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

We study supersymmetric AdSn×Σ7-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_n\times \Sigma ^{7-n}$$\end{document}, n=2,3,4,5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2,3,4,5$$\end{document} solutions in seven-dimensional maximal gauged supergravity with CSO(p,q,5-p-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,5-p-q)$$\end{document} and CSO(p,q,4-p-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,4-p-q)$$\end{document} gauge groups. These gauged supergravities are consistent truncations of eleven-dimensional supergravity and type IIB theory on Hp,q∘T5-p-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{p,q}\circ T^{5-p-q}$$\end{document} and Hp,q∘T4-p-q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{p,q}\circ T^{4-p-q}$$\end{document}, respectively. Apart from recovering the previously known AdSn×Σ7-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_n\times \Sigma ^{7-n}$$\end{document} solutions in SO(5) gauge group, we find novel classes of AdS5×S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_5\times S^2$$\end{document}, AdS3×S2×Σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_3\times S^2\times \Sigma ^2$$\end{document} and AdS3×CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_3\times CP^2$$\end{document} solutions in non-compact SO(3, 2) gauge group together with a class of AdS3×CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_3\times CP^2$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_7$$\end{document} vacuum to the AdSn×Σ7-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_n\times \Sigma ^{7-n}$$\end{document} fixed points and singular geometries in the form of curved domain walls with Mkwn-1×Σ7-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Mkw_{n-1}\times \Sigma ^{7-n}$$\end{document} slices. In many cases, the singularities are physically acceptable and can be interpreted as non-conformal phases of (n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-1)$$\end{document}-dimensional SCFTs obtained from twisted compactifications of N=(2,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=(2,0)$$\end{document} 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×Σ7-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_n\times \Sigma ^{7-n}$$\end{document} fixed points and curved domain walls while, in CSO(p,q,4-p-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CSO(p,q,4-p-q)$$\end{document} gauge group, RG flows interpolating between asymptotically locally flat domain walls and curved domain walls are given.


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 theoa e-mail: parinya.ka@hotmail.com (corresponding author) b e-mail: danai.nuchino@hotmail.com ries on the worldvolume of the unwrapped branes to lowerdimensional field theories on the worldvolume of the branes wrapped on internal compact manifolds. For supersymmetric theories, the latter are obtained from twisted compactifications of the former on the internal manifolds. Some amount of supersymmetry is preserved by performing a topological twist along the internal manifolds [7].
We will study this type of solutions within the maximal N = 4 gauged supergravity constructed in [34] using the embedding tensor formalism, see also [35,36] for an earlier construction. Unlike the previously known results mentioned above, we will consider more general gauge groups of the form C SO( p, q, 5 − p − q) and C SO( 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 eleven-dimensional supergravity and type IIB theory on H p,q • T 5− p−q and H p,q • T 4− p−q , respectively, see [37] and [38]. The manifold H p,q • T r is a product of a ( p+q −1)-dimensional hyperbolic space and an r -torus. The former is in general non-compact unless p = 0 or q = 0 in which the H p,0 or H 0,q become a ( p −1)-sphere (S p−1 ) or a (q − 1)-sphere (S q−1 ). To the best of our knowledge, supersymmetric Ad S n × 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 Ad S 7 vacuum dual to N = (2, 0) superconformal field theory (SCFT) in six dimensions. The Ad S n × 7−n solutions describe conformal fixed points in n − 1 dimensions. In this case, these fixed points correspond to (n − 1)dimensional SCFTs obtained from twisted compactifications of N = (2, 0) SCFT in six dimensions on 7−n . For all other gauge groups, the vacua are given by half-supersymmetric domain walls dual to six-dimensional N = (2, 0) nonconformal field theories. We accordingly interpret the resulting Ad S n × 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 nonconformal field theories.
The paper is organized as follows. In Sect. 2, we briefly review the maximal gauged supergravity in seven dimensions. The study of supersymmetric Ad S n × 7−n solutions in gauged supergravities with C SO( p, q, 5 − p − q) and C SO( p, q, 4 − p − q) gauge groups is presented in Sects. 3 and 4, respectively. Conclusions and comments on the results are given in Sect. 5. For convenience, we also collect all bosonic field equations of the maximal seven-dimensional 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 formulae 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 (2.10) The covariant derivative of the supersymmetry parameters is defined by (2.11) with ∇ μ being the space-time covariant derivative. The composite connection Q μa b and the vielbein on the SL(5)/SO (5) coset P μab cd are obtained from The gauge generators in the representation 5 of SL (5) can be written in term of the embedding tensor as The fermion shift matrices A 1 and A 2 are given by  [gh] . (2.19) The "dressed" components of the embedding tensor are defined by together with its inverse The scalar potential is given by Unlike in the ungauged supergravity in which all threeform fields can be dualized to two-form fields, the field content of the gauged supergravity can incorporate massive twoand 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 threeform 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 (2.29) where the covariant field strengths of the three-form fields are given by All of these fields interact with each other via the vectortensor topological term L V T whose explicit form can be found in [34].

