On new maximal supergravity and its BPS domain-walls

We revise the SU(3)-invariant sector of $\mathcal{N}=8$ supergravity with dyonic SO(8) gaugings. By using the embedding tensor formalism, analytic expressions for the scalar potential, superpotential(s) and fermion mass terms are obtained as a function of the electromagnetic phase $\omega$ and the scalars in the theory. Equipped with these results, we explore non-supersymmetric AdS critical points at $\omega \neq 0$ for which perturbative stability could not be analysed before. The $\omega$-dependent superpotential is then used to derive first-order flow equations and obtain new BPS domain-wall solutions at $\omega \neq 0$. We numerically look at steepest-descent paths motivated by the (conjectured) RG flows.


Motivation and outlook
Dimensional reduction of higher-dimensional field theories down to four dimensions (4d) has proven a very successful road towards the unification of gravitational and Yang-Mills interactions [1,2]. The first modern constructions go back to the seminal papers [3] and [4] in the 70's where the dimensional reductions of super Yang-Mills theory (SYM) and supergravity (SUGRA) were discussed. Both cases, even though fundamentally different in what concerns the theories to be reduced, display some universal features: i) appearance of a scalar potential V ii) generation of fermion mass terms iii) modification of the supersymmetry transformation rules for the fermions in the theory. The reduced Lagrangian can schematically be viewed as where the concrete expressions for the fermi mass terms in L fermi and the scalar potential V depend on the theory which is to be reduced.

Fetching ideas from SYM
Reducing N = 1, d = 10 SYM on a six-torus produces N = 4, d = 4 SYM. The reduced Lagrangian is of the form 1 [3] L 10d = − 1 4 F 2 + iψ Dψ −→ L 4d = − 1 4 F 2 + iψ i Dψ i + 1 2 (Dφ ij ) 2 where g is the gauge coupling constant, f represents the structure constants of the gauge group and i = 1, .., 4 is a fundamental index of the R-symmetry group SU(4) ∼ SO (6) emerging from the reduction. The four Weyl fermions ψ i descend from the original Majorana-Weyl fermion ψ in ten dimensions, whereas the scalar fields φ ij = φ [ij] correspond to the six internal components of the 10d gauge fields and are subject to the reality condition A key observation [3] is that the reality condition (1.3) prevents the R-symmetry group of the reduced theory to be extended to U(4) = U(1) × SU (4). The additional U(1) is simply not compatible with this condition.
The interaction in the reduced theory stems from the non-abelian structure of the theory in higher-dimensions (gf = 0). The L 4d in the r.h.s of (1.2) matches the general form (1.1) for dimensionally reduced theories: the first line contains the kinetic terms for the different fields, the second one corresponds to (scalar dependent) fermi mass terms which are of order gf , and the last line is identified with a scalar potential of order (gf ) 2 . Last but not least, the supersymmetry (SUSY) transformation for the fermions in the reduced theory reads so the last term implies a modification (linear order in gf like the fermi mass terms) with respect to the standard transformation rule.
The SUGRA side of the story Kaluza-Klein reductions of N = 2, d = 10 supergravity on a six-torus and of 11d supergravity on a seven-torus produces N = 8 (maximal) ungauged supergravity in four dimensions [5,6]. The resulting theory possesses an abelian G = U(1) 28 gauge symmetry under which all the scalars coming from the reduction of the higher-dimensional fields are neutral.
As a consequence, no scalar potential or fermion mass terms are generated However, certain background fluxes for the higher-dimensional fields can be turned on in a way still compatible with N = 8 supersymmetry [7][8][9][10][11] inducing what is called a gauging.
The result of the reduction is then a gauged supergravity with a non-abelian gauge group G and coupling constant g. The background fluxes are identified with the structure constants of G and the situation becomes similar to the SYM reduction (1. 2) with f ≡ fluxes. As a consequence of the gauging, the scalars in the theory become charged under G and both a scalar potential (quadratic on g · fluxes) and fermi mass terms (linear on g · fluxes) are generated L 10d/11d The supersymmetry transformation rules for the gravitini and the dilatini get also modified in a similar fashion to (1.4). In analogy to the SYM condition (1.3), the scalar fields in maximal supergravity can be arranged into a tensor φ IJKL = φ [IJKL] subject to the reality whereas ω = π 2 corresponds to a purely magnetic choice.
This time the index I = 1, ..., 8 refers to the fundamental representation of the SU (8) R-symmetry group emerging from the reduction. As for SYM, the reality condition (1.7) prevents the R-symmetry group to be extended to U(8) = U(1) × SU (8).
The lack of the U(1) factor both in N = 4 SYM and N = 8 SUGRA in four dimensions relates to the fact that these are CPT-self-conjugate multiplets of the supersymmetry algebra 2 . They satisfy the condition λ M AX = N /4 with λ M AX being the maximum helicity state inside the supermultiplet. Because of self-conjugacy, CPT doubling is not necessary. Then the scalars sit in a real representation of the R-symmetry group and the reality conditions (1.3) and (1.7) preventing the U(1) factor have to be imposed.

A novel U(1) in maximal supergravity
The existence of a relevant U(1) in maximal supergravity lying outside the SU(8) Rsymmetry group but still inside the Sp(56) electromagnetic group of the theory, was exploited in ref. [13] to build a one-parameter family of gauged supergravities with G = SO (8) gauging.
This parameter was identified with an electromagnetic phase ω which specifies the linear combination of electric (28 of them) and magnetic (28 of them) vectors entering the gauging G (see Figure 1), namely A G µ = cos ω A elec µ + sin ω A magn µ . (1.8) The G 2 -invariant sector of this new family of SO (8) gauged supergravities was analysed in refs [13,14] and found to contain genuinely dyonic critical points at ω = 0 with no counterpart in the standard electric case 3 of ω = 0 [5,17,18]. Similar results followed from the analysis of the SU(3)-invariant sector in ref. [19] based on a conjectured ω-dependent superpotential compatible with the N = 2 structure of the truncated theory [20][21][22] as well as with ω → −ω and ω → ω + π 4 identifications of the electromagnetic phase [13,16,19,23]. In this way, an ω-dependent superpotential could be envisaged (up to an overall phase) and the structure of SU(3)-invariant critical points investigated.
However, a supergravity derivation of the ω-dependent L 4d including fermi mass terms L fermi and scalar potential V for the SU(3) truncation, as done in refs [20,21] for the ω = 0 case, remains to be done. As we will see later, the precise knowledge of L fermi happens to be crucial for computing full mass spectra at ω = 0 and will allow us to check the stability of critical points of V which could not be analysed in ref. [19]. The derivation of L 4d can be carried out within the framework of the embedding tensor [8,24,25] and we will present it here. This is a purely four-dimensional supergravity formalism, so inverting the arrow in (1.6) might not necessarily be possible. In other words, the connection to reductions of a higher-dimensional theory is lost. Nevertheless, the knowledge of L fermi (more concretely of the T -tensor to be introduced later) as a function of ω and the scalar fields in the truncated theory, might help in finding new reduction Ansätze for a higher-dimensional origin of the electromagnetic phase along the lines of refs [26][27][28].

BPS domain-walls and flows equations
A second interest in deriving the explicit form of the ω-dependent L 4d is in the light of the AdS/CFT correspondence [29][30][31]. In its purely electric version (ω = 0), the SU(3)-invariant sector of the SO(8) gauged supergravity has played a central role in constructing RG flows dