with α ± defined by The solution preserves N = 2 supersymmetry and breaks the SO(6) Rsymmetry to SO(3). The solution is singular when This givesΦ → ±∞ or Φ → ±∞ which in turn leads to V → g 2 e 4|Φ| cos 2 θ sin 2 η sin 2 θ .
We can see that the scalar potential is unbounded from above, V → +∞, unless θ = 0 or η = 0 both of which give the N = 6 solution as previously mentioned. Therefore, the IR singularities of the N = 2 solutions are unphysical by the criterion of [76].

Supersymmetric Janus solutions
In this section, we consider supersymmetric Janus solutions in the form of curved domain walls. The solutions can be obtained from an AdS 3 -sliced domain wall ansatz Since the analysis closely follows that given in [51], see also [52], we will not repeat all the detail here but mainly review relevant results for deriving the corresponding BPS equations. Compared to the RG flow case, the BPS equations will be modified by the curvature of the three-dimensional slices. In addition, as pointed out in [51], the existence of Janus solutions requires non-vanishing pseudoscalars resulting in more complicated set of BPS equations in constrast to the simple flat domain wall or RG flow case. We begin with the conditions δψ î µ = 0 forμ = 0, 1 which give This leads to the following BPS equation for W = |W|. We still use the γr projection given in (38). Imposing the γξ projection of the form with κ 2 = 1, we can solve the condition (113) for the γr projector leading to the phase factor for real W and for complex W. We also note that the constant κ = ±1 corresponds to the chiralities of the Killing spinors on the two-dimensional conformal defects described by the AdS 3 -slices. Finally, the conditions δψ Â ξ = 0 and δψ Â r = 0 can be solved to obtain the explicit form of the Killing spinors, see details in [51], in which the constant spinors ε (0) A could possibly have an r-dependent phase and satisfy γrε It turns out that among the previously considered cases only SO(2) × SO(4) and SO(3) symmetric scalars can possibly possess supersymmetric Janus solutions. This is mainly a consequence of the consistency in turning on nonvanishing pseudoscalars. For the SO(3) case, the analysis is highly complicated as already seen in the case of RG flows considered in the previous section. Therefore, we will only give the Janus solution with SO(2) × SO(4) symmetry. This case is more traceable, and it turns out that the solution can be analytically obtained.
With the superpotential we find the following BPS equations It should be noted that, for ℓ → ∞, these equations reduce to those of the RG flow studied in the previous section. Furthermore, these equations take a very similar form to the SO(4) symmetric Janus solution in N = 5 gauged supergravity studied in [50]. By taking ϕ as an independent variable, we can solve for A(ϕ) and ζ(ϕ).
The complete solution is given by for constants C and r 0 . This solution takes the same form as the solution given in [51], [50] and [52] in N = 8, 5, 3 gauged supergravities. We also note the unbroken supersymmetries on the conformal defects in these cases as follows: N = (4, 4), N = (4, 1) and N = (2, 1). All of these solutions should be related by truncations of N = 8 gauged supergravity to N = 3 and N = 5, 6 theories. This indicates that the N = (4, 4) Janus solution of [51] survives in the truncation to N = 3, 5, 6 gauged supergravities. According to the AdS/CFT correspondence, we then expect the dual N = 3, 5, 6 SCFTs to possess the same two-dimensional conformal defect as in the N = 8 SCFT. We end this section by giving a brief comment on the possible SO(3) symmetric Janus solution. A partial analysis shows that there appears to be no obstruction in obtaining the BPS equations for Janus solutions in this case as in a similar analysis of N = 5 theory in [50]. Therefore, we expect a supersymmetric N = 2 Janus solution with SO(3) symmetry to exist in N = 6 gauged supergravity as well. Since the analysis is far more complicated than the SO(2) × SO(4) case, we refrain from giving any definite result here.

Supersymmetric AdS 4 black holes
In this section, we look for supersymmetric AdS 4 black holes in the form of curveddomain wall solutions interpolating between a locally asymptotically AdS 4 and AdS 2 × Σ 2 geometries. The AdS 2 × Σ 2 space describes the near horizon geometry of the black holes. In the following analysis, we will consider only the cases of Σ 2 being a two-sphere (S 2 ) and a hyperbolic space (H 2 ).
The metric ansatz is taken to be with the function F (θ) defined by It is straightforward to derive non-vanishing components of the spin connection with F ′ (θ) = dF dθ . In general, the curvature of the Σ 2 part on the world-volume of the domain wall will completely break supersymmetry. However, it is well-known that some amount of supersymmetry can be preserved by performing a topological twist. This can be achieved by turning on some gauge fields along Σ 2 in such a way that the corresponding spin connection, ωθφ in the above metric, is cancelled. This turns the covariant derivative of ǫ A along Σ 2 into a partial one. The resulting Killing spinors are accordingly given by spinors that are independent of the Σ 2 coordinates.

SO(2) × SO(2) × SO(2) symmetric solutions
We first consider the SO(2) × SO(2) × SO(2) twist with the following ansatz for the gauge fields The constants p i , i = 1, 2, 3, are identified with magnetic charges. The corresponding field strengths are given by We have introduced a parameter κ with κ = 1, −1 for Σ 2 being S 2 or H 2 , respectively. We also note that F ′′ (θ) = −κF (θ). With the SO(2) × SO(2) × SO(2) singlet scalars given by (62), we find non-vanishing components of the composite connection With the component Qφ A B , the spin connection ωθφ can be cancelled by imposing the following projector and the twist conditions These conditions imply that p 1 = p 2 = p 3 . The twist is then obtained from the diagonal subgroup SO(2) diag ⊂ SO(2) × SO(2) × SO(2) as in the pure N = 4 and N = 5 gauged supergravities studied in [77] and [50], respectively. For consistency, we will also set A 3 = A 2 = A 1 . For all p i non-vanishing, the twists allow all the supersymmetries corresponding to ǫ A , A = 1, 2, . . . , 6 to be unbroken subject to the projector (133). We also note some useful relations for deriving the full set of BPS equations. Using ǫ0rθφ = 1 and γ 5 ǫ A = −ǫ A , we find It turns out that, we need to turn on the SO(2) ∼ U(1) gauge field of U(6) ∼ SU(6) × U(1) in order to find a consistent set of BPS equations. We similarly take the ansatz for this U(1) gauge field to be We also note that both A 0 and A IJ can appear in the BPS equations due to the off-diagonal element of the scalar coset representative. In particular, we have the relationsF However, A 0 does not participate in the twist since ǫ A are not charged under the U(1) factor outside SU (6). It is also useful to define the "central charge" matrix In the present case, it turns out that Z AB is proportional to the identity matrix, With all these and the projector (38), we find the following BPS equation, from δψθ A and δψφ A , which leads to With the projectors (38) and ( The latter fixes the time component of the gauge fields. Finally, as in the case of domain walls and Janus solutions, the condition δψr A = 0 determines the rdependence of the Killing spinors giving rise to ǫ A = e f 2 ǫ A(0) . Similar to the RG flow case, it turns out that we need to set ζ 1 = ζ 2 = ζ 3 = 0 for consistency. This gives real W and Z resulting in e iΛ = ±1 and A 1 = 0. We will also set A 0 (r) = 0 for simplicity although it is not constrained by the previously obtained conditions. In addition, the compatibility between the BPS equations coming from δχ A and δχ ABC requires We also note that, in this case, the choice p 0 = 0 breaks all supersymmetry. This implies that the SO(2) × SO(2) × SO(2) twist needs to be accompanied by the U(1) gauge field A 0 µ . With all these, we find a consistent set of BPS equations given by For an AdS 2 × Σ 2 fixed point to exist, we require that ϕ ′ . It can be easily verified that the above equations do not admit any AdS 2 × Σ 2 fixed points.
Although there is no supersymmetric AdS 2 × Σ 2 fixed point, we are able to analytically obtain the complete solution to these BPS equations. Since it might be useful for some holographic studies, we will present the solution here. By changing to a new radial coordinate ρ using the relation dρ dr = e ϕ 3 , we can form the following linear combinations and The first equation can be solved by with an integraton constant ρ 0 . Using this result in the second equation, we find the solution with another integration constant C.