Solutions from gaugings in 15 representation
We now consider supersymmetric Ad S n × 7−n solutions with gauge group C SO( 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. 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 We begin with solutions of the form Ad S 5 × 2 with the metric ansatz given by 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 four-dimensional Minkowski space Mkw 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 em = e U dx m , er = dr, we find the following non-vanishing components of the spin connection ωmr (1) = U em, ωˆir (1) (3.10) The indexî =θ,φ is a flat index on 2 k , and 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.
To perform a topological twist on 2 k , we turn on the following SO(2) × SO(2) gauge fields (3.18) and set all the other fields to zero. By imposing the twist condition together with the following projection conditions (3.20) and γ r a = a , (3.21) we can derive the BPS equations  (3.26) 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 Ad S 5 fixed point solutions given by
For SO (5) gauge group, there exist Ad S 5 fixed points when gp 2 = −1, 0, gp 2 = 0 and gp 2 < 0 ∪ gp 2 > 1, (3.32) for 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 Ad S 5 × R 2 fixed point preserves sixteen supercharges while the Ad S 5 × H 2 and Ad S 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 Ad S 5 × 2 fixed points, dual to four-dimensional SCFTs from M5-branes, together with the corresponding RG flows from the supersymmetric Ad S 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 Ad S 7 critical point to Ad S 5 fixed points and then to singular geometries in the form of curved domain walls with Mkw 4 × 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 fourdimensional SCFTs dual to Ad S 5 × 2 fixed points. The latter are in turn obtained from twisted compactifications of N = (2, 0) SCFT in six dimensions dual to the N = 4 Ad S 7 vacuum. Examples of these RG flows are given in Figs. 1, 2, and 3 for the cases of Ad S 5 × H 2 , Ad S 5 ×R 2 and Ad S 5 × S 2 fixed points, respectively. In these solutions, we have chosen the position of the Ad S 5 × 2 fixed points to be r = 0 and set g = 16. We note here that one of the magnetic charges is fixed by the twist condition (3.19). In the numerical solutions, we have chosen p 2 to be the independent parameter.
We can use the explicit uplift formulae given in [39,40] to determine whether these IR singularities are physical by considering the (00)-component of the eleven-dimensional metric given bŷ g 00 = 1 3 g 00 . (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) , (3.35) we find the behavior ofĝ 00 along the flows as shown in Fig.  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 2 .
We now consider SO (3,2) gauge group. In this case, we find new Ad S 5 × S 2 fixed points in a small range, with g > 0, As in the previous case, these Ad S 5 × S 2 solutions also preserve eight supercharges and are dual to N = 1 SCFTs in four dimensions. In contrast 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 Ad S 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 Fig. 5, we give examples of RG flows between the Ad S 5 × S 2 fixed points and curved domain walls with the worldvolume given by Mkw 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 . In all of these flow solutions, we have set g = 16.
In Fig. 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 (3.37) The tensor η M N = diag(1, . . . , 1, −1, . . . , −1) is the SO( p, q) invariant tensor, and μ P are coordinates on H p,q satisfying 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 Fig. 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
we find non-vanishing components of the spin connection as follow whereî =ψ,θ,φ is a flat index on 3 k . We will perform the twist by turning on SO (
We now impose a simple twist condition 52) (3.53) From these BPS equations, we find an Ad S 4 × H 3 fixed point only for SO(5) gauge group given by This is the Ad S 4 × H 3 solution studied in [11]. The solution preserves eight supercharges and corresponds to N = 2 SCFT in three dimensions. As in the previous case, in addition to the holographic RG flows from the supersymmetric N = 4 Ad S 7 vacuum to this Ad S 4 × H 3 geometry, we also consider more general RG flows from the Ad S 4 × H 3 fixed point to curved domain walls with a Mkw 3 × H 3 slice dual to N = 2 non-conformal field theories in three dimensions. There are many possible RG flows of this type. The simplest possibility is given by RG flows with φ 2 = φ 3 = 0 along the entire flows. Examples of these RG flows are given in Figs. 7 and 8 in which φ 1 → ∞ and φ 1 → −∞, respectively. Both types of the singularities are physically acceptable as can be seen from the behavior of the (00)-component of the eleven-dimensional metricĝ 00 given in Fig. 9. These singular geometries are then dual to N = 2 non-conformal field theories in three dimensions obtained from twisted com- Although φ 2 and φ 3 vanish at both Ad S 7 and Ad S 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 Fig. 10. The behavior ofĝ 00 near the singularities,ĝ 00 → ∞, indicates that these singularities are unphysical by the criterion of [8].

Solutions with S O(3) + twists
We now consider another twist given by turning on SO(3) + gauge fields. In this case, the (5). We will accordingly turn on the following gauge fields The gauge groups containing SO(3) + ⊂ SO(4) are given by SO (5), SO(4, 1), and C SO(4, 0, 1). These groups can be gauged altogether by the following embedding tensor There is only one SO(3) + singlet scalar corresponding to the SL(5) non-compact generator Y = e 1,1 + e 2,2 + e 3,3 + e 4,4 − 4e 5,5 . (3.57) It should be noted that this generator is invariant under a larger symmetry SO (4).
the scalar potential is given by As expected, there is an N = 4 supersymmetric Ad S 7 critical point at φ = 0 and a non-supersymmetric, unstable, Ad S 7 critical point at φ = 1 10 ln 2. To implement the twist, we impose the following projection conditions given in (3.48) and (3.60) 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 Ad S 4 ×H 3 fixed point only for SO (5) gauge group. This Ad S 4 × H 3 vacuum is given by which does not seem to appear in the previously known results.
Unlike the previous case, this Ad S 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 Ad S 7 vacuum to this Ad S 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 Fig. 11. It can be seen again that the IR singularities are physical sincê g 00 → 0 near the singularities. For C SO(4, 0, 1) gauge group, we can analytically solve the BPS equations. The resulting solution is given by 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 Mkw 3 .
Setting C =C = 0, we find the leading behavior of the solution at larger φ ∼r 2 and U ∼ V ∼ 2φ. (3.68) 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 Ad S 7 space, we will call this limit an asymptotically locally flat domain wall. On the other hand, asr → 0, we find φ ∼ − 3 20 ln(4 pr ), V ∼ 7 10 ln(4 pr ), 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 be completely 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, Using this result, we find that asr → 0,ĝ 00 → ∞, so, in this case, the IR singularity is unphysical.

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

Ad S 3 × M 4 solutions with S O(4) twists
As in the previous section, we will consider SO(4) symmetric solutions for SO (5), SO(4, 1) and C SO(4, 0, 1) gauge groups with the embedding tensor given in (3.56). To find Ad S 3 × M 4 k solutions, we use the following ansatz for the seven-dimensional 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 With the vielbein basis of the form em = e U dx m , er = dr, we obtain the following non-vanishing components of the spin connection withî,ĵ =3, . . . ,6 =χ,ψ,θ,φ being flat indices on M 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 threeform field strengths cannot vanish in order to satisfy the Bianchi's identity for H (3) μνρ M since the above gauge fields lead to non-vanishing M N P Q R H N P terms. To preserve the residual SO(4) symmetry, only 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 (3) μνρ5 is obtained by turning on the massive three-form S (3) μνρ5 . On the other hand, for C SO(4, 0, 1) gauge group, we have Y 55 = 0, so the contribution to H (3) 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 C SO(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 Ad S 3 fixed point only for k = −1 and ρ = 1. The resulting Ad S 3 × H 4 solution is given by This is the Ad S 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 Ad S 7 vacuum to this Ad S 3 × H 4 fixed point and curved domain walls. Examples of these RG flows are given in Fig. 12.
Unlike the previous cases, the IR singularities in this case are unphysical due to the behaviorĝ 00 → ∞.

Ad S
In this section, we consider the manifold 4 in the form of a product of two Riemann surfaces 2 k 1 × 2 k 2 . The ansatz for the metric takes the form of in which the metrics on 2 k 1 and 2 k 2 are given in (3.7). Using the following choice for the vielbein 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 2 k 1 and 2 k 2 , respectively. As in other cases, we consider all gauge groups of the form C SO( 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 C SO(2, 2, 1) and C SO(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 second-ordered field equations.
Using the projection conditions (3.21) and together with the twist conditions we obtain the following BPS equations  (3.105) Among all the gauge groups with an SO(2) × SO(2) subgroup, it turns out that Ad S 3 × 2 × 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 Ad S 3 × 2 × 2 solutions given here are new.
Following [14], we define the following two parameters to characterize the possible Ad S 3 × 2 × 2 solutions where we have set ρ = 1. In order for the Ad S 3 fixed points to exist in SO(5) gauge group with σ = 1, one of the Riemann surfaces must be negatively curved, and Ad S 3 × H 2 × 2 solutions can be found within the regions in the parameter space (z 1 , z 2 ) shown in Fig. 13. These regions are the same as those given in [14]. The Ad S 3 × 2 × 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 Ad S 7 to Ad S 3 × H 2 × 2 fixed points and curved domain walls in the IR are shown in Figs. 14, 15 and 16 for 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 Ad S 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 Ad S 3 ×S 2 × H 2 , Ad S 3 ×S 2 ×R 2 and Ad S 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 Ad S 3 vacua to exist in SO(3, 2) gauged maximal supergravity as shown in Fig. 17.
For SO (3,2) gauge group, there is no asymptotically locally Ad S 7 geometry. We will consider RG flows between the Ad S 3 × S 2 × 2 fixed points and curved domain walls with Mkw 3 × S 2 × 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 half-supersymmetric domain wall of the seven-dimensional gauged supergravity. A number of these RG flows with A general ansatz for the seven-dimensional metric takes the form of in which the explicit form for the metric on the Kahler fourcycle will be specified separately in each case.

Ad S 3 × K 4 k solutions with S O(3) twists
We begin with a twist along the SU (2) ∼ SO(3) part of the spin connection and choose the following form of the metric on K 4 k ds 2 with ψ ∈ [0, π 2 ], and f k (ψ) is the function defined in (3.8).
we find non-vanishing components of the spin connection whereî =3, . . . ,6 is the flat index on K 4 k . ωˆiˆj (1) are the SU (2) spin connections.  To perform the twist, we turn on SO(3) gauge fields with the following ansatz with the modified two-form field strengths given by  Using the scalar coset representative (3.44) and the projection (3.21), we find the following BPS equations 120)

Ad S 3 × K 4 k solutions with S O(3) + twists
We now move to Ad S 3 × K 4 k solutions with the twist given by identifying the SU (2) part of the spin connection with the self-dual SO(3) +

⊂ SO(3) + × SO(3) − ∼ SO(4). We begin with the SO(3) × SO(3) gauge fields of the form
The self-dual SO(3) gauge fields can be defined as In this case, we perform the twist by imposing the twist condition (3.47) and the three projections given in (3.117) together with an additional projection for the self-duality of SO(3) 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 representative (3.58), we find the following BPS equations which is the Ad S 3 × C H 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 Ad S 7 vacuum to this Ad S 3 × C H 2 fixed point and curved domain walls in the IR are given in Fig. 24. The IR singularities are physically acceptable as indicated by the behaviorĝ 00 → 0.

Ad S 3 × K 4 k solutions with S O(2) × SO(2) twists
As a final case for Ad S 3 × K 4 k solutions, we will perform another twist on the Kahler four-cycle by cancelling 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 the following vielbein em = e U dx m , er = dr, e3 = e V ψ all non-vanishing components of the spin connection are given by together with the twist condition (3.19). The associated twoform gauge field strengths are given by (2) and H 34 (2) = F 34 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 Ad S 3 fixed point solutions in which we have taken g > 0 for convenience. Up to some differences in notations, these Ad S 3 × C H 2 fixed points are the same as the solutions studied in [14]. As in the previous cases, we study RG flows from the supersymmetric Ad S 7 vacuum to the Ad S 3 × C H 2 fixed points and curved domain walls in the IR. Some examples of these flows are given in Fig. 25 for g = 16 and different values of p 2 . The behaviors of the eleven-dimensional metric componentĝ 00 for these RG flows are shown in Fig. 26 which indicates that the singularities are physical. Apart from these Ad S 3 × C H 2 fixed points, we find new Ad S 3 × C P 2 fixed points in SO(4, 1) and SO (3,2) gauge groups respectively in the following ranges, with g > 0,

