Holographic RG Flow in a New $SO(3)\times SO(3)$ Sector of $\omega$-Deformed $SO(8)$ Gauged ${\cal N}=8$ Supergravity

We consider a certain ${\cal N}=1$ supersymmetric, $SO(3)\times SO(3)$ invariant, subsector of the $\omega$-deformed family of $SO(8)$-gauged ${\cal N}=8$ four-dimensional supergravities. The theory contains two scalar fields and two pseudoscalar fields. We look for stationary points of the scalar potential, corresponding to AdS vacua in the theory. One of these, which breaks all supersymmetries but is nonetheless stable, is new. It exists only when $\omega\ne 0$. We construct supersymmetric domain wall solutions in the truncated theory, and we give a detailed analysis of their holographic dual interpretations using the AdS/CFT correspondence. Domain walls where the pseudoscalars vanish were studied previously, but those with non-vanishing pseudoscalars, which we analyse numerically, are new. The pseudoscalars are associated with supersymmetric mass deformations in the CFT duals. When $\omega$ is zero, the solutions can be lifted to M-theory, where they approach the Coulomb-branch flows of dielectric M5-branes wrapped on $S^3$ in the deep IR.


Introduction
For thirty years after its construction in 1982 [1], the four-dimensional SO(8) gauged maximally supersymmetric N = 8 supergravity was widely considered to be a unique theory.
Interestingly, using the embedding tensor formulation [2], it was recently realized that there exists a family of deformations of the theory, characterised by a single parameter commonly called ω, associated with a mixing of the electric and magnetic vector fields employed in the SO(8) gauging [3,4]. Inequivalent N = 8 theories are parameterised by values of ω in the interval 0 ≤ ω ≤ π/8. This development has raised numerous interesting questions, such as its possible higher-dimensional string/M-theory origin and the consequences of the ω deformation for the holographic dual theory.
The potential for the 70 scalar fields in the N = 8 theory depends non-trivially on the ω parameter, and the structure of the stationary points, which is already rich in the original undeformed theory, becomes even more involved in the deformed theories. As in the undeformed case, the investigation of the stationary points in the complete theory is extremely complicated, and in order to render the problem tractable, one has to consider consistent truncations in which only subsets of the scalar fields are retained. There have been a number of studies in which truncations of the new ω-deformed maximal supergravity have been performed, typically with the focus being on finding scalar-field truncations in which the scalar potential still has a non-trivial dependence on the parameter ω, leading to a richer structure of anti-de Sitter (AdS) stationary points, with the nature of the vacuum states now being dependent on ω. One can also then look for domain-wall solutions that approach the AdS stationary points asymptotically at infinity.  [10]. These various truncations are parallel to the consistent scalar-field truncations performed for the the original de Wit-Nicolai theory [11][12][13][14][15][16][17][18][19]. Consistent truncations retaining U (1) gauge fields have also been considered, giving rise to an ω-deformed version of the ST U supergravity [20], and a one-parameter extension [21] of an Einstein-Maxwell-scalar system [22] previously obtained via a reduction from eleven dimensions on a seven-dimensional Sasaki-Einstein manifold. The latter has been used in a study of holographic condensed matter systems.
In this paper, we consider a new consistent truncation of SO (8)  Within the de Wit-Nicolai theory, this sector does not yield new critical points. However, it does provide two new critical points in the ω-deformed theories. (These are absent in the undeformed theory because the value of the scalar potential at these points goes to infinity in the limit when ω goes to zero.) By construction, the two new critical points preserve SO(3) D × SO(3) R global symmetry. Moreover, one of them also preserves N = 3 supersymmetry in the full N = 8 theory [23], while the other one, which had not been found previously, is non-supersymmetric but nonetheless stable.
Via the AdS/CFT correspondence, stable AdS solutions in supergravity theories correspond to local conformal field theories (CFT) living on the boundary of AdS. Two AdS critical points may be connected by a domain-wall solution, which is interpreted as the holographic description of an RG flow from one CFT in the ultra-violet (UV) to another CFT in the infra-red (IR). There are also interesting classes of holographic flows starting from AdS in the UV and flowing to a non-AdS spacetime in the deep IR. In such solutions, the scalar fields flow to infinite values at the IR end of the flow, thus rendering the IR geometry singular. In fact, most of the known domain-wall solutions belong to this class. It seems natural to interpret these solutions as RG flows to non-conformal IR quantum field theories. A proper understanding of the nature of the IR singularities of the geometry and the corresponding QFT in the IR requires embedding the lower-dimensional solution into the UV-complete string or M-theory. From the higher-dimensional perspective, the singularities are physically allowable if they are associated with branes of positive tension. Examples are holographic Coulomb-branch flows, such as those studied in [19,24,25]. In some other examples, involving brane polarization [26], the singularities are placed at the locus of the dielectric branes [27]. A very recent paper suggests that a singular lower-dimensional solution can lift to a smooth higher-dimensional solution [28]. Properties of the IR QFT are also revealed by the study of the higher-dimensional brane configuration. Depending on the nature of the sources triggering the flow, the structure of the IR theories can take diverse forms.
In the context of the AdS 4 /CFT 3 correspondence and its uplift to M-theory, supersymmetric domain-wall solutions flowing to non-AdS spacetimes in the IR have not been well studied and only very few examples are known. This is a consequence of the complexity of the N = 8 supergravity theory, which contains 70 scalar fields . The main purpose of this paper is to explore new domain-wall solutions captured by the consistently-truncated We begin with the study of such supersymmetric domain-wall solutions in the original undeformed de Wit-Nicolai theory, since in this case embedding within M-theory is known, and furthermore the dual CFT is known to be the ABJM theory. A proper interpretation in terms of a UV-complete framework can therefore be achieved. Near the boundary AdS, the leading fall-off coefficients of the scalars and pseudoscalars in the truncated theory are interpreted as the vacuum expectation values (VEVs) of dimension-1 primary operators and the supersymmetric mass terms in the dual CFT respectively. When the pseudoscalars are turned off, the supersymmetric domain-wall solutions were found analytically, describing the Coulombbranch flows on M2-branes spreading out into six possible distributions in the transverse space. When the pseudoscalars are turned on, the complexity of the flow equations is such that we are only able to obtain the solutions numerically, by integrating the flow equation from the IR to the UV. The solutions we find correspond to flows driven by both the VEV and the mass terms. The competition between the VEV and mass terms leads to a variety of possible IR singularities in the geometry. The physical solutions approach the Coulomb branch flow of dielectric M5-branes wrapping on S 3 in the deep IR.
We then turn to the supersymmetric domain-wall solutions in the ω-deformed theories, within the same truncated scalar sector. We are interested in supersymmetric holographic RG flows, and for these it now turns out that the pseudoscalars are necessarily active. The singular IR behaviors of the solutions are similar to those arising in the ω = 0 case. However, since the higher-dimensional origin of the ω-deformed theories is currently unknown, we must necessarily postpone for now any attempt to give a complete interpretation of the ωdeformed supersymmetric domain-wall solutions. In this regard, we note that a recent paper [29] contains a no-go theorem showing that the ω-deformed gauged supergravities cannot be realised via a compactification that is locally described by ten or eleven dimensional supergravity.
The plan of the paper is as follows. In section 2 we discuss the consistent truncation of the ω-deformed N = 8 supergravities to the SO(3) D × SO(3) R invariant sector, focusing in particular on the four scalar fields, and their scalar potential. We obtain the first-order equations implied by imposing the requirement of N = 1 supersymmetry on the domain-wall solutions, and we show how the scalar potential may be written in terms of a superpotential.
In section 3 we study the critical points of the scalar potential. These include a variety of critical points that were found previously in truncations with larger invariant symmetry groups containing SO(3) D × SO(3) R , in addition to the new non-supersymmetric critical point that arises in our truncation when ω = 0. In section 4 we discuss in detail the N = 1 supersymmetric domain-wall solutions that are asymptotic to the maximally-symmetric N = 8 AdS solution in the UV, in the case when ω = 0 so that we can lift the solutions to M-theory and thus give a holographic dual interpretation via the ABJM model. We discuss this both for the case of vanishing pseudoscalars, for which the domain-wall solutions had been found analytically in earlier work, and also when the pseudoscalars are non-vanishing, in which case we have to resort to numerical analysis. In section 5 we extend our discussion to the case where the ω deformation parameter is non-zero. Supersymmetric domain walls must now necessarily have non-vanishing pseudoscalar fields, and hence all our discussion in this section is based on the numerical analysis of the solutions. After presenting our conclusions in section 6, we include two appendices in which we give details of the embedding of SO(3) D × SO(3) R in SO (8), and some of the conventions for gamma matrices and uplift formulae that we employ in the paper.
The scalar potential of the ω-deformed family of SO(8) gauged N = 8 supergravities can be described conveniently in the symmetric gauge, where the E 7 /SU (8) scalar coset representative is parameterized as Here φ ijkℓ are complex scalar fields, totally antisymmetric in the rigid SU (8) indices, and obeying the complex self-duality constraint Note that in the symmetric gauge SU (8) and SO(8) indices are identified. Introducing coordinates x I on R 8 (where I is an 8 s index), the 35 complex scalar fields can be written The SO(3) D × SO(3) R invariant subset that we shall be considering in this paper are given by (2.4) (A detailed derivation of the invariant 4-forms can be found in Appendix A.) Here Ψ 1 and Ψ 2 parameterise an SL(2,R) SO(2) × SL(2,R) SO (2) coset. Having obtained the form of the scalar 56-bein V for the consistent truncation we are considering, it is a mechanical, if somewhat involved, procedure to substitute it into the expressions given in [4] for the various terms in the Lagrangian of the ω-deformed N = 8 gauged supergravity. Introducing four real scalar fields by writing the Einstein and scalar sectors of the N = 1 truncation are described by the Lagrangian The potential V can be written as V = g 2 V , where g is the gauge coupling constant and 64 V = −256 cosh 4 φ 2 + 2 cosh φ 1 cosh 2 φ 2 (−57 − 20 cosh 2φ 2 + 13 cosh 4φ 2 + 24 sinh 4 φ 2 cos 4σ 2 ) Note that the potential is invariant under the transformations and so inequivalent theories are characterised by the parameter ω lying in the interval [0, π/8].
The potential can be expressed in terms of a superpotential W , with and where we have defined We are interested in N = 1 supersymmetric domain-wall solutions, of the form The existence of a Killing spinor requires that the first-order equations should be satisfied, where a prime denotes a derivative with respect to ρ and we use the The solutions of these first-order equations also obey the second-order equations of motion that follow from the Lagrangian (2.6). This can be seen easily as follows. The action (including the Gibbons-Hawking term) evaluated on the domain-wall ansatz is given by 15) and this is clearly extremised by the solutions of (2.14). (Here we adopt the same notation as [13].) Using γ 2 ǫ 8 = (ǫ 8 ) * , and the solutions to the Killing spinor equation are given by where ε 1 and ε 1 are two real constants. Utilizing (2.16), eqs (2.14) can be rewritten as where φ I denotes all four real scalars, and K IJ is the inverse metric for the kinetic terms in the scalar coset. Eq. (2.18) implies that the BPS equations in the bulk describe a gradient flow in the scalar coset manifold, with −g|W | being the "potential" whose gradient drives the flow. When the solutions to (2.18) are asymptotically-AdS domain walls and therefore correspond to an RG flow in the dual CFT, Eq. (2.18) also implies a holographic strong a-theorem 1 1 Recently, gradient flows and a strong a-theorem were studied in three dimensions [30], demonstrating the existence of a candidate a-function for renormalisable Chern-Simons matter theories at two-loop order.
The monotonic behavior of the a-function along renormalisation group flows is related to the β-function via a gradient flow equation involving a positive-definite metric similar to that in our holographic discussion.
The function A defined in [30] is equivalent to our |W | here.

Critical Points of the Scalar Potential
The symmetry group SO(3) D × SO(3) R can be embedded in SO(8) through the chain On the other hand the SU (3) invariant sector of N = 8 supergravity has been thoroughly studied both in the original de Wit and Nicolai theory [11,12,18] and in the ω-deformed case [6], with the group embedding Using the Newton-Raphson method, with the potential (2.7), we scanned for its critical points. We found all the previously-known critical points with G 2 or SO(7) symmetry, and also two critical points with SO(3) D × SO(3) R symmetry. One of them, preserving N = 3 supersymmetry, was first discovered in [23]. For this critical point, the dependence of the two complex scalars, and the associated cosmological constant, on ω are displayed in Fig.   1. This was the first example of an N = 3 supersymmetric vacuum in SO(8) gauged N = 8 supergravity. The mass spectrum of the fluctuations around this vacuum is given by [23]:  Table 1, we list all the critical  Three remarks are in order: a) The two transformations (2.8) and (2.9) combine into a symmetry in the case that ω = π/8: This can be seen in the table.
b) Points related by have the same location in the complex plane, and hence are equivalent.
c) It is interesting to see that there are two critical points with the same cosmological constant, but with different residual symmetry; one with G 2 and the other with

Holographic N = 1 RG Flows on M2-branes
In this section, we study the domain-wall solutions to Eq.(2.14). In particular, we are interested in solutions approaching the trivial N = 8 AdS vacuum in the UV. We shall first restrict our discussion to the ω = 0 case, in which the supergravity is just the original de Wit-Nicolai theory. The boundary CFT corresponding to the trivial N = 8 vacuum is the ABJM theory [32] with Chern-Simons level k = 1 or k = 2, for which the supersymmetry is enhanced from N = 6 to N = 8 [33][34][35]. Therefore, our domain-wall solutions describe the RG flows on M2-branes driven by N = 1 deformations.
Owing to the fact that SO(8) is manifest in the gravity theory but is not manifest in the ABJM theory under the large N limit, it is not straightforward to map the bulk scalars to the boundary primary operators. However, we recall that the SU (2) × SU (2) ABJM theory is equivalent to the BLG [36,37] theory, which is manifestly SO (8)  ABJM theory.
We divide the 70 bulk scalars into real (self-dual) and imaginary (anti-self dual) parts, by writing Table 1: Critical points in the ω = π/8 theory. We use ϑ to denote the number 3 + 2 √ 3.
The mass spectra of fluctuations around the critical points are independent of ω, except for the last one. Points marked with "*" disappear when ω → 0, while the two points with SO(7) − symmetry become degenerate in energy.
The 70 scalars can be mapped into two 35-dimensional symmetric traceless tensors as follows: matrices are expressible in terms of triality rotation matrices. In our conventions, the SO (8) gamma matrices take the form The details of the SO (8) ijkl in which the two modes are both renormalizable [38]. Boundary conditions preserving N = 8 supersymmetry require [39,40] This set of boundary conditions amounts, in the dual holographic picture, to S ijkl and P (2) ijkl being the VEVs of 35 dimension-1 scalar operators and 35 dimension-2 pseudoscalar operators in the dual SCFT respectively. In terms of the fields in the BLG theory, these operators take the form The expansion coefficients S ijkl and P (1) ijkl then correspond to the sources for these operators, according to the standard AdS/CFT dictionary. However, we shall see below that besides the source terms, the coefficients S (2) ijkl can depend on terms quadratic in the VEVs of the 35 v operators. This phenomenon has been seen previously, in the study of a continuous distribution of branes (see [41] for a review). As far as we are aware, there is no clear way to related these terms to the VEVs of higher-dimension operators. Also, we emphasize that when the dual SCFT is deformed by some operators, the operators in (4.5) will receive corrections.
We consider the domain-wall solutions asymptotic to the trivial N = 8 vacuum in the UV at z = 0. Denoting ζ 1 (z) = S(z) + iP (z) and ζ 2 (z) = S(z) + i P (z), we find that near the UV boundary, the solutions to Eq. (2.14) take the form where and In general, O (1) and O (2) will differ from each other by a similarity transformation generated by Γ 8 Iα , which is the identity matrix in our conventions. The expressions (4.9) suggest that the 35 v operators receive VEVs proportional to O (1) , and that also a fermion mass term of the form O (2) is turned on in the dual SCFT. The supersymmetric completion of the fermion mass term must include a bosonic mass term whose coefficient should be quadratic in the fermionic mass parameter. This implies that the quadratic β terms in (4.7) are related to the boson mass parameter, since β 1 and β 2 correspond to the fermion mass parameters.
We can check the relation between the boson and fermion mass parameters explicitly, by utilizing the N = 1 formulation of the BLG theory [42]. As mentioned previously, the gauge group of the BLG theory is SU (2) × SU (2), and therefore one cannot take a large-N limit of the theory with N being the rank of the gauge group. Nonetheless, we shall see that a naive application of the BLG theory gives rise to a result matching with the gravity dual.
The N = 1 formulation of the BLG theory is expressed in terms of the superfield where the gauge group indices and the SO(1, 2) spinor indices are suppressed. In order to have the fermion mass term of the form O (2) in (4.9), the BLG action must be deformed by since in our conventionsΓ 8 Iα = δ Iα . In the component language, this deformation contains terms of the form O (2) IJ Tr(φ I F J ). The supersymmetric kinetic term contains a contribution 1 2 Tr(F I F I ) quadratic in the auxiliary field F I . Integrating out F I generates a positivedefinite mass term Decomposing the mass matrix into its traceless part and its trace, we find that the traceless part is given by whilst the trace part takes the form From Eq. to the Konishi multiplet which is dual to the shortest stringy mode. However, as we have seen, the non-chiral operator Tr(φ I φ I ) appears in the N = 1 mass terms. In fact, there is no contradiction. As pointed out in [43], when the CFT is deformed the form of the chiral operators changes, and they mix with other operators. In the case of the supersymmetric mass deformation, the chiral operator (4.5) mixes with the non-chiral operator Tr(φ I φ I ), giving the scalars a positive-definite mass.
In the following, we briefly discuss how to reformulate the 35 v and 35 s operators in the framework of the ABJM theory. The global symmetry of the ABJM theory is SU (4)×U (1) b , rather than SO (8).
which implies Group theoretically, (4.15) means that the 35 v operators in (4.5) should be replaced by three sets of operators in the ABJM theory, namely where A and B label the 4 of SU (4), while M −2 and M 2 are the monopole operators in the proper representations of the gauge group needed for gauge invariance of the operators [44].
When the gauge group is SU (2) × SU (2), the bifundamental scalars of the ABJM theory are related to the original BLG variables with SO(4) indices through where the SO (8)

Uplift to eleven dimensions
The 35 v dilatons parameterise the coset SL(8, R)/SO (8), and one can use the local SO (8) symmetry to diagonalise the coset, so that the scalar Lagrangian can be written in terms of seven dilatons ϕ where The eight X i , subject to the constraint are parameterized by the seven dilatons as where b i are the weight vectors of the fundamental representation of SL(8, R), satisfying and u is an arbitrary vector. Setting the Lagrangian (4.18) becomes equivalent to our Lagrangian (2.6) in the case that σ 1 = σ 2 = 0.
In [19], a set of domain-wall solutions was found, given by where The transformation is a diffeomorphism, and so the inequivalent solutions are parameterized by only seven, rather than eight, of the constants ℓ 2 i . In particular, η can be chosen so that the smallest of the ℓ 2 i is set to zero, while keeping the remaining ones non-negative. The domain-wall solutions are singular in the IR, and the nature of the singularity depends on the number of ℓ 2 i that can be set to zero by means of the shift symmetry (4.25). In terms of the new coordinates, where the smallest of the ℓ 2 i have been shifted to zero, the singular IR behavior of the metric is given by where k is the number of ℓ 2 i that are set to zero by the shift. To compare with (4.6), we expand S ≡ tanh 1 2 φ 1 and S ≡ tanh 1 2 φ 2 in terms of z, which is related to r by We find that By setting we see that (4.29) reproduces (4.6). From (4.28), one can see that the leading coefficients are invariant under the shift (4. 25), and that they therefore have an invariant physical meaning.
Below, we shall show that in fact they are related to the VEVs of the 35 v operators.
The solution in (4.24) can be uplifted to eleven dimensions, where it describes a continuous distribution of M2-branes [19]. The uplifted solution is given by where The transverse-space metric ds 2 8 is given by where i µ 2 i = 1 defines a unit S 7 in R 8 . The metric (4.32) can be expressed as a flat Euclidean 8-metric ds 2 8 = dy m dy m by making the coordinate transformation In terms of these Euclidean coordinates, the harmonic function H takes the form where σ is the normalized distribution function of the M2-branes.
Besides the trivial coincident branes case described by a delta-function distribution, solutions in our truncated theory correspond to six possible distributions of M2-branes, depending on the relative magnitudes of the constants ℓ 2 i , which are given in (4.28).
• ℓ 2 8 is the smallest among the ℓ 2 i . In this case, using the shift symmetry (4.25), ℓ 2 8 can be set to zero, and the M2-branes are distributed in a 7-ellipsoid. The explicit form of the distribution function can be found in [19], which suggests the existence of branes with negative tension. Thus, solutions in this class are unphysical.
are the smallest. This case is contained in the G 2 truncation of N = 8 supergravity [14]. The domain-wall solution is sourced by positive-tension M2-branes distributed in a segment.
To summarize, except for the case where the M2-branes are distributed in a 7-ellipsoid, the domain-wall solutions are all sourced by M2-branes with positive tension, and therefore they may be considered to be physical.
