Twisted compactifications of 6D field theories from maximal 7D gauged supergravity

We study supersymmetric AdSn×Σ, n = 2, 3, 4, 5 solutions in sevendimensional maximal gauged supergravity with CSO(p, q, 5 − p − q) and CSO(p, q, 4 − p − q) gauge groups. These gauged supergravities are consistent truncations of eleven-dimensional supergravity and type IIB theory on Hp,q ◦ T 5−p−q and Hp,q ◦ T 4−p−q, respectively. Apart from recovering the previously known AdSn × Σ7−n solutions in SO(5) gauge group, we find novel classes of AdS5×S, AdS3×S×Σ and AdS3×CP 2 solutions in non-compact SO(3, 2) gauge group together with a class of AdS3×CP 2 solutions in SO(4, 1) gauge group. In SO(5) gauge group, we extensively study holographic RG flow solutions interpolating from the SO(5) supersymmetric AdS7 vacuum to the AdSn × Σ7−n fixed points and singular geometries in the form of curved domain walls with Mkwn−1×Σ slices. In many cases, the singularities are physically acceptable and can be interpreted as non-conformal phases of (n− 1)-dimensional SCFTs obtained from twisted compactifications of N = (2, 0) SCFT in six dimensions. In SO(3, 2) and SO(4, 1) gauge groups, we give a large number of RG flows between the new AdSn×Σ fixed points and curved domain walls while, in CSO(p, q, 4−p−q) gauge group, RG flows interpolating between asymptotically locally flat domain walls and curved domain walls are given. 1 ar X iv :1 91 2. 04 80 7v 1 [ he pth ] 1 0 D ec 2 01 9


Introduction
Wrapped branes play an important role in the study of the AdS/CFT correspondence [1,2,3] and its generalization to non-conformal field theories DW/QFT correspondence [4,5,6]. In particular, these brane configurations describe RG flows across dimensions from supersymmetric field theories on the worldvolume of the unwrapped branes to lower-dimensional field theories on the worldvolume of the branes wrapped on internal compact manifolds. For supersymmetric theories, the latter are obtained from the former by twisted compactifications on the internal manifolds. Some amount of supersymmetry is preserved by performing a topological twist along the internal manifolds [7].
In this paper, we are interested in solutions describing wrapped 5-branes in string/M-theory. Rather than searching directly for wrapped brane solutions in string/M-theory, finding supersymmetric solutions of seven-dimensional gauged supergravities in the form of domain walls interpolating between AdS 7 and AdS n × Σ 7−n geometries, with Σ 7−n being a (7 − n)-dimensional compact manifold, is a more traceable task. In many cases, the resulting solutions can be embedded in ten or eleven dimensions by using consistent truncation ansatze. Solutions of this type in the maximal N = 4 gauged supergravity with SO(5) gauge group in seven dimensions have been extensively studied in previous works [8,9,10,11,12,13,14,15]. see also [16,17,18,19] for similar solutions in N = 2 gauged supergravity. For similar solutions in other dimensions, see [20,21,22,23,24,25,26,27,28,29,30,31,32,33] for an incomplete list.
We will study this type of solutions within the maximal N = 4 gauged supergravity constructed in [34] using the embedding tensor formalism, see also [35] and [36] for an earlier construction. Unlike the previously known results mentioned above, we will consider more general gauge groups of the form CSO(p, q, 5− p−q) and CSO(p, q, 4−p−q) obtained respectively from the embedding tensor in 15 and 40 representations of the global symmetry SL (5). Gauged supergravities with these gauge groups can be obtained from consistent truncations of elevendimensional supergravity and type IIB theory, respectively, see [37] and [38]. To the best of our knowledge, supersymmetric AdS n × Σ 7−n solutions in N = 4 gauged supergravity with non-compact and non-semisimple gauge groups have not been considered in the previous studies.
For the aforementioned gaugings of N = 4 supergravity, only SO(5) gauge group admits a fully supersymmetric AdS 7 vacuum dual to N = (2, 0) superconformal field theory (SCFT) in six dimensions. In this case, the AdS n × Σ 7−n solutions describe conformal fixed points in n − 1 dimensions. These fixed points correspond to (n − 1)-dimensional SCFTs obtained from twisted compactifications of N = (2, 0) SCFT in six dimensions on Σ 7−n . For all other gauge groups, the vacua are given by half-supersymmetric domain walls dual to six-dimensional N = (2, 0) non-conformal field theories. We accordingly interpret the resulting AdS n × Σ 7−n solutions as conformal fixed points in lower-dimensions of these N = (2, 0) non-conformal field theories. We will study various possible RG flows from both conformal and non-conformal field theories in six dimensions to these lower-dimensional SCFTs as well as to non-conformal field theories.
The paper is organized as follows. In section 2, we briefly review the maximal gauged supergravity in seven dimensions. The study of supersymmetric AdS n × Σ 7−n solutions in gauged supergravities with CSO(p, q, 5 − p − q) and CSO(p, q, 4 − p − q) gauge groups is presented in sections 3 and 4, respectively. Conclusions and comments on the results are given in section 5. For convenience, we also collect all bosonic field equations of the maximal sevendimensional gauged supergravity in the appendix.

N = 4 gauged supergravity in seven dimensions
In this section, we briefly review seven-dimensional N = 4 gauged supergravity in the embedding tensor formalism constructed in [34]. We will omit all the detail and only collect relevant fomulae involving the bosonic Lagrangian and fermionic supersymmetry transformations which are essential for finding supersymmetric solutions. The reader is referred to [34] for more detail. The only N = 4 supermultiplet in seven dimensions is the supergravity multiplet with the field content where C denotes the charge conjugation matrix obeying These fermionic fields transform in representations of the local SO(5) ∼ U Sp(4) R-symmetry with U Sp(4) fundamental or SO(5) spinor indices a, b, . . . = 1, ..., 4. Accordingly, the four gravitini ψ a µ and the spin-1 2 fields χ abc transform as 4 and 16, respectively. χ abc satisfy the following conditions and Ω ab χ abc = 0 (2.4) with Ω ab = Ω [ab] being the U Sp(4) symplectic form satisfying (Ω ab ) * = Ω ab and Ω ac Ω bc = δ b a . (2.5) Raising and lowering of U Sp(4) indices by Ω ab and Ω ab correspond to complex conjugation. The fourteen scalars are described by the SL (5) The inverse of V M A denoted by V A M will be written as V ab M with The bosonic Lagrangian of the N = 4 seven-dimensional gauged supergravity can be written as νρλM (γ µ νρλ − 9 2 δ ν µ γ ρλ )Ω ab V bc M c

9)
δχ abc = 2Ω cd P µde ab γ µ e + gA d,abc (2.10) The covariant derivative of the supersymmetry parameters is defined by The fermion shift matrices A 1 and A 2 are given by The "dressed" components of the embedding tensor are defined by Unlike in the ungauged supergravity in which all three-form fields can be dualized to two-form fields, the field content of the gauged supergravity can incorporate massive two-and three-form fields. The degrees of freedom in the vector and tensor fields of the ungauged theory will be redistributed among massless and massive vector, two-form and three-form fields after gaugings. In general, with a proper gauge fixing of various tensor gauge transformations, there can be t self-dual massive three-form and s massive two-form fields for s ≡ rank Z and t ≡ rank Y . In addition, there are 10 − s massless vectors and 5 − s − t massless two-form fields. It should be noted that t + s ≤ 5 by the quadratic constraint which ensures that the embedding tensor leads to gauge generators for a closed subalgebra of SL (5). Furthermore, more massive gauge fields can arise from broken gauge symmetry.
The field strength tensors of vector and two-form fields are defined by with the usual non-abelian gauge field strength These field strengths satisfy the following Bianchi identities where the covariant field strengths of the three-form fields are given by All of these fields interact with each other via the vector-tensor topological term L V T whose explcit form can be found in [34].

Solutions from gaugings in 15 representation
We now consider supersymmetric AdS n ×Σ 7−n solutions with gauge group CSO(p, q, 5− p − q) obtained from gaugings in 15 representation. In this case, non-vanishing components of the embedding tensor can be written as We will use the following choice of SO(5) gamma matrices to convert an SO(5) vector index to a pair of antisymmetric spinor indices where σ i are the usual Pauli matrices. Γ A satisfy the following relations and the symplectic form of U Sp(4) is chosen to be The coset representative of the form V M ab and the inverse V ab M are then given by 3.1 Supersymmetric AdS 5 ×Σ 2 solutions with SO(2)×SO(2) symmetry We begin with solutions of the form AdS 5 × Σ 2 with the metric ansatz given by (3.6) Σ 2 k is a Riemann surface with the metric given by and dx 2 1,3 = η mn dx m dx n with m, n = 0, .., 3 is the flat metric on the fourdimensional Minkowski space M kw 4 . The function f k (θ) is defined by with k = 1, 0, −1 corresponding to S 2 , R 2 and H 2 , respectively.
With the following choice of vielbein we find the following non-vanishing components of the spin connection The indexî =θ,φ is a flat index on Σ 2 k , and f k (θ) = df k (θ) dθ . The r-derivatives will be denoted by while a on any function with an explicit argument refers to the derivative of the function with respect to that argument.
With all these, we can straightforwardly compute the scalar potential For SO(5) gauge group, this potential admits two AdS 7 critical points given by and The former preserves N = 4 supersymmetry with SO(5) symmetry while the latter is a non-supersymmetric AdS 7 vacuum with SO(4) symmetry. We note here that for φ 1 = φ 2 the SO(2) × SO(2) symmetry is enhanced to SO(4). These two AdS 7 vacua have been identified long ago in [36].
To perform a topological twist on Σ 2 k , we turn on the following SO(2) × SO(2) gauge fields and set all the other fields to zero. By imposing the twist condition together with the following projection conditions we can derive the following BPS equations It can be readily verified that these BPS equations together with the twist condition (3.93) imply the second-ordered field equations. The radial component of the gravitino variations δψ a r gives the usual solution for the Killing spinors in which a (0) are constant spinors. By imposing the conditions V = φ 1 = φ 2 = 0 and U = 1 L AdS 5 on the BPS equations, we find a class of AdS 5 fixed point solutions given by (3.29) The AdS 5 fixed points do not exist in the case of ρ = 0. Furthermore, it turns out that good AdS 5 fixed points exist only in SO(5) and SO(3, 2) gauge groups with ρ = σ = 1 and ρ = −σ = 1, respectively. For SO(5) gauge group, there exist AdS 5 fixed points when for Σ 2 = H 2 , R 2 , S 2 , respectively. In deriving the above conditions, we have chosen g > 0 for convenience. We emphasize here that the AdS 5 × R 2 fixed point preserves sixteen supercharges while the AdS 5 × H 2 and AdS 5 × S 2 preserve only eight supercharges. This is due to the fact that no spin connection on R 2 needs to be cancelled by performing a twist. In this case, the projector involving γθφ is not needed. All these AdS 5 × Σ 2 fixed points, dual to four-dimensional SCFTs from M5-branes, together with the corresponding RG flows from the supersymmetric AdS 7 vacuum have recently been discussed in [13].
In this work, we will extend the study of these RG flows by considering more general RG flows from the N = 4 AdS 7 critical point to AdS 5 fixed points and then to singular geometries in the form of curved domain walls with M kw 4 × Σ 2 slices. According to the usual holographic interpretation, these geometries should be dual to non-conformal field theories in four dimensions arising from the RG flows from four-dimensional SCFTs dual to AdS 5 × Σ 2 fixed points. The latter are in turn obtained from twisted compactification of N = (2, 0) SCFT in six dimensions dual the N = 4 AdS 7 vacuum. Examples of these RG flows are given in figures 1, 2, and 3 for the case of AdS 5 × H 2 , AdS 5 × R 2 and AdS 5 × S 2 fixed points, respectively. In these solutions, we have chosen the position of the AdS 5 × Σ 2 fixed points to be r = 0 and set g = 16.
We can use the explicit uplift formulae given in [39,40] to determine whether these singularities are physical by considering the (00)-component of the eleven-dimensional metric given bŷ (3.33) The warped factor ∆ is defined by with µ M , M = 1, 2, . . . , 5, being the coordinates on S 4 and satisfying µ M µ M = 1.
Using the coset representative given in (3.13) and the S 4 coordinates µ M = (cos ξ, sin ξ cos ψ cos α, sin ξ cos ψ sin α, sin ξ sin ψ cos β, sin ξ sin ψ sin β) ,         we find the behavior ofĝ 00 along the flows as shown in figure 4. As can be seen from the figure,ĝ 00 → 0 near the singularities. These singularities are then physical according to the criterion given in [8]. Therefore, the singularities can be interpreted as holographic duals of non-conformal phases of the four-dimensional SCFTs obtained from twisted compactifications of six-dimensional N = (2, 0) SCFT on Σ 2 . We now consider SO(3, 2) gauge group. In this case, we find new AdS 5 × S 2 fixed points in a small range, with g > 0, As in the previous case, these AdS 5 ×S 2 solutions also preserve eight supercharges and are dual to N = 1 SCFTs in four dimensions. In constrast to SO(5) gauge group, the vacuum solution in this case is given by a half-supersymmetric domain wall, see the solutions given in [41]. According to the DW/QFT correspondence, these solutions are expected to describe N = (2, 0) non-conformal field theories in six dimensions. The above AdS 5 × S 2 fixed points can be regarded as conformal fixed points in four dimensions arising from twisted compactifications of the N = (2, 0) field theories in six dimensions on S 2 . In figure 5, we give examples of RG flows between the AdS 5 × S 2 fixed points and curved domain walls with the worldvolume given by M kw 4 × S 2 . The latter should describe non-conformal phases of the N = 1 SCFTs in four dimensions. The two ends of the flows represent two possible non-conformal phases with (φ 1 → ∞, φ 2 → −∞) and (φ 1 → −∞, φ 2 → ∞). In all of these flow solutions, we have set g = 16.
In figure 6, we give the behavior of the eleven-dimensional metric componentĝ 00 along the flows. This is obtained by using the consistent truncation of eleven-dimensional supergravity on H p,q given in [37]. The explicit form ofĝ 00 is similar to that given in (3.33) but with the warped factor ∆ given by  µ P are coordinates on H p,q satsifying µ P µ Q η P Q = 1. For SO(3, 2), we have a truncation of eleven-dimensional supergravity on H 3,2 with η M N = diag(1, 1, 1, −1, −1). From figure 6, we see thatĝ 00 → 0 on both sides of the flows. Therefore, all of these singularities are physically acceptable. We accordingly interpret these solutions as RG flows between N = 1 SCFTs and non-conformal field theories in four dimensions obtained from twisted compactifications of N = (2, 0) field theory on S 2 .

Supersymmetric AdS 4 × Σ 3 solutions with SO(3) symmetry
We now carry out a similar analysis for supersymmetric solutions of the form AdS 4 × Σ 3 with Σ 3 being a 3-manifold with constant curvature. The ansatz for the metric takes the form of where dx 2 1,2 = η mn dx m dx n , m, n = 0, 1, 2 is the metric on the three-dimensional Minkowski space. The metric on Σ 3 k is given by with the function f k defined in (3.8).
Although φ 2 and φ 3 vahish at both AdS 7 and AdS 4 × H 3 fixed points, we can consider RG flows to curved domain walls with non-vanishing φ 2 and φ 3 . Examples of various possible RG flows are given in figure 10. The behavior ofĝ 00 near the singularities,ĝ 00 → ∞, indicates that these singularities are unphysical by the criterion of [8].
With SL(5)/SO(5) coset representative of the form the scalar potential is given by As expected, there is an N = 4 supersymmetric AdS 7 critical point at φ = 0 and a non-supersymmetric, unstable, AdS 7 critical point at φ = 1 10 ln 2. To implement the twist, we impose the following projection conditions given in (3.48) and Together with the twist condition (3.47) and the γ r projection condition (3.21), we find the following BPS equations As in the SO(3) twist, the BPS equations admit an AdS 4 × H 3 fixed point only for SO(5) gauge group. This AdS 4 × H 3 vacuum is given by which does not seem to appear in the previously known results. Unlike the previous case, this AdS 4 × H 3 fixed point preserves only four supercharges and corresponds to N = 1 SCFT in three dimensions. We can similarly study numerical RG flows from the supersymmetric AdS 7 vacuum to this AdS 4 × H 3 fixed point and to curved domain walls dual to N = 1 non-conformal field theories in three dimensions. Examples of these RG flows are given in figure  11. It can be seen again that the IR singularities are physical sinceĝ 00 → 0 near the singularities.
For CSO(4, 0, 1) gauge group, we can analytically solve the BPS equations. The resulting solution is given by (3.67) The new radial coordinater is defined by dr dr = e −V . The integration constantsC and C can be neglected by shifting the coordinater and rescaling the coordinates x m on M kw 3 .
Setting C =C = 0, we find the leading behavior of the solution at largẽ r φ ∼r 2 and In this limit, the contribution from the gauge fields to the BPS equations is highly suppressed due to V → ∞. The asymptotic behavior is then identified with the standard, flat, domain wall found in [41]. Similar to the case of solutions with an asymptotically locally AdS 7 space, we will call this limit an asymptotically locally flat domain wall.
On the other hand, asr → 0, we find We also note that, in this case, the complete truncation ansatz in term of type IIA theory on S 3 has been constructed in [42]. Therefore, the solution can fully be embedded in type IIA theory. In this paper, we are only interested in the time component of the ten-dimensional metric given by, see for example [41] for more detail,ĝ Using this result, we find that asr → 0,ĝ 00 → ∞, so, in this case, the IR singularity is unphysical.

Supersymmetric AdS 3 × Σ 4 solutions
In this section, we move on to the analysis of AdS 3 ×Σ 4 solutions. We will consider two types of the internal manifold Σ 4 namely a Riemannian four-manifold M 4 with a constant curvature and a product of two Riemann surfaces Σ 2 × Σ 2 with SO(4) and SO(2) × SO(2) twists, respectively.

AdS 3 × M 4 solutions with SO(4) twists
As in the previous section, we will consider SO(4) symmetric solutions for SO(5), SO(4, 1) and CSO(4, 0, 1) gauge groups with the embedding tensor given in (3.56). To find AdS 3 × M 4 k solutions, we use the following ansatz for the sevendimensional metric with dx 2 1,1 = η mn dx m dx n for m, n = 0, 1 being the metric on two-dimensional Minkowski space. The explicit form of the metric on M 4 k is given by we obtain the following non-vanishing components of the spin connection withî,ĵ =3, ...,6 =χ,ψ,θ,φ being flat indices on Σ 4 k . We will perform a twist on M 4 k by turning on SO(4) gauge fields to cancel the spin connection as follow The corresponding two-form field strengths are given by For the SO(4) singlet scalar, we use the same coset representative given in (3.58). However, in this case, the three-form field strengths cannot vanish in order to satisfy the Bianchi's identity for H µνρ5 is allowed. We also note that for SO(5) and SO(4, 1) gauge groups, the corresponding embedding tensor Y M N is non-degenerate. There are in total t = rankY = 5 massive three-form fields, so for these gauge groups, H is obtained by turning on the massive three-form S (3) µνρ5 . On the other hand, for CSO(4, 0, 1) gauge group, we have Y 55 = 0, so the contribution to H µνρ5 comes from a massless two-form field B µν5 . However, we are not able to determine a suitable ansatz for B µν5 in order to find a consistent set of BPS equations that are compatible with the second-ordered field equations. Accordingly, we will not consider the non-semisimple CSO(4, 0, 1) gauge group in the following analysis.
For SO(5) and SO(4, 1) gauge groups, the appropriate ansatz for the modified three-form field strength is given by Imposing the twist condition (3.47) and the projector in (3.21) together with additional projectors of the form we find the BPS equations From these equations, we find an AdS 3 fixed point only for k = −1 and ρ = 1. The resulting AdS 3 × H 4 solution is given by This is the AdS 3 × H 4 fixed point given in [11] for the maximal SO(5) gauged supergravity. The solution preserves four supercharges and corresponds to N = (1, 1) SCFT in two dimensions with SO(4) symmetry. As in the previous cases, we will consider RG flows from the supersymmetric AdS 7 vacuum to this AdS 3 × H 4 fixed point and curved domain walls. Examples of these RG flows are given in figure 12. Unlike the previous cases, the IR singularities in this case are unphysical due to the behaviorĝ 00 → ∞.

AdS
In this section, we consider the manifold Σ 4 in the form of a product of two Riemann surfaces Σ 2 The ansatz for the metric takes the form of in which the metrics on Σ 2 k 1 and Σ 2 k 2 are given in (3.7). Using the following choice for the vielbein em = e U dx m , er = dr, eθ 1 = e V dθ 1 , we obtain all non-vanishing components of the spin connection as follow withî 1 =θ 1 ,φ 1 andî 2 =θ 2 ,φ 2 being flat indices on Σ 2 k 1 and Σ 2 k 2 , respectively. As in other cases, we consider all gauge groups of the form CSO(p, q, r) with an SO(2) × SO(2) subgroup. These gauge groups are obtained from the embedding tensor given in (3.14). To perform the twist, we turn on the SO(2) × SO(2) gauge fields The corresponding modified two-form field strengths are given by We also need to turn on the following three-form field strength in which α is a constant related to the magnetic charges by the relation For ρ = 0 corresponding to CSO(2, 2, 1) and CSO(4, 0, 1) gauge groups, we need to impose a relation on the magnetic charges to ensure that the resulting BPS equations are compatible with all the secondordered field equations. Using the projection conditions (3.21) and together with the twist conditions g(p 11 + σp 21 ) = k 1 and we obtain the following BPS equations In deriving these equations, we have used the coset representative given in (3.13) for SO(2) × SO(2) singlet scalars. From the BPS equations, we find a class of AdS 3 × Σ 2 × Σ 2 fixed point solutions  Among all the gauge groups with an SO(2) × SO(2) subgroup, it turns out that AdS 3 × Σ 2 × Σ 2 solutions are possible only for SO(5) and SO(3, 2) gauge groups with ρ = σ = 1 and ρ = −σ = 1, respectively. For SO(5) gauge group, the solutions have been extensively studied in [14]. For SO (3,2) gauge group, all the AdS 3 × Σ 2 × Σ 2 solutions given here are new. Following [14], we define the following two parameters to characterize the possible AdS 3 × Σ 2 × Σ 2 solutions and where we have set ρ = 1. In order for the AdS 3 fixed points to exist in SO (5) gauge group with σ = 1, one of the Riemann surfaces must be negatively curved, and AdS 3 × H 2 × Σ 2 solutions can be found within the regions in the parameter space (z 1 , z 2 ) shown in figure 13. These regions are the same as those given in [14]. The AdS 3 × Σ 2 × Σ 2 fixed points preserve four supercharges and correspond to N = (2, 0) SCFTs in two dimensions with SO(2)×SO(2) symmetry. Examples of RG flows with g = 16 from the N = 4 supersymmetric AdS 7 to AdS 3 ×H 2 ×Σ 2 fixed points and curved domain walls in the IR are shown in figures 14, 15 and 16 for Σ 2 = H 2 , R 2 and S 2 , respectively. All the IR singularities are physical witĥ g 00 → 0 near the singularities. We now carry out a similar analysis for the case of SO(3, 2) gauge group with ρ = −σ = 1. It turns out that in this case, the AdS 3 fixed points exist only for at least one of the two Riemann surfaces is positively curved. For definiteness, we will choose k 1 = 1 and k 2 = −1, 0, 1 corresponding to AdS 3 × S 2 × H 2 , AdS 3 × S 2 × R 2 and AdS 3 × S 2 × S 2 fixed points, respectively. Using the parameters z 1 and z 2 defined in (3.106) with σ = −1, we find regions in the parameter space (z 1 , z 2 ) for AdS 3 vacua to exist in SO(3, 2) gauged maximal supergravity as shown in figure 17.  For SO(3, 2) gauge group, there is no asymptotically locally AdS 7 geometry. We will consider RG flows between the AdS 3 × S 2 × Σ 2 fixed points and curved domain walls with M kw 3 × S 2 × Σ 2 slices. These curved domain walls have SO(2) × SO(2) symmetry and are expected to describe non-conformal field theories in two dimensions obtained from twisted compactifications of N = (2, 0) non-conformal field theory in six dimensions. The latter is dual to the halfsupersymmetric domain wall of the seven-dimensional gauged supergravity. A number of these RG flows with g = 16 and different values of z 1 and z 2 are given in figures 18, 19 and 20. We see that all singularities in the flows from AdS 3 × S 2 × R 2 fixed points are unphysical while only the singularities on the right (left) with φ 1 → ∞ and φ 2 → −∞ (φ 1 → −∞ and φ 2 → ∞) in the flows from AdS 3 × S 2 × S 2 (AdS 3 × S 2 × H 2 ) fixed points are physical.

AdS 3 vacua from Kahler four-cycles
In this section, we consider twisted compactifications of six-dimensional field theories on Kahler four-cycles K 4 k . The constant k = 1, 0, −1 characterizes the constant curvature of K 4 k and corresponds to a two-dimensional complex projective space CP 2 , a four-dimensional flat space R 4 and a two-dimensional complex hyperbolic space CH 2 , respectively. The manifold K 4 k has a U (2) ∼ U (1)×SU (2) spin connection. Therefore, we can perform a twist by turning on SO(2) ∼ U (1) or SO(3) ∼ SU (2) gauge fields to cancel the U (1) or SU (2) parts of the spin , and f k (ψ) is the function defined in (3.8). τ i are SU (2) leftinvariant one-forms satisfying we find non-vanishing components of the spin connection whereî =3, ...,6 is the flat index on K 4 k . ωˆiĵ (1) are the SU (2) spin connections. To perform the twist, we turn on SO(3) gauge fields with the following ansatz A i−2,j−2 with the modified two-form field strengths given by Unlike the previous case, we do not need to turn on the three-form field strengths since, in this case, M N P QR H (2)N P ∧ H (2)QR = 0. We then impose the twist condition (3.47) together with the following three projection conditions Using the scalar coset representative (3.44) and the projection (3.21), we find the following BPS equations It turns out that only SO(5) gauge group admits an AdS 3 × CH 2 fixed point given by This is the AdS 3 × CH 2 solution found in [11].
The self-dual SO(3) gauge fields can be defined as  As in the case of SO(4) symmetric solutions, we will consider only SO(5) and SO(4, 1) gauge group with ρ = 0. Using the embedding tensor (3.56) and the SO(4) invariant coset represenvative (3.58), we find the following BPS equations in which we have also imposed the γ r projection (3.21). From these equations, an which is the AdS 3 × CH 2 fixed point found in [11]. The solution preserves two supercharges and corresponds to N = (1, 0) SCFT in two dimensions with SO(3) symmetry. Supersymmetric RG flows from the N = 4 AdS 7 vacuum to this AdS 3 × CH 2 fixed point and curved domain walls in the IR are given in figure  24. The IR singularities are physically acceptableas indicated by the behavior of g 00 → 0.

AdS
twists As a final case for AdS 3 × K 4 k solutions, we will perform another twist on the Kahler four-cycle by cancellng the U (1) part of the spin connection. To make this U (1) part manifest, we write the metric on K 4 k as with τ i being the SU (2) left-invariant one-forms given in (3.110). The sevendimensional metric is still given by (3.107) with ds 2  With the following vielbein all non-vanishing components of the spin connection are given by To perform the twist, we turn on the SO(2) × SO(2) gauge fields together with the twist condition (3.19). The associated two-form gauge field strengths are given by where J (2) is the Kahler structure defined by With the above non-vanishing SO(2) × SO(2) gauge fields, we need to turn on the modified three-form field strength of the form with ρ being the parameter in the embedding tensor (3.14) for gauge groups with an SO(2) × SO(2) subgroup. As in the previous cases, the appearance of ρ in (3.139) implies that the resulting BPS equations are not compatible with the field equations for the case of ρ = 0. In subsequent analysis, we will accordingly consider only gauge groups with ρ = 0. With the γ r projector (3.21) and the scalar coset representative (3.13), the corresponding BPS equations read From these equations, we find the following AdS 3 fixed point solutions .
For SO(5) gauge group, there exist AdS 3 × CH 2 fixed points in the range in which we have taken g > 0 for convenience. Up to some differences in notations, these AdS 3 ×CH 2 fixed points are the same as the solutions studied in [14]. As in the previous cases, we study RG flows from the supersymmetric AdS 7 vacuum to the AdS 3 × CH 2 fixed points and curved domain walls in the IR. Some examples of these flows are given in figure 25 for g = 16 and different values of p 2 . The behaviors of the eleven-dimensional metric componetĝ 00 for these RG flows are shown in figure 26 which indicates that the singularities are physical. Apart from these AdS 3 × CH 2 fixed points, we find new AdS 3 × CP 2 fixed points in SO(4, 1) and SO(3, 2) gauge groups respectively in the following ranges, with g > 0, Since there is no supersymmetric AdS 7 critical point for SO(4, 1) and SO(3, 2) gauge groups, we will study supersymmetric RG flows between these AdS 3 × CP 2 fixed points and curved domain walls with SO(2) × SO(2) symmetry. Examples of these RG flows in SO(4, 1) and SO(3, 2) gauge groups are shown respectively in figures 27 and 28 with g = 16 and different values of p 2 . From the behaviors of g 00 in figure 29, we find that the singularities on the left (right) with φ 1 → ±∞ and φ 2 → ∓∞ (φ 1 → ∞ and φ 2 → −∞) of the flows in SO(4, 1) (SO(3, 2)) gauge group are physical.

Supersymmetric AdS 2 × Σ 5 solutions
We end this section by considering solutions of the form AdS 2 × Σ 5 . AdS 2 × Σ 5 solutions for the manifold Σ 5 being S 5 or H 5 have been given in [11]. The twist     is performed by turning on SO(5) gauge fields. This is obviously possible only for SO(5) gauge group. In addition, no scalars in SL(5)/SO(5) coset are singlets under SO (5), so the solutions are given purely in term of the seven-dimensional metric. The corresponding RG flows from the supersymmetric AdS 7 vacuum and the AdS 2 × H 5 or AdS 2 × S 5 fixed points have already been analytically given in [11]. We will not repeat the analysis for this case here. However, if we consider Σ 5 as a product of three-and two-manifolds Σ 3 ×Σ 2 , it is possible to perform a twist by turning on SO(3)×SO(2) gauge fields along Σ 3 × Σ 2 . In this case, there are two gauge groups with an SO(3) × SO(2) subgroup namely SO(5) and SO (3,2). The ansatz for the seven-dimensional metric takes the form of ds 2 7 = −e 2U (r) dt 2 + dr 2 + e 2V (r) ds 2 (3.150) The explicit form of the metrics on the Σ 3 k 1 and Σ 2 k 2 are given in (3.39) and (3.7), respectively.
(3.153) Therefore, the SL(5)/SO(5) coset representative is parametrized by We now turn on the SO(3) × SO(2) gauge fields of the form with the corresponding two-form field strengths given by With all these gauge fields non-vanishing, we also need to turn on the three-form field strengths for σ = ±1 and the scalar coset representative (3.154), we can derive the following BPS equations in which we have also used the γ r projector in (3.21). From these equations, we find an AdS 2 fixed point only for k 1 = k 2 = −1 and σ = 1. The resulting AdS 2 × H 3 × H 2 fixed point is given by which is the solution found in [12]. The three projectors in (3.159) imply that this solution preserves four supercharges. The solution is dual to the superconformal quantum mechanics. Examples of RG flows from the supersymmetric AdS 7 vacuum to the AdS 2 × H 3 × H 2 fixed point and curved domain walls in the IR are given in figure 30. From the behavior of the eleven-dimensional metric componentĝ 00 , we see that the singularity is physically acceptable. Therefore, this singularity is expected to describe supersymmetric quantum mechanics obtained from a twisted compactification of N = (2, 0) SCFT in six dimensions.

Solutions from gaugings in 40 representation
In this section, we repeat the same analysis for gaugings from 40 representation. The gauge groups are of the form CSO(p, q, r) with p + q + r = 4. In this case, the embedding tensor is given by with w M N = w (M N ) . The SL(5) symmetry can be used to fix v M = δ M 5 . Follow [34], we will also split the index M = (i, 5) and set w 55 = w i5 = 0. The remaining SL(4) ⊂ SL(5) symmetry can be used to diagonalize w ij in the form This generates CSO(p, q, r) gauge groups for p+q +r = 4 with the corresponding gauge generators given by With the SL(5) index splitting M = (i, 5), it is also useful to parametrize the SL(5)/SO(5) coset in term of the SL(4)/SO(4) submanifold as V is the SL(4)/SO(4) coset representative, and t 0 , t i refer respectively to SO(1, 1) and four nilpotent generators in the decomposition SL(5) → SL(4) × SO(1, 1). The unimodular matrix M M N decomposes accordingly The scalar potential for the embedding tensor (4.1) reads It can be straightforwardly verified that the nilpotent scalars b i appear at least quadratically in the Lagrangian. Therefore, these scalars can always be consistently truncated out. In the following analysis, we will consider only supersymmetric solutions with b i = 0 for simplicity. As in the case of gaugings in 15 representation, when the compact manifold Σ d has dimension d > 3, the modified three-form field strengths need to be turned on in order to satisfy the corresponding Bianchi's identity. However, with Y M N = 0, there are no massive three-form fields. In this case, the contribution to H (3) µνρM arises from the two-form fields. There are respectively 5 − s and s, for s = rank Z, massless and massive two-form fields. The latter also appear in the modified gauge field strengths H in which B µνj are massive two-form fields. However, we are not able to find a consistent set of BPS equations that are compatible with the field equations for non-vanishing massive two-form fields. Therefore, in the following analysis, we will truncate out all the massive two-form fields. Finally, we point out here that the CSO(p, q, r) with p + q + r = 4 gauge group is not large enough to accomodate SO(5) or SO(3) × SO(2) subgroups. It is accordingly not possible to have AdS 2 × Σ 5 or AdS 2 × Σ 3 × Σ 2 solutions.

Solutions with the twists on Σ 2
We first look for AdS 5 × Σ 2 solutions for Σ 2 being a Riemann surface. The ansatz for the seven-dimensional metric is given in (3.6). We will consider solutions obtained from SO(2) × SO(2) and SO (2) twists on Σ 2 . The procedure is essentially the same as in the gaugings in 15 representation, so we will not give all the details here to avoid a repetition.

Solutions with SO(2) × SO(2) twists
Gauge groups with an SO(2) × SO(2) subgroup can be obtained from the embedding tensor of the form with the parameter σ = 1, −1 corresponding to SO(4) and SO(2, 2) gauge groups, respectively. There is only one SO(2) × SO(2) singlet scalar from SL(4)/SO(4) coset described by the coset representative The scalar potential is given by In this case, there is no supersymmetric AdS 7 fixed point. The supersymmetric vacuum is given by half-supersymmetric domain walls dual to N = (2, 0) nonconformal field theories in six dimensions. We now perform the twist by turning on the following SO(2) × SO(2) gauge fields and imposing the projection conditions given in (3.20) and together with the twist condition (3.19).
With all these, we find the following BPS equations 14) . In subsequent analysis, we will consider interpolating solutions between an asymptotically locally flat domain wall and curved domain walls in SO(4) gauge group.
For large V , the contribution from the gauge fields is highly suppressed. In this limit, we find which implies that U ∼ V → ∞ as r → ∞. Examples of flow solutions with this asymptotic behavior are given in figures 31, 32 and 33 for Σ 2 = S 2 , R 2 , H 2 , respectively. In these solutions, we have set g = 16. We note here that the flows to the flat M kw 4 × R 2 -sliced domain walls given in figure 32 are possible provided that we set p 2 = −p 1 as required by the twist condition. It should also be pointed out that the green curve in figure 32 is simply the usual flat domain wall since k = p 1 = p 2 = 0. In this case, the solution preserves the full SO(4) gauge symmetry due to the vanishing of the SO(2) × SO(2) singlet scalar φ. This solution has already been given analytically in [41]. As shown in [38], the maximal gauged supergravity in seven dimensions with CSO(p, q, 4 − p − q) gauged group obtained from the embedding tensor in 40 representation can be embedded in type IIB theory via a truncation on H p,q • T 4−p−q . For the present discussion, we only need the ten-dimensional metric which, for SO(p, 4 − p) gauge group, is given bŷ η ij is the SO(p, 4 − p) invariant tensor, and µ i are coordinates on H p,q satisfying µ i µ j η ij = 1. In term of the parametrization (4.4), κ is identified as follows For CSO(p, q, 4−p−q) gauge group, we decompose the SL(4) indices i, j, . . . into (î,ĩ) withî = 1, . . . p+q andĩ = p+q+1, . . . , 4−p−q. The ten-dimensional metric and the warped factor are still given by (4.18) and (4.19) but with η ij = (ηˆiĵ, η˜ij) replaced by the SO(p, q) invariant tensor ηˆiĵ and η˜ij = 0. In this case, µˆi become coordinates on H p,q satisfying ηˆiĵµˆiµĵ = 1 while µ˜i are coordinates on T 4−p−q . For the present case of SO(4) gauge group, we simply have η ij = δ ij for i, j = 1, 2, 3, 4. The behavior of the time component of the ten-dimensional metriĉ g 00 for the flow solutions in figures 31, 32 and 33 is shown in figure 34. Near the IR singularities, we findĝ 00 → 0, so the singularities are physically acceptable.

Solutions with SO(2) twists
We then move to another twist on Σ 2 by turning on only an SO(2) gauge field. This can be achieved from the SO(2) × SO(2) gauge fields given in (4.11) by setting p 2 = 0 and p 1 = p. In this case, the SL(4)/SO(4) coset representative is given by in which Y 1 , Y 2 , and Y 3 are non-compact generators commuting with the SO(2) symmetry generated by X 12 . The explicit form of these generators is given by we obtain the BPS equations Although these equations are more complicated than those of the SO(2) × SO(2) twist, there do not exist any AdS 5 × Σ 2 fixed points.

Solutions with the twists on Σ 3
In this section, we repeat the same analysis for solutions with the twists on Σ 3 . We will consider two different twists by turning on SO(3) and SO(3) + gauge fields.

Solutions with SO(3) twists
In this case, the gauge groups with an SO(3) subgroup are described by the embedding tensor w ij = diag(1, 1, 1, ρ) (4.31) with ρ = −1, 0, 1 corresponding to SO(3, 1), CSO(3, 0, 1) and SO(4) gauge groups, respectively. There is one SO(3) singlet scalar from the SL(4)/SO(4) coset with the coset representative given by (4.32) leading to the scalar potential To perform the twist, we turn on the SO(3) gauge fields and impose the projectors (3.48) on the Killing spinors together with the twist condition (3.47). With all these and the γ r projector (4.12), the resulting BPS equations read (4.39) We have neglected an additive integration constant for U which can be absorbed by rescaling the coordinates on M kw 3 . With ρ = 0, we find that φ 0 + 3 5 φ = 0 which gives with an integration constant C 0 . Taking a linear combination V + 6 5 φ and changing to a new radial coordinater given by dr dr = e − 4 5 φ , we find The integration constant C can also be set to zero by shifting the coordinater. With all these results, the equation for φ gives in which we have set C = 0 for simplicity, andC is another integration constant. Asr → 0, we find that the above solution becomes a locally flat domain wall with U ∼ V → ∞. The asymptotic behavior is given by Forr → ∞, we find lnr . The complete solutions can be obtained numerically. As r → ∞, we find

Solutions with SO(3) + twists
We now consider another twist by turning on the gauge fields for self-dual SO(3) + gauge symmetry. Only the scalar field φ 0 is SO(3) + singlet, so we simply have M ij = δ ij . Furthermore, we consider only SO(4) gauge group since this is the only gauge group that contains the SO(3) + subgroup.  The SO(3) + gauge fileds are given by With the projectors (3.48), (3.126) and (4.12) together with the twist condition (3.47), the resulting BPS equations are given by As in the case of SO(3) twist, we do not find any AdS 4 fixed points from these equations.
From the BPS equations, we again find that U = 2φ 0 . By taking a linear combination V − 2φ 0 and defining a new radial coordinater by dr dr = e −V , we obtain V = ln(pr + C) + 2φ 0 . (4.52) Using φ 0 from (4.52) in equation for V , we find, after changing to the coordinatẽ r, V = g 5p (pr + C) 2 + 7 10 ln(pr + C). (4.53) Asr → ∞, we find, with C set to zero by shiftingr, which is identified as a domain wall solution given in [41]. On the other hand, as r → 0, the solution becomes This singularity is also unphysical since the ten-dimensional metric giveŝ We note that this solution is the same as that given in section 3.2.2 for CSO(4, 0, 1) gauge group. The two gauged supergravities can be obtained from truncations of type IIB and type IIA theories on S 3 , respectively. In fact, there is a duality between these solutions as pointed out in [38].

Solutions with the twists on Σ 4
We finally look for supersymmetric solutions obtained from the twists on a fourmanifold Σ 4 . As in the case of gaugings in 15 representation, we will consider two types of Σ 4 in terms of a product of two Riemann surfaces Σ 2 × Σ 2 and a Kahler four-cycle K 4 .

Solutions with
For solutions with SO(2) × SO(2) twists, there are two gauge groups to consider, SO(4) and SO(2, 2) with the embedding tensor (4.8). The ansatz for the metric is given in (3.83). To cancel the spin connection on Σ 2 k 1 × Σ 2 k 2 , we turn on the following SO(2) × SO(2) gauge fields with the following two-form field strengths Following a similar analysis for gaugings in 15 representation, we also turn on the modified three-form field strength using the ansatz in which β is a constant. We now impose the twist conditions g(σp 11 + p 21 ) = k 1 and g(σp 12 + p 22 ) = k 2 (4.60) and the projection conditions in (3.92) together with (4.12). With all these, the resulting BPS equations are given by U = − e 2φ 0 10 e −2(V +φ) (e 4φ p 11 + p 21 ) + e −2(W +φ) (e 4φ p 12 + p 22 )  Unlike the similar analysis for gaugings in 15 representation, it turns out that compatibility between these BPS equations and the second-ordered field equations requires p 12 p 21 + p 11 p 22 = 0 (4.66) for any values of β. This implies that the constant β is a free parameter in this case. However, we do not find any AdS 3 fixed points from the BPS equations. For SO(4) gauge group, examples of flow solutions between asymptotically locally flat domain walls and curved domain walls for various forms of Σ 2 × Σ 2 are shown in figures 38 to 43. In these solutions, we have chosen particular values of g = 16 and β = 2. The green curve in figure 41 is the flat domain wall solution given in [41]. All of the IR singularities are physical as can be seen from the behavior of the ten-dimensional metric given in figure 44.
We have also considered SO(2) twists on Σ 2 × Σ 2 by setting p 11 = p 12 = 0 and obtain more complicated BPS equations. However, we do not find any AdS 3 fixed points either. Therefore, we will not give further detail on this analysis.