By treating f and h as functions of ϕ 3 , we find the following solutions for f and h Finally, the solution ϕ 3 (ρ) can be given implicitly in the following equation 4C 0 e 4gρ (e 8gρ 0 + e 8gρ + 8gCe 4gρ ) = β 0 + β 1 ln e 4g(ρ 0 −ρ) + 1 e 4g(ρ 0 −ρ) − 1 +β 2 ln e 8g(ρ−ρ 0 ) − 1 1 + e 8g(ρ−ρ 0 ) + 8gCe 4gρ (156) in which C 0 is a constant, and the coefficients β 0 , β 1 and β 2 are defined by Since there is no AdS 2 × Σ 2 fixed point in the IR, the solution describes a flow from the locally supersymmetric AdS 4 vacuum to a curved domain wall with world-volume R × Σ 2 . According to the AdS/CFT correspondence, the solution is expected to describe an RG flow from the N = 6 SCFT in three dimensions to a supersymmetric quantum mechanics in the IR. The latter arises from the former by a twisted compactification on Σ 2 .

SO(2) × SO(4) symmetric solutions
We now look at a truncation of the previous result by setting p 2 = p 3 = 0 and ϕ 2 = ϕ 3 = 0. The resulting solutions will preserve SO(2) × SO(4) symmetry with the twist performed along the SO(2) factor. In this case, the supersymmetry corresponding to ǫ 3,4,5,6 will be broken since we cannot perform a twist along these directions. With ǫ 3,4,5,6 = 0, we find the BPS equations in which we have set ϕ 1 = ϕ. We also note that with only the SO(2) twist, it is not necessary to set p 0 = κp 1 . However, the existence of an AdS 2 × Σ 2 fixed point requires vanishing p 0 . For p 0 = 0, we find a fixed point for constant ϕ 0 . This is an AdS 2 × H 2 fixed point since the reality of h implies κ = −1.
The complete flow solution can be obtained by using the same procedure as in the previous sections. The resulting solution is given by in which we have defined the new radial coordinate ρ by dρ dr = e ϕ . We have neglected the integration constant of f by absorbing it in the rescaling of the time coordinate t.
Near r ∼ ρ → ∞, we find which gives an asymptotically locally AdS 4 critical point. On the other hand, by which is the AdS 2 × H 2 fixed point identified above.

U (3) symmetric solutions
As a final case, we consider U(3) symmetric solutions with a twist performed along the SO(2) ∼ U(1) factor. The corresponding gauge generator of this U(1) factor is given by X 14 + X 25 + X 36 . We then turn on the following gauge fields With the U(3) singlet scalar given in (51), we find the composite connection The twist is implemented by imposing We also note that, similar to the SO(2) × SO(2) × SO(2) twist, all ǫ A can be non-vanishing. In addition, we also need non-vanishing A 0 which we will again use the ansatz (136). In this case, consistency requires p 0 = −κp.
As in the RG flow case, we need to set ζ = 0 for consistency between the BPS equations and the field equations. This again results in A(r) = 0. Repeating the same analysis as in the previous cases, we find the following BPS equations It is easily verified that there is no AdS 2 × Σ 2 fixed point in these equations. In this case, we are not able to obtain the analytic flow solution.

Conclusions and discussions
In this paper, we have studied N = 6 gauged supergravity in four dimensions with SO(6) gauge group. The gauged supergravity can be obtained from a truncation of the maximal N = 8 theory with SO(8) gauge group. There is a unique N = 6 supersymmetric AdS 4 vacuum preserving the full SO(6) gauge symmetry. This can be identified with AdS 4 × CP 3 geometry in type IIA theory dual to an N = 6 SCFT in three dimensions. We have found a number of RG flow solutions with various symmetries from this N = 6 SCFT to possible non-conformal phases in the IR. In particular, there is one solution, breaking the SO(6) R-symmetry to SO(2) × SO(4), with unbroken N = 6 Poincare supersymmetry. This is precisely in agreement with the field theory result on mass deformations of N = 6 SCFTs given in [24]. Other solutions preserve U(3), SO(3) and SO(2) × SO(2) × SO (2) symmetries. While most of the solutions preserve N = 6 supersymmetry, in the case of SO(3) symmetry, it is possible to find N = 2 supersymmetric solutions.