The harmonic function H has a Taylor expansion at large | y | given by whereŷ i ≡ y i /| y |, and the partial-wave expansion coefficients d i 1 i 2 ···n is ∆ = n/2 rather than ∆ = n, since at large y, | y | 2 is asymptotic to the standard radial coordinate of AdS. In general, the higher-order coefficients d (n) i 1 ···in with n > 2 are present, which means there are VEVs for higher-dimension operators as well as for those in the 35 v . Therefore, although the consistent truncation keeps only a finite number of fields, the profile of the Coulomb branch flow captures infinitely many VEVs. For our case, from (4.33), we find

Holographic RG flow with N = 1 mass deformations
In this section, we study the solutions to eq. (2.14) when the pseudoscalars are turned on. In this case, besides the non-trivial AdS 4 G 2 critical point found in [11], the other interesting solutions are domain walls asymptotic to AdS 4 in the ultraviolet. Specifically, for the domain-wall solutions we shall consider 2 , the pseudoscalars P and P behave like We shall use the coordinates defined in (2.13), in which the IR cutoff z IR corresponds to ρ = ρ IR . Under the diffeomorphism ρ → ρ + λ, the IR cutoff ρ IR and Fefferman-Graham expansion coefficients are changed to Therefore, the combinations e −ρ IR (α 1 , β 1 , α 1 , β 1 ) are invariant under the shift of ρ, and they characterise the solution. By shifting ρ one can choose a special coordinate in which ρ IR = 0, and the Fefferman-Graham coefficients (α 1 , β 1 , α 1 , β 1 ) are then equal to the shiftinvariant quantities. We shall present the domain-wall solutions using this particular choice of coordinate. For technical convenience, we work with the redefined the scalar fields Infinity in the complex ψ plane is mapped into the unit circle in the complex ζ plane.
We first discuss the solutions in which |ζ 1 | and |ζ 2 | both approach 1 in the IR at ρ = 0.
Near the unit circle, we may perform a perturbative expansion for ζ 1 and ζ 2 , writing When ζ 1 and ζ 2 approach the unit circle from generic angles specified by σ 1IR and σ 2IR , the perturbations at leading order satisfy the equations where the non-vanishing coefficients f 1 , f 2 and f 3 are given by An example of a solution in this class is given in Fig. 3, where σ 1IR = 11 8 π and σ 2IR = 13 8 π. Near the IR region ρ = 0, solutions for the perturbations take the form from which we can obtain the IR behavior of the scale factor e 2A(ρ) , and therefore the IR behavior of the metric. It turns out that the metric shares a similar singular behavior with the 4D Coulomb branch flow metric whose 11D uplift describes a continuous distribution of M2-branes on a 7-ellipsoid, namely, An explicit example of a solution for the σ 1IR = 5π/4 case is given in Fig. 4. Near the which we leads to the following singular behavior of the metric near the IR cutoff: which leads to a novel singular behavior of the metric which has not been observed in the Coulomb-branch flow metric, with The UV expansion is characterised by the coefficients Having shown several examples in which both ζ 1 and ζ 2 approach the unit circle in the IR, we now present an example in which ζ 2 limits to a point inside the unit circle; in other words, φ 2 stays finite at the IR cutoff. In general, this imposes a very complicated functional relation among |ζ 2 |, σ 1IR and σ 2IR . However, we find that if σ 2IR = π/2, this expression reduces to the rather simple form For |ζ 2IR | = 0.5, the numerical solution is given in Fig. 6. The IR expansion of the solution for which the singular IR behavior of the metric takes the form In this case, it should be noted that the convenient variables to study the perturbation of ζ 2 are (δReζ 2 , δImζ 2 ) rather than (δr 2 , δσ 2 ). This IR singularity is similar to the one appearing in the 4D Coulomb-branch flow metric whose 11D uplift describes a continuous distribution of M2-branes on a 4-ellipsoid. The UV expansion is specified by the coefficients where the inverse of the internal seven-dimensional metric is given by In the above formula, the warp factor is given by whereg mn (y) is the metric on the unit 7-sphere and the 28 Killing vectorsK m IJ are those of the unit 7-sphere. These may be described in terms of the coordinates X A on an 8-dimensional Euclidean space, subject to the constraint X A X A = 1, as we find that the uplift of the solution given in Fig. 3 can be brought to the form, near This singular behavior is the same as that appearing in the metric describing M2-branes distributed in a 7-ellipsoid, which involves M2-branes with negative tension. Thus, the class of flow solutions represented by the example in Fig. 3 is not physically acceptable.
For M5-branes distributed in an n-ellipsoid, near the IR singularity the metric behaves like dŝ 2 11 ≃ H −1/3 ds 2 6 + H 2/3 ds 2 5⊥ , H = r n−3 . (4.62) We find that after being uplifted to 11D, the metrics corresponding to the solutions given in Fig. 4 and Fig. 6 can be brought to the above forms with n = 2 and n = 1 respectively, by identifying ρ as the appropriate power of r. The six directions on the world-volume of the M5-branes consist of three directions on the world-volume of the M2-branes, and three directions from the 7-sphere. Since there are only three flat directions in the world-volume of the M5-branes, they must wrap on the remaining three directions. We are comparing such a configuration of M5-branes with one in which the M5-branes are flat and infinitely large. This comparison is valid, as long as we focus on the region infinitesimally close to brane. From the results given in [19], it can be deduced that for n ≤ 3, the geometry (4.62) is sourced by M5-branes with positive tension. Thus solutions possessing similar singular behaviors to the ones shown in Fig. 4 and Fig. 6 are physically allowed, with the singularity being balanced by normal positive-tension brane sources.
We also computed the uplift for the class of solutions represented by Fig. 5, for which the warp factor is given by The 11D metric for this solution does not approximate any Coulomb-branch metric of M-branes given in [19] in the near horizon region, indicating that a new supersymmetric solution should exist in 11D supergravity.

Holographic N = 1 RG Flows in the ω-Deformed Theory
In the ω-deformed theory, it is straightforward to verify that in the second-order equations of motion, σ 1 and σ 2 can be consistently to set zero. However, one also can show that and so there does exist a set of first-order equations which resembles the one governing the holographic Coulomb-branch flow on M2-branes. This is therefore a sharp distinction between the ω-deformed theory and the undeformed theory. Since we are interested in the supersymmetric holographic RG flows, it follows that pseudoscalars will always play a non-trivial role in the domain-wall solutions. A supersymmetric domain-wall solution interpolating between the SO(8) point and the G 2 point in the ω-deformed theory has been studied in [8].
Owing to the lack of any known higher-dimensional origin for the ω-deformed SO (8) gauged N = 8 supergravity, it is unclear which 3D CFT should be its holographic dual.
However, if we assume that such a dual CFT exists, and that it belongs to a certain type of Chern-Simons matter theory, the bulk scalar fields should still be dual to scalar operators that are bilinear in the boundary scalars or fermions, as was discussed in Sect. 3 for the undeformed case.
This assumption leads us to a preferred redefinition of the bulk scalar fields, for the following reason. Using the current definition of the scalar fields, near the boundary of AdS 4 the functional relations amongst the Fefferman-Graham coefficients will ostensibly depend on the ω parameter. However, as in the ω = 0 case, via the AdS/CFT correspondence these functional relations should match with those relating the bosonic mass parameters to the fermionic ones. These are determined by the boundary N = 1 supersymmetry, and should therefore be independent of ω. This suggests that the ostensible ω-dependence of the functional relations amongst the Fefferman-Graham coefficients should be a technical artifact that can be removed by redefining the bulk scalar fields. Indeed, in terms of the new complex scalar fields the equalities (4.7) are maintained. As a consistency check, if we write then in order that S ′ , S ′ and P ′ , P ′ should be dual to ∆ = 1 and ∆ = 2 primary operators respectively, the boundary conditions should preserve the N = 1 supersymmetry of ω-deformed supergravity in the bulk. In terms of the original fields, the above boundary conditions amount to These are consistent with the N = 1 boundary conditions given in [40] for the ω-deformed SO(8) gauged N = 8 supergravity.
Similarly to discussion we gave in the ω = 0 case, in the study of supersymmetric domain-wall solutions, when ω = 0 we shall now work with the fields ζ ′ 1 and ζ ′ 2 . Infinity in the complex ψ ′ plane is mapped into the unit circle in the complex ζ ′ plane. From now on, for simplicity of presentation, we shall remove the primes from the fields; all the fields below should be interpreted as the ω-rotated ones defined in (5.2). Flow solutions in ωdeformed theory are driven by the ω-dependent "potential" |W |, and so the perturbative IR expansion of the equations of motion depends on ω. However, the singular IR behaviors of the solutions are similar to those arising in the ω = 0 case. For instance, for a generic solution in which both ζ 1 and ζ 2 attain the unit circle, the perturbative IR expansion is given by where the non-vanishing ω-dependent coefficientsf 1 ,f 2 andf 3 are given bỹ 3 )|,  For ω = π/8, a numerical solution belonging to this branch is plotted in Fig. (8). Near the When (σ 1IR , σ 2IR ) = (π − 2ω 3 , 2ω 3 ) but still withf 1 vanishing, one obtains the second branch of solutions, which is analogous to those in the ω = 0 case with f 0 = 0 but σ 2IR = 0.