Supersymmetric Ad S 2 × 5 solutions
We end this section by considering solutions of the form Ad S 2 × 5 . Ad S 2 × 5 solutions for the manifold 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) have already been analytically given in [11]. We will not repeat the analysis for this case here. However, if we consider 5 as a product of three-and two-manifolds 3 × 2 , it is possible to perform a twist by turning on SO(3)×SO(2) gauge fields along 3 × 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 (3.150) The explicit form of the metrics on the 3 k 1 and 2 k 2 are given in (3.39) and (3.7), respectively.
Using the vielbein et = e U dt, er = dr, eθ 2 = e W dθ 2 , we find non-vanishing components of the spin connection as follow

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 in which we have also used the γ r projector in (3.21).
From these equations, we find an Ad S 2 fixed point only for k 1 = k 2 = −1 and σ = 1. The resulting Ad S 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 superconformal quantum mechanics. Examples of RG flows from the supersymmetric Ad S 7 vacuum to the Ad S 2 × H 3 × H 2 fixed point and curved domain walls in the IR are given in Fig. 30. From the behavior of the (a) (b) Fig. 29 Profiles of the eleven-dimensional metric componentĝ 00 for the RG flows between Ad S 3 × C P 2 fixed points and curved domain walls in SO(4, 1) and SO(3, 2) gauge groups 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. These result in gauge groups of the form C SO( 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 . Following [34], we will 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 i j in the form This generates C SO( 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 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) μνρ M arises solely 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 μν j 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 C SO( p, q, r ) with p + q + r = 4 gauge group is not large enough to accommodate SO(5) or SO(3) × SO(2) subgroups. Consequently, it is not possible to have Ad S 2 × 5 or Ad S 2 × 3 × 2 solutions.

Solutions with the twists on 2
We first look for Ad S 5 × 2 solutions for 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 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 S O(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 Ad S 7 fixed point. The supersymmetric vacuum is given by half-supersymmetric domain walls dual to N = (2, 0) non-conformal 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. Similar solutions can also be found in SO(2, 2) gauge group.
For large V , the contribution from the gauge fields is highly suppressed. In this limit, we find Examples of flow solutions with this asymptotic behavior are given in Figs. 31, 32 and 33 for 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 Mkw 4 ×R 2 -sliced domain walls given in Fig.  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 Fig. 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 C SO( p, q, 4 − p − q) gauge 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 tendimensional metric which, for SO( p, 4 − p) gauge group, is given bŷ For C SO( p, q, 4 − p − q) gauge group, we decompose the SL(4) indices i, j, . . . into (î,ĩ) withî = 1, . . . p + q and i = p + q + 1, . . . , 4. The ten-dimensional metric and the warped factor are still given by (4.18) and (4.19) but with η i j = (ηˆiˆj , η˜i˜j ) replaced by the SO( p, q) invariant tensor ηˆiˆj and η˜i˜j = 0. In this case, μˆi become coordinates on H p,q satisfying ηˆiˆj μˆi μˆj = 1 while μ˜i are coordinates on T 4− p−q .
For the present case of SO(4) gauge group, we simply have η i j = δ i j for i, j = 1, 2, 3, 4. The behavior of the time component of the ten-dimensional metricĝ 00 for the flow solutions in Figs. 31, 32 and 33 is shown in Fig. 34. Near the IR singularities, we findĝ 00 → 0, so the singularities are physically acceptable.

Solutions with S O(2) twists
We then move to another twist on 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

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

Solutions with S O(3) twists
In this case, the gauge groups with an SO(3) subgroup are described by the embedding tensor  (4.32) leading to the scalar potential To perform the twist, we turn on the SO(3) gauge fields (4.34) 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 We have neglected an additive integration constant for U which can be absorbed by rescaling the coordinates on Mkw 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 gr −C p pr (4.42) 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 Computing the type IIB metric, we obtain g 00 ∼ e 2U + 3 asr → ∞, which implies that the IR singularity is unphysical.
For ρ = 0, we can only partially solve the BPS equations. As in the ρ = 0 case, the BPS equations give U = 2φ 0 . Taking a linear combination φ 0 + 3 5 φ and defining a new coordinater via dr dr = e − 24 5 φ , we find The complete solutions can be obtained numerically. As r → ∞, we find

Solutions with S O(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 i j = δ i j . 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 fields 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 Ad S 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 coordinater , Asr → ∞, we find, with C set to zero by shiftingr , which is identified with a domain wall solution given in [41]. On the other hand, asr → 0, the solution becomes This singularity is unphysical since the ten-dimensional metric giveŝ We note that this solution is the same as that given in Sect. 3 We finally look for supersymmetric solutions obtained from the twists on a four-manifold 4 . As in the case of gaugings in 15 representation, we will consider two types of 4 in terms of a product of two Riemann surfaces 2 × 2 and a Kahler four-cycle K 4 .

Solutions with S O(2) × SO(2) twists on 2 × 2
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 2 k 1 × 2 k 2 , we turn on the following SO(2) × SO(2) gauge fields with the corresponding 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(σ p 11 + p 21 ) = k 1 and g(σ p 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 +e −2(W +φ) (e 4φ p 12 + p 22 ) −e −2(W +φ) (e 4φ p 12 + p 22 ) +e −2(W +φ) (e 4φ p 12 + p 22 )  Fig. 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 Fig. 44.
We have also considered SO(2) twists on 2 × 2 by setting p 11 = p 12 = 0 and obtain more complicated BPS equations. However, we do not find any Ad S 3 fixed points either. Therefore, we will not give further detail on this analysis.

Solutions with S O(3) twists on K 4
For 4 being a Kahler four-cycle K 4 k , we perform an SO(3) 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 find the following BPS equations in which we have used the twist condition (3.47). We do not find any Ad S 3 fixed points from these equations. Examples of supersymmetric flows for β = −2 are given in Figs. 45 and 46 for k = 1 and k = −1, respectively. From the behavior of the ten-dimensional metric given in Fig. 47, we find that the IR singularities for k = −1 are physical. In the case of

Solutions with S O(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 (4.74) 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 threeform 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 Ad S 3 fixed points from these equations.

Conclusions and discussions
In this paper, we have extensively studied supersymmetric Ad S n × 7−n solutions of the maximal gauged supergravity in seven dimensions with C SO( p, q, 5 − p − q) and C SO( 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/M-theory. 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 Ad S n × 7−n fixed points with n = 2, 3, 4, 5. We have provided numerical RG flows interpolating between the supersymmetric N = 4 Ad S 7 vacuum dual to N = (2, 0) SCFT in six dimensions and all these Ad S n × 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 take the form of curved domain walls with Mkw n−1 × 7−n slices and can be interpreted as non-conformal field theories in n − 1 dimensions. The flow solutions suggest that they describe 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 Ad S 5 × S 2 , Ad S 3 × S 2 × 2 and Ad S 3 × C P 2 solutions in non-compact SO(3, 2) gauge group. Unlike in SO(5) gauge group, there do not exist any supersymmetric Ad S 7 fixed points in this gauge group. The maximally supersymmetric vacua are given by half-supersymmetric 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 Ad S 3 ×C P 2 solutions has been found. For convenience, we summarize all the Ad S n × 7−n fixed points identified here together with all the previously known results in SO(5) gauge group in Table 1.
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 spacetime. 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 Ad S 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 higherdimensional perspective as black strings in Ad S 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 Ad S 8 spacetime due to the absence of supersymmetric Ad S d vacua for d > 7.
For C SO( 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 C SO( p, q, 5 − p − q) gauge group but have not found any Ad S 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 Mkw n−1 × 7−n worldvolume. By the standard DW/QFT correspondence, these solutions should be interpreted as RG flows across dimensions between non-conformal 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 analysis of this paper by using more general ansatz in particular with non-vanishing massive two-form fields and find new classes of Ad S n × 7−n solutions of seven-dimensional gauged supergravity.