We have analytically given all of these solutions and also checked that, except for the N = 2 solution, the resulting IR singularities are physical by the criterion given in [76].
We have also considered more complicated solutions by generalizing the flat domain walls to the curved ones. In the case of AdS 3 -sliced domain walls, we have found a supersymmetric Janus solution, describing a two-dimensional conformal defect within the N = 6 SCFT, with SO(2) × SO(4) symmetry and N = (2, 4) supersymmetry on the defect. The resulting solution takes the same form as those given in N = 8, N = 5 and N = 3 gauged supergravities studied in [51], [50] and [52]. We therefore argue that these solutions are related to the N = 8 solution by truncations. In order for Janus solutions to exist, it is necessary that pseudoscalars are non-vanishing as pointed out in [51]. It turns out that, among the remaining cases studied in this work, only the SO(3) invariant sector could possibly admit supersymmetric Janus solutions. Futhermore, we have studied supersymmetric solutions of the form AdS 2 × Σ 2 and the interpolating solutions between these geometries and the N = 6 AdS 4 vacuum. We have found one AdS 2 × H 2 fixed point with SO(2) × SO(4) symmetry from SO(2) twist. The solution interpolating between this fixed point and the AdS 4 vacuum preserves two supercharges while the IR fixed point AdS 2 × H 2 has four supercharges. Holographically, this solution corresponds to an RG flow from the N = 6 SCFT to superconformal quantum mechanics which is useful in computing black hole entropy along the ling of [66,67,68].
For SO(2)×SO(2)×SO(2) twist, the BPS equations are more complicated but admit no AdS 2 × Σ 2 fixed point. However, in this case, we are able to obtain a complete flow solution between the AdS 4 critical point to a curved domain wall with world-volume R × Σ 2 in the IR. The solution preserves N = 6 supersymmetry in three dimensions, or twelve supercharges, and SO(2) × SO(2) × SO (2) symmetry. This should be dual to a twisted compactification on Σ 2 of the UV N = 6 SCFT to a supersymmetric quantum mechanics in the IR. We have also looked for AdS 2 × Σ 2 geometries from an SO(2) ∼ U(1) twist in the case of U(3) symmetric solutions, but there do not exist any AdS 2 × Σ 2 fixed points.
Since all the solutions presented here are fully analytic, we hope they could be useful in the study of gauge/gravity holography and other related aspects. We also note that most of the structures of the solutions are very similar to those of the N = 5 gauged supergravity studied in [50]. In particular, the N = 6 Poincare supersymmetry in three dimensions is unbroken on the domain wall solutions if there are no non-vanishing pseudoscalars. Unlike in the N = 5 theory, we are not able to find a definite conclusion on whether this is true in general due to a more complicated scalar coset manifold. However, many similarities in the structures of various types of supersymmetric solutions suggest that this should be the case.
There are a number of directions to extend the present work which is clearly only the first-step in classifying supersymmetric solutions of N = 6 gauged supergravity. First of all, it would be interesting to uplift the RG flow solutions to M-theory via the embedding in N = 8 gauged supergravity which in turn can be obtained from a truncation of M-theory on S 7 . The time component g 00 of the eleven-dimensional metric can be used to determine whether the four-dimensional singularities, in particular the N = 2 case, are physically acceptable in M-theory using the criterion given in [78]. This would lead to a complete holograhic description of mass deformations of N = 6 CSM theory and possible related M-brane configurations.
We have only considered gauged supergravity with SO(6) gauge group electrically embedded in the global SO * (12) symmetry. It would be interesting to study magnetic and dyonic gaugings involving also magnetic gauge fields. In particular, performing a similar study in the case of N = 6 gauged supergravity with the electric-magnetic phase ω, see [79] and [80,81], could be of particular interest since in the omega deformed N = 8 theory, the structure of vacua and domain walls are much richer than the electric counterpart, see [82,83,84,85] for more detail. In addition, the study of genuine N = 6 gaugings which cannot be embedded in the N = 8 theory is worth considering. In this case, the gaugings do not satisfy extra quadratic constraints coming from the truncation of the N = 8 theory, see a discussion in [86], so the corresponding solutions cannot be embedded in the maximal theory.