A solution belonging to this branch in the ω = π/8 theory is found when σ 1IR = 19π 12 and σ 2IR = 0.319382π. The numerical solution is shown in Fig. 9 with the IR expansion Similarly to the ω = 0 case, the ω-deformed theory also admits solutions in which ζ 2 limits to a point inside the unit circle; in other words φ 2 stays finite at the IR cutoff. In Figure 9: A typical flow obtained in the ω = π/8 theory by choosing σ 1IR = 19π 12 and σ 2IR = 0.319382π.
However, we find that if σ 2IR = 2ω 3 , this expression becomes the simpler one An example of such a flow in ω = π/8 theory is plotted in Fig. 10, for which the IR behavior In this case, it should be noted that the convenient variables to study the perturbation of ζ 2 are (δReζ 2 , δImζ 2 ) rather than (δr 2 , δσ 2 ). When ρ → 0, the metric appears to be of the same singular form as (4.55). The UV expansion of this solution is characterised by To achieve a proper understanding of the nature of the IR singularities of the geometry, it seems to be indispensable to embed the lower-dimensional solution into the UV-complete string or M-theory. Some tentative lower-dimensional criteria for characterising a physicallyallowable IR singularity without reference to string theory were proposed in [41]. However, as pointed out by [49], there exist solutions that satisfy these lower-dimensional criteria but which nonetheless lift to higher-dimensional solutions with unphysical singularities.
Since the higher-dimensional origin of the ω-deformed theories is currently unknown, we must necessarily postpone attempting to give a complete interpretation of the ω-deformed supersymmetric domain-wall solutions which generically have IR singularities.

Conclusions
This paper has addressed the problem of finding new solutions in the ω-deformed SO (8) gauged consists of four scalar fields, parameterising an SL(2,R) SO(2) × SL(2,R) SO (2) coset. The scalar potential depends on the ω-parameter explicitly, and can be reformulated using an ω-dependent superpotential in the standard way. We gave an extensive discussion of the stationary points of the scalar potential. In addition to the previously-known G 2 or SO(7)-invariant points, there are two SO(3) D × SO(3) R -invariant critical points captured by the SO(3) D × SO(3) Rinvariant sector. One of them preserves N = 3 supersymmetry in the full N = 8 theory [23], while the other one, which had not been found previously, is non-supersymmetric but nonetheless stable. The cosmological constants of these two critical points depend on the value of the ω parameter. In each case the value of the cosmological constant diverges in the ω → 0 limit, indicating the absence of these two new stationary points in the original de Wit-Nicolai theory.
We then looked for supersymmetric domain-wall solutions, which satisfy a set of firstorder equations required by the N = 1 supersymmetry. These equations describe a gradient flow in the scalar coset manifold, with the superpotential being the "potential" whose  (8) gauged N = 8 supergravity, in addition to those that were found within the truncation to the ST U supergravity model [50][51][52][53], and the SO(3) × SO(3)-invariant sector that was studied in [21,22]. The inclusion of both electric and magnetic charges will lead to a richer structure in the phase diagrams for AdS black holes.
Interestingly, the dyonic ISO(1, 7) gauged maximal supergravity can arise from a consistent reduction of massive type IIA supergravity [54], allowing a stringy interpretation of the physical consequences of the dyonic gauging, from which many details of the CFTs dual to the supersymmetric AdS vacua can be deduced. Therefore, it should be worthwhile to study the supersymmetric domain-wall solutions, and other type of solutions, in N = 8 supergravities with different dyonic gaugings.

A Branching Rules and Invariant 4-Forms
The embedding of the SO(3) D × SO(3) R that we are considering in this paper into SO (8) can be described via the chain of embeddings  (4) gives Note that i is now taken to be an SO(3) D index, and has been contracted. The 8-dimensional Hodge dual forms are also invariant, corresponding to scalars that will be retained. They can be written as We take the 't Hooft matrices to be given by The 't Hooft matrices are invariant under so(3) R , whose generators are chosen to be where the R ij are the so(4) generators, with (R ij ) ij = −(R ij ) ji = 1, and all other elements equal to zero.

B Conventions
The 8D gamma matrices admits a real representation: where σ 1 , σ 2 and σ 3 are the standard Pauli matrices, and σ 0 denotes the 2 × 2 identity matrix. The gamma matrices are all block off-diagonal: The 8 × 8 matricesΓ i Iα are the triality tensors, which map between the three 8-dimensional representations of SO (8). Note that sinceΓ 8 Iα = δ Iα , the I and α indices are equivalent under SO (7).