RG flows from (1,0) 6D SCFTs to N=1 SCFTs in four and three dimensions

We study $AdS_5\times \Sigma_2$ and $AdS_4\times \Sigma_3$ solutions of $N=2$, $SO(4)$ gauged supergravity in seven dimensions with $\Sigma_{2,3}$ being $S^{2,3}$ or $H^{2,3}$. The $SO(4)$ gauged supergravity is obtained from coupling three vector multiplets to the pure $N=2$, $SU(2)$ gauged supergravity. With a topological mass term for the 3-form field, the $SO(4)\sim SU(2)\times SU(2)$ gauged supergravity admits two supersymmetric $AdS_7$ critical points, with $SO(4)$ and $SO(3)$ symmetries, provided that the two $SU(2)$ gauge couplings are different. These vacua correspond to $N=(1,0)$ superconformal field theories (SCFTs) in six dimensions. In the case of $\Sigma_2$, we find a class of $AdS_5\times S^2$ and $AdS_5\times H^2$ solutions preserving eight supercharges and $SO(2)\times SO(2)$ symmetry, but only $AdS_5\times H^2$ solutions exist for $SO(2)$ symmetry. These should correspond to some $N=1$ four-dimensional SCFTs. We also give RG flow solutions from the $N=(1,0)$ SCFTs in six dimensions to these four-dimensional fixed points including a two-step flow from the $SO(4)$ $N=(1,0)$ SCFT to the $SO(3)$ $N=(1,0)$ SCFT that eventually flows to the $N=1$ SCFT in four dimensions. For $AdS_4\times\Sigma_3$, we find a class of $AdS_4\times S^3$ and $AdS_4\times H^3$ solutions with four supercharges, corresponding to $N=1$ SCFTs in three dimensions. When the two $SU(2)$ gauge couplings are equal, only $AdS_4\times H^3$ are possible. The uplifted solutions for equal $SU(2)$ gauge couplings to eleven dimensions are also given.

1 Introduction tioned AdS 7 vacua and these AdS 5 or AdS 4 geometries will describe RG flows from N = (1, 0) SCFTs to lower dimensional SCFTs. Previously, this type of solutions has mainly been studied within the framework of the maximal N = 4 gauged supergravity. The solutions provide gravity duals of twisted compactifications of the N = (2, 0) SCFTs. A number of these AdS 5 solutions have been studied in [19,20,21,22], and the resulting solutions can be uplifted to elevendimensional supergravity by using the reduction ansatz given in [23]. In addition, compactifications of N = (1, 0) SCFT has recently been explored from the point of view of massive type IIA theory in [24].
We will give another new solutions to this class from N = 2 SO(4) gauged supergravity. It has been pointed out in [20] that the AdS 5 × S 2 solution preserving SO(2) × SO (2) symmetry and N = 2 supersymmetry in five dimensions, eight supercharges, cannot be obtained from pure minimal N = 2 gauged supergravity. We will show that this solution is a solution of N = 2 SO(4) gauged supergravity obtained from coupling pure N = 2 gauged supergravity to three vector multiplets. We will additionally give new AdS 5 × H 2 solutions which are different from those given in [20] and [21] in the sense that the two SU(2) gauge couplings are different, and the residual symmetry is only the diagonal subgroup of SO(2)×SO (2). This case is not a truncation of the N = 4 SO(5) gauged supergravity, and the embedding of these solutions in higher dimensions are presently unknown. We will also study holographic RG flow solutions interpolating between AdS 7 vacua and these AdS 5 fixed points. The solutions describe deformations of N = (1, 0) SCFTs in six dimensions to the IR N = 1 SCFT in four dimensions.
On AdS 4 solutions from seven-dimensional gauged supergravity, a class of AdS 4 × H 3 and AdS 4 × S 3 solutions have been obtained in [25]. A number of extensive studies of these solutions in terms of wrapped M5-branes on various supersymmetric cycles in special holonomy manifolds have been given in [26,27,28]. In particular, the solution studied in [28] has been obtained from the maximal gauged supergravity and preserves N = 2 superconformal symmetry in three dimensions. In this work, we will look for AdS 4 solutions in the N = 2 SO(4) gauged supergravity preserving only four supercharges. The corresponding solutions should then correspond to some N = 1 SCFTs in three dimensions. We will show that there exist AdS 4 × S 3 and AdS 4 × H 3 solutions in this SO(4) gauged supergravity with four supercharges when the two SU(2) gauge couplings are different. For equal SU(2) gauge couplings, only AdS 4 × H 3 solutions exist and can be uplifted to eleven dimensions using the reduction ansatz given in [29].
The paper is organized as follow. In section 2, relevant information on N = 2 SO(4) gauged supergravity in seven dimensions and supersymmetric AdS 7 critical points are reviewed. AdS 5 × S 2 and AdS 5 × H 2 solutions together with holographic RG flows from AdS 7 critical points to these AdS 5 fixed points will be given in section 3. We present AdS 4 × S 3 and AdS 4 × H 3 solutions in section 4 and give the embedding of some AdS 5 × Σ 2 and AdS 4 × Σ 3 solutions in eleven dimensions in section 5. We finally give some comments and conclusions in section 6.
2 Seven-dimensional N = 2 SO(4) gauged supergravity and AdS 7 critical points In this section, we give a description of the SO(4) N = 2 gauged supergravity in seven dimensions and the associated supersymmetric AdS 7 critical points. These critical points preserve N = 2 supersymmetry in seven dimensions and correspond to six-dimensional N = (1, 0) SCFTs. All of the notations used throughout the paper are the same as those in [16] and [17].

SO(4) gauged supergravity
The SO(4) N = 2 gauged supergravity in seven dimensions is constructed by gauging the half-maximal N = 2 supergravity coupled to three vector multiplets. The supergravity multiplet (e m µ , ψ A µ , A i µ , χ A , B µν , σ) consists of the graviton, two gravitini, three vectors, two spin-1 2 fields, a two-form field and the dilaton. We will use the convention that curved and flat space-time indices are denoted by µ, ν, . . . and m, n, . . ., respectively. Each vector multiplet (A µ , λ A , φ i ) contains a vector field, two gauginos and three scalars. The bosonic field content of the matter coupled supergravity then consists of the graviton, six vectors and ten scalars parametrized by the R + ×SO(3, 3)/SO(3)×SO(3) ∼ R + ×SL(4, R)/SO(4) coset manifold. In the following, we will consider the supergravity theory in which the two-form field B µν is dualized to a three-form field C µνρ . The latter admits a topological mass term, so the resulting gauged supergravity admits an AdS 7 vacuum.
The SO(4) gauged supergravity is obtained by gauging the SO(4) ∼ SO(3) × SO(3) subgroup of the global symmetry group SO (3,3). One of the All spinor fields, including the supersymmetry parameter ǫ A , are symplectic-Majorana spinors transforming as a doublets of the SU(2) R R-symmetry. From now on, the SU(2) R douplet indices A, B = 1, 2 will not be shown explicitly. The bosonic Lagrangian of the N = 2 gauged supergravity is given by where the scalar potential and the Chern-Simons term are given by with the gauge field strength defined by The structure constants f K IJ of the gauge group include the gauge coupling associated to each simple factor in a general gauge group G 0 ⊂ SO (3,3).
We are mainly interested in supersymmetric solutions. Therefore, the supersymmetry transformations of fermions are necessary. However, we will not consider bosonic solutions with the three-form field turned on. We will accordingly set C µνρ = 0 throughout. The fermonic supersymmetry transformations, with all fermions and the three-form field vanishing, are given by Various quantities appearing in the Lagrangian and supersymmetry transformations are defined by the following relations where γ m are space-time gamma matrices satisfying {γ m , γ n } = 2η mn with η mn = diag(−1, 1, 1, 1, 1, 1, 1).

Supersymmetric AdS 7 critical points
We will now briefly review supersymmetric AdS 7 critical points found in [17]. There are two critical points preserving the full N = 2 supersymmetry in seven dimensions. The two critical points however have different symmetries namely one critical point, at which all scalars vanishing, preserves the full SO(4) gauge symmetry while the other is only invariant under the diagonal subgroup gauge group, the gauge structure constants can be written as [16] f IJK = (g 1 ǫ ijk , −g 2 ǫ rst ).
Before discussing the detail of the two critical points, we give an explicit parametrization of the SO(3, The non-compact generators corresponding to 9 scalars take the form of Accordingly, the coset representative can be obtained by an exponentiation of the appropriate Y ir generators. Y ir generators and the 9 scalars transform as ( The coset representative is then given by The scalar potential for the dilaton σ and the SO(3) diag singlet scalar φ can be straightforwardly computed. Its explicit form reads [17] V = 1 32 e −σ (g 2 1 + g 2 2 ) (cosh(6φ) − 9 cosh(2φ)) + 8g 1 g 2 sinh 3 (2φ) +8 g 2 2 − g 2 1 + 64h 2 e 5σ + 32e There are two supersymmetric AdS 7 vacua given by where we have chosen g 1 = −16h in order to make the SO(4) critical point occurs at σ = 0. This is achieved by shifting σ. The value of the cosmological constant has been denoted by V 0 . The two critical points correspond to N = (1, 0) SCFTs in six dimensions with SO(4) and SO(3) symmetries, respectively. An RG flow solution interpolating between these two critical points has already been studied in [17]. In the next sections, we will study supersymmetric RG flows from these SCFTs to other SCFTs in four and three dimensions providing holographic descriptions of twisted compactifications of these N = (1, 0) SCFTs.

Flows to N = 1 SCFTs in four dimensions
In this section, we look for solutions of the form AdS 5 ×S 2 or AdS 5 ×H 2 in which S 2 and H 2 are a two-sphere and a two-dimensional hyperbolic space, respectively.
In the case of S 2 , we take the seven-dimensional metric to be with dx 2 1,3 being the flat metric on the four-dimensional spacetime. By using the vielbein we can compute the following spin connections where ′ denotes the r-derivative. Hatted indices are tangent space indices.
In the case of H 2 , we take the matric to be With the vielbein the spin connections are found to be

AdS 5 solutions with SO(2) × SO(2) symmetry
We now construct the BPS equations from the supersymmetry transformations of fermions. We first consider the S 2 case. In order to preserve supersymmetry, we make a twist by turning on the SO(2) × SO(2) ⊂ SO(4) gauge fields, among the six gauge fields A I , such that the spin connections on S 2 is cancelled by these gauge connections. The Killing spinor corresponding to the unbroken supersymmetry is then a constant spinor on S 2 . We begin with the solutions preserving the full SO(2) × SO(2) residual gauge symmetry generated by J (1) 12 and J (2) 12 . Scalars which are singlet under SO(2) × SO(2) are the dilaton and the scalar corresponding to the SO(3, 3) non-compact generators Y 33 . We will denote this scalar by Φ. By considering the variation of the gravitino along S 2 directions, we find that the cancellation between the spin and gauge connections imposes the twist condition Using the projection conditions we find the following BPS equations In the H 2 case, we choose the gauge fields to be which can be verified that the spin connection ωxŷ in (22) is cancelled by virtue of the twist condition (24) and the projection conditions By an analogous computation, we find a similar set of BPS equations as in (26), (27), (28) and (29) with (a, b) replaced by (−a, −b). At large r, solutions to the above BPS equations should approach the SO(4) AdS 7 critical point with Φ ∼ σ ∼ 0 and F ∼ G ∼ r. This is the UV (1, 0) SCFT. As r → −∞, we look for the solution of the form AdS 5 × S 2 or AdS 5 × H 2 such that φ ′ = σ ′ = G ′ = 0 and F ′ = constant. We find that there is an AdS 5 solution given by This solution is given for Σ 2 = S 2 . The solution in the H 2 case is given similarly by flipping the signs of a and b.
It should be noted that, in this fixed point solution with SO(2) × SO(2) symmetry, the coupling g 2 does not appear. The solution can then be taken as a solution of the gauged supergravity with g 2 = g 1 . Therefore, the solution can be uplifted to eleven dimensions by using the reduction ansatz in [29]. This will be done in section 5. The uplifted solution is however not new since similar solutions have been found previously in [20,21], and supergravity solutions interpolating between AdS 7 and AdS 5 × S 2 or AdS 5 × H 2 have also been investigated. The solutions have an interpretation in terms of RG flows from the UV SCFT in six dimensions to four-dimensional SCFTs with SO(2) × SO(2) symmetry.
Note also that, in this case, it is not possible to find an RG flow from the SO(3) AdS 7 point to any of these four-dimensional SCFTs since this AdS 7 critical point is not accessible from the BPS equations given above.