Solutions with SO(3) twists on K 4
For Σ 4 being a Kahler four-cycle K 4 k , we perform the twist to cancel the SU (2) part of the spin connection, given in (3.112), by turning on the SO(3) gauge fields with the two-form field strengths given by These field strengths do not lead to any problematic terms in the modified Bianchi's identity for the three-form field strengths. However, we can have a non-vanishing three-form field strength by using the following ansatz  which is a manifestly closed three-form for a constant β.
With the SL(4)/SO(4) coset representative and the embedding tensor given in (4.32) and (4.31) together with the projections (3.117) and (4.12), we in which we have used the twist condition (3.47). We do not find any AdS 3 fixed points from these equations. Examples of supersymmetric flows for β = −2 are given in figures 45 and 46 for k = 1 and k = −1, respectively. From the behavior of the ten-dimensional metric given in figure 47, we find that the IR singularities for k = −1 are physical. In the case of flat K 4 k with k = 0, we have p = 0 by the twist condition resulting in the standard flat domain wall solutions.

Solutions with SO(2) twists on K 4
As the final case, we briefly consider the SO(2) twist by turning on an SO(2) gauge field to cancel the U (1) part of the spin connection for the metric (3.132).  This gauge field is given by The SO(2) singlet scalars from SL(4)/SO(4) coset are described by the coset representative (4.21), and the embedding tensor is given in (4.23). We can also turn on the three-form field strength (4.69). With the twist condition (3.47) and the projections (4.12) together with As in all of the previous cases for gaugings in 40 representation, there are no AdS 3 fixed points from these equations.

Conclusions and discussions
In this paper, we have extensively studied supersymmetric AdS n × Σ 7−n solutions of the maximal gauged supergravity in seven dimensions with CSO(p, q, 5−p−q) and CSO(p, q, 4−p−q) gauge groups. These gauged supergravities can be embedded respectively in eleven-dimensional and type IIB supergravities on H p,q •T 5−p−q and H p,q • T 4−p−q . Therefore, all the solutions given here have higher dimensional origins and could be interpreted as different brane configurations in string/Mtheory. This makes applications of these solutions in the holographic context more interesting. Accordingly, we hope our results would be useful along this line of research. For a particular case of SO(5) gauge group, we have recovered all the previous results on AdS n × Σ 7−n fixed points with n = 2, 3, 4, 5. We have provided numerical RG flows interpolating between the supersymmetric N = 4 AdS 7 vacuum dual to N = (2, 0) SCFT in six dimensions and all these AdS n × Σ 7−n fixed points. Some of these flows have not previously been discussed, so our results could complete the list of already known flow solutions. Furthermore, we have extended all these RG flows to singular geometries in the IR. These singularities describe curved domain walls with M kw n−1 × Σ 7−n slices and can be interpreted as non conformal field theories in n − 1 dimensions. The flow solutions suggest that they are non-conformal phases of the (n − 1)-dimensional SCFTs obtained from twisted compactifications of N = (2, 0) SCFT in six dimensions.
We have also discovered novel classes of AdS 5 × S 2 , AdS 3 × S 2 × Σ 2 and AdS 3 × CP 2 solutions in non-compact SO(3, 2) gauge group. Unlike in SO(5) gauge group, there do not exist any supersymmetric AdS 7 fixed points in this gauge group. The maximally supersymmetric vacua are given by halfsupersymmetric domain walls. In this case, we have studied RG flow solutions between these new fixed points and curved domain walls. We have also examined the behavior of the time component of the eleven-dimensional metric and found that many of the singularities are physically acceptable. The singular geometries identified here can be then interpreted as a holographic description of non-conformal field theories obtained from twisted compactifications of N = (2, 0) six-dimensional field theories. A similar study has been carried out for SO(4, 1) gauge group in which a new class of AdS 3 × CP 2 solutions has been found.
Flow solutions for non-compact SO(3, 2) and SO(4, 1) gauge groups can also be interpreted as black 3-brane and black strings in asymptotically curved domain wall space-time. These solutions are similar to four-dimensional black holes studied in [43]. In [44], these black hole solutions have been shown to arise from a dimensional reduction of the AdS 5 black strings studied in [14]. It has also been pointed out in [44] that the four-dimensional black holes, with curved domain wall asymptotics, should be seen from a higher-dimensional perspective as black strings in AdS 5 . However, this is not the case for the solutions given in this paper. Our solutions cannot be related to any supersymmetric black objects in eight dimensions with asymptically AdS 8 space-time due to the absence of supersymmetric AdS d vacua for d > 7.
For CSO(p, q, 4−p−q) gauge group which is obtained from a truncation of type IIB theory, we have performed a similar analysis as in the CSO(p, q, 5−p−q) gauge group but have not found any AdS n fixed points. The resulting gauged supergravity admits half-supersymmetric domain walls as vacuum solutions which, upon uplifted to ten dimensions, describe 5-branes in type IIB theory. We have given supersymmetric flow solutions interpolating between asymptotically locally flat domain walls, in which the effect of magnetic charges are small compared to the superpotential of the domain walls, and curved domain walls with M kw n−1 × Σ 7−n worldvolume. By the standard DW/QFT correspondence, these solutions should be interpreted as RG flows across dimensions between nonconformal field theories in six and n − 1 dimensions. It could be interesting to study these field theories on the worldvolume of the 5-branes in type IIB theory. Our results suggest that these N = (2, 0) field theories have no conformal fixed points in lower dimensions. It could be interesting to have a definite conclusion whether this is true in general. On the other hand, if this is not the case, it would also be interesting to extend the anylysis of this paper by using more general ansatz in particular with non-vanishing massive two-form fields and find new classes of AdS n × Σ 7−n solutions of seven-dimensional gauged supergravity.