AdS 5 solutions with SO(2) symmetry
We now consider AdS 5 solutions with SO(2) symmetry. We will study two possibilities namely the SO (2)

Flows with SO(2) diag symmetry
We begin with the SO(2) diag symmetry generated by J The three singlets correspond to the non-compact generators The coset representative describing these singlets can be written as Since we have not found any AdS 5 × S 2 solution, we will give only the result for the H 2 case. The SO(2) diag gauge field can be obtained from the SO(2) × SO(2) gauge fields in (23) with the condition that As in the previous case, the twist imposes the condition g 1 a = 1 which in the present case also implies g 2 b = 1.
Using the projection conditions (31), we find the following BPS equations In this case, there are a number of possible AdS 5 fixed point solutions, and it is possible to have a solution interpolating between the SO(3) AdS 7 critical points and the AdS 5 in the IR. We will investigate each of them in the following discussion.
We first look at the AdS 5 × H 2 critical point with g 2 = g 1 since this can be uplifted to eleven dimensions. When g 2 = g 1 , the fixed point solution exists only for Φ 1 = Φ 3 = 0, and the corresponding solution is given by The AdS 5 solution preserves eight supercharges corresponding to N = 1 superconformal field theory in four dimensions with SO(2) symmetry. A flow solution  interpolating between this AdS 5 × H 2 fixed point and the SO(4) AdS 7 given in (15) for h = 1 is shown in Figure 1.
It should be noted here that this fixed point can be obtained from the SO(2)×SO(2) fixed points given in the previous section by setting the parameter b = a. It can be readily verified that, for b = a, solution in (32) is valid only for the upper sign and Σ 2 = H 2 . The resulting solution is precisely that given in (43).
We now move to solutions with g 2 = g 1 . The solution given in (43) is a special case of a more general solution, with Φ 1 = Φ 3 = 0 and g 2 = g 1 , which is given by where we have used the relation g 1 = −16h in the solutions for σ and G to simplify the expressions. An example of the corresponding flow solutions from the UV N = (1, 0) SO(4) SCFT to this critical point, with g 2 = −2g 1 and h = 1, is given in Figure 2.
In all of the above solutions, it is not possible to have a flow from the SO(3) AdS 7 critical point (16). To find this type of flows, we look for AdS 5 fixed points with Φ 3 = 0 but Φ 1 = 0 and Φ 2 = 0. In this case, the AdS 5 × H 2 solution  is given by Note that at the values of Φ 1 and Φ 2 are the same as the SO(3) AdS 7 point. In equation (16), we have Actually, there are two equivalent values of Φ 1 namely either Φ 1 = Φ 0 or Φ 1 = −Φ 0 . The two choices are equivalent in the sense that they give rise to the same value of the cosmological constant and the same scalar masses. The difference between the two is the generators of SO(3) under which the SO(3) singlet scalar φ in (16) 13 and J 23 . This difference does not affect the result discussed here since, in both cases, the residual SO(2) diag is still generated by J   [17].
We now consider RG flows from N = (1, 0) SCFTs in six dimensions to four-dimensional SCFTs identified with the critical point (45). In order to give some explicit examples, we choose particular values of the two couplings g 1 and g 2 . In the following solutions, we will set g 2 = −2g 1 and h = 1. With these, the IR AdS 5 × H 2 is given by The SO(4) UV point (15) is given by We have chosen Φ 1 = Φ 2 at the IR fixed points for definiteness. There exist an RG flow from the SO(4) N = (1, 0) SCFT in the UV to the N = 1 four-dimensional SCFT in the IR as shown in Figure 3. With a particular boundary condition, we can find an RG flow from the SO(4) AdS 7 to the SO(3) AdS 7 critical points and then to the AdS 5 critical point as shown in Figure 4. This solution is similar to the flow from SO(6) AdS 5 to Khavaev-Pilch-Warner (KPW) AdS 5 critical point and continue to a two-dimensional N = (2, 0) SCFT in [30].

Flows with SO(2) R symmetry
We then move on and briefly look at the SO(2) R symmetry. There are three singlet scalars from the SO(3, 3)/SO(3) × SO(3) coset.These scalars will be denoted by Φ 1 , Φ 2 and Φ 3 corresponding to the non-compact generators Y 31 , Y 32 and Y 33 , respectively.
In this case, the gauge field corresponding the SO(2) R generator is given by By using the same procedure, we find that, in order to have a fixed point, all of the Φ i 's must vanish, and only      There exist RG flows from the SO(4) N = (1, 0) SCFT to these fourdimensional SCFTs. The BPS equations describing theses flows are given by Examples of the solutions with some values of the parameter a are shown in Figure 5. This critical point is also a solution of pure N = 2 gauged supergravity studied in [19].

Flows to N = 1 SCFTs in three dimensions
In this section, we look for AdS 4 vacua of the form AdS 4 × S 3 or AdS 4 × H 3 with S 3 and H 3 being a three-sphere and a three-dimensional hyperbolic space, respectively. These solutions will correspond to some SCFTs in three dimensions. In order to identify these AdS 4 vacua with the IR fixed points of the six-dimensional SCFTs corresponding to both of the AdS 7 vacua given in (15) and (16), we consider the scalars which are singlets under SO(3) diag subgroup of the full SO(4) gauge group. The relevant scalar from the SO(3, 3)/SO(3) × SO(3) coset is the one corresponding to the generator (12) with the coset representative given in (13). In the S 3 case, we will take the metric ansatz to be ds 2 7 = e 2F dx 2 1,2 + dr 2 + e 2G dψ 2 + sin 2 ψ(dθ 2 + sin 2 θdφ 2 ) .
From the above metric, we find the spin connections which accordingly suggest to turn on the following SO(3) diag gauge fields Note that at the beginning, the parameter a of each gauge field needs not be equal. However, the twist condition requires that all of the parameters in front of A i must be equal. The corresponding field strengths are, after using (58), To set up the BPS equations, we impose the projection conditions For the H 3 case, we take the metric to be with the spin connections given by We then turn on the following gauge fields, to cancel the above spin connections on H 3 , with A i+3 = g 1 g 2 A i , i = 1, 2, 3. These gauge fields then become SO(3) diag gauge fields.
We will also impose the projection conditions The twist condition is still given by (58). In both cases, the last projector in (60) and (64) is not independent from the second and the third ones, so the fixed point solution will preserve four supercharges corresponding to N = 1 superconformal symmetry in three dimensions.
With all of the above conditions, we find the following BPS equations, for the H 3 case, The corresponding equations for the S 3 case are similar with a replaced by −a. We now look for a fixed point solution at which G ′ = φ ′ = σ ′ = 0 and F ′ = constant. For g 2 = g 1 , only AdS 4 × H 3 solutions exist and are given by This solution can be uplifted to eleven dimensions using the ansatz of [29]. When g 2 = g 1 , we also find AdS 4 × H 3 solutions , G = 1 2 ln 5a(g 2 2 − g 2 1 ) This solution can be connected to both AdS 7 critical points in (15) and (16) by some RG flows. In this g 2 = g 1 case, there can be both AdS 4 ×S 3 and AdS 4 ×H 3 solutions. The solution however takes a more complicated form depending on the values of g 1 and g 2 . The AdS 4 × H 3 and AdS 4 × S 3 solutions are given respectively by and In both cases, the scalar φ 0 is a solution to the equation The explicit form of φ 0 can be obtained but will not be given here due to its complexity. There are many possible solutions for φ 0 depending on the values of g 1 , g 2 and a. An example of AdS 4 × S 3 solutions is, for g 2 = 1 2 g 1 , given by φ = −0.9158, σ = 0.5493, One of the AdS 4 × H 3 solutions is, for g 2 = 1 2 g 1 , given by Numerical solutions for RG flows from the UV N = (1, 0) SCFTs in six dimensions to these three-dimensional N = 1 SCFTs can be found in the same way as those given in the previous section. And, with suitable boundary conditions, the flow from SO(4) AdS 7 point to the SO(3) AdS 7 point and then to AdS 4 × S 3 or AdS 4 × H 3 in the case of g 2 = g 1 should be similarly obtained. We will however not give these solutions here.

Uplifting the solutions to eleven dimensions
In this section, we will uplift some of the AdS 5 and AdS 4 solutions found in the previous sections to eleven dimensions using a reduction ansatz given in [29].
Only solutions with equal SU(2) gauge couplings, g 2 = g 1 , can be uplifted by this ansatz. Therefore, we will consider only this case in the remaining of this section.
The reduction ansatz given in [29] is naturally written in terms of SL(4, R)/SO(4) scalar manifold rather than the SO(3, 3)/SO(3) × SO(3) we have considered throughout the previous sections. It is then useful to change the parametrization of scalars from the SO(3, 3)/SO(3) × SO(3) to SL(4, R)/SO(4) cosets. For conveniences, we will repeat the supersymmetry transformations of fermions with the three-form field fermions vanishing where In the above equations, V R α denotes the SL(4, R)/SO(4) coset representative. For the explicit form of the eleven-dimensional metric and the four-form field including the notations used in the above equations, we refer the reader to [29]. We now consider the AdS 5 and AdS 4 solutions separately.

Uplifting the AdS 5 solutions
For AdS 5 solutions, the seven-dimensional metric is given by (17) and (20). We will restrict ourselves to AdS 5 fixed points with SO(2) × SO(2) symmetry. The non-zero gauge fields are A αβ = (A 12 , A 34 ) whose explicit form is given by The U(1) × U(1) singlet scalar from SL(4, R)/SO(4) coset is parametrized by the coset representative from which theT RS = diag(e −Φ , e −Φ , e Φ , e Φ ) follows. Note that the parameter a and b here are different from those in section 3 since the gauge fields A i and A r correspond respectively to the anti-self-dual and self-dual parts of the SO(4) gauge fields A αβ . Using the above supersymmetry transformations and imposing the projection conditions γrǫ = ǫ and γθφΓ 12 ǫ = ǫ, we obtain the BPS equations In the above equations, we have used Γ 34 ǫ = −Γ 12 ǫ which follows from the condition Γ 1234 ǫ = ǫ. The latter is part of the truncation from the maximal SO(5) gauged supergravity to the half-maximal SO(4) gauged supergravity studied in [29]. We have also used the twist condition given by which comes from the requirement that the gauge connection cancels the spin connection. For large r, the solution should approach X = 1, Φ = 0 and F ∼ G ∼ r giving AdS 7 background with SO(4) symmetry. This corresponds to the UV N = (1, 0) SCFT in six dimensions. In the IR with the boundary condition F ∼ r and G, Φ, σ ∼ constant, there is a class of solutions given by This gives AdS 5 × S 2 background preserving U(1) × U(1) symmetry and eight supercharges since only the projector γθφΓ 12 ǫ = ǫ is needed at the fixed point.
Therefore, this solution corresponds to N = 1 SCFT in four dimensions. This solution is the same as in [20] with the identification (m 1 , m 2 ) → (−b, a) up to some field redefinitions. So, we conclude that the AdS 5 × Σ 2 solutions found in [20] is a solution of the N = 2 SO(4) gauged supergravity.
For the H 2 case, the above analysis can be repeated in a similar manner. The resulting BPS equations are, as expected, given by (84), (85), (86) and (87) with (a, b) replaced by (−a, −b). It can also be verified that for both AdS 5 × S 2 and AdS 5 × H 2 solutions given in (89), solutions with the positive sign are valid for g > 0 and a > 0 while solutions with the negative sign are valid for g < 0 and a < 0.
It should also be noted that we can truncate the above BPS equations to those of SO(2) R symmetry, generated by the anti-selfdual gauge field A 12 − A 34 , by setting b = −a. Since the twist condition in this case becomes 2ga = −1 which implies that ga < 0, only the AdS 5 × H 2 exists. This precisely agrees with the result of section 3.2.2. The corresponding solution is given by The AdS 5 ×H 2 with SO(2) diag symmetry found in section 3.2.1 for g 2 = g 1 can also be uplifted using the formulae given here by truncating the SO(2)×SO(2) symmetry to SO(2) diag as remarked previously in section 3.2.1. The SO(2) diag corresponds to the gauge field A 12 since the A 3 and A 6 , in section 3.2, are related to the anti-self-dual, 1 2 (A 12 −A 34 ), and self-dual, 1 2 (A 12 + A 34 ), fields, respectively. So, the SO(2) diag gauge field is given by A 12 . As in section 3.2, only solutions with the upper sign in the solution (89) and AdS 5 × H 2 are possible. The result is given by This is consistent with the twist condition (88) which, for b = 0, becomes ga = −1.
The quantities X 0 , Φ 0 and G 0 are the values of the corresponding fields at the fixed point (89). The quantity ∆ is defined by which, in the present case, gives ∆ = X 0 cos 2 ξ e −Φ 0 sin 2 ψ + e Φ 0 cos 2 ψ + X −4 0 sin 2 ξ .
To set up the associated BPS equations, we use the seven-dimensional metric (61) and the following gauge fields The twist condition is given by ga = 1. We will also impose the projection conditions With all of the above conditions, we obtain the following BPS equations These equations admit a fixed point solution The parametrization of the µ α coordinates can be chosen to be µ α = (cos Ψμ a , sin Ψ) withμ a satisfyingμ aμa = 1. The SO(3) diag symmetry corresponds to the gauge fields A ab . In the following, we accordingly set A 4a = 0 for a = 1, 2, 3 and find that Dµ a = cos ΨDμ a − sin Ψμ a dΨ, Dµ 4 = cos ΨdΨ where Dμ a = dμ a + gA abμb .

Conclusions
We have studied AdS 5 × Σ 2 and AdS 4 × Σ 3 solutions of N = 2 gauged supergravity in seven dimensions with SO(4) gauge group. We have found that there exist both AdS 5 × S 2 and AdS 5 × H 2 solutions with the gauge fields for SO(2) × SO(2) turned on. With SO(2) R or SO(2) diag gauge fields, only AdS 5 × H 2 solution is possible. This is consistent with the results given in [19] and [21]. We recover AdS 5 × S 2 and AdS 5 × H 2 solutions studied in [20] and [21] with SO(2) × SO (2) symmetry. In the case of equal SU(2) gauge couplings, the solutions can be uplifted to eleven dimensions, and the uplifted solutions have explicitly given. We have also considered RG flow solutions interpolating between supersymmetric AdS 7 critical points in the UV and these AdS 5 solutions in the IR. In the case of SO(2) diag symmetry, there exist flow solutions from SO(4) AdS 7 critical point to AdS 5 as well as flows from SO(4) AdS 7 to SO(3) AdS 7 and then continue to AdS 5 fixed points similar to the flows from four-dimensional SCFTs to two-dimensional N = (2, 0) SCFTs studied in [30]. Other results of this paper are a number of new AdS 4 ×S 3 and AdS 4 ×H 3 solutions for unequal SU(2) gauge couplings. With equal SU(2) couplings, only AdS 4 × H 3 geometry is possible, and the resulting solutions can be uplifted to eleven dimensions.
The results obtained in this paper should be relevant in the holographic study of N = (1, 0) SCFTs in six dimensions. These would also provide new AdS 5 and AdS 4 solutions, corresponding to new SCFTs in four and three dimensions, within the framework of seven-dimensional gauged supergravity. The embedding of the solutions in the case of unequal SU(2) gauge couplings (if possible) would be interesting to explore. It would also be interesting to compare the AdS 5 and AdS 4 solutions obtained here and the solutions found recently in [31,32] in the context of massive type IIA theory. Finally, it is of particular interest to find an interpretation of all these solutions in terms of wrapped M5-branes on Σ 2 and Σ 3 . Along this line, it would also be useful to find an implication of the AdS 4 solutions in terms of the M2-brane worldvolume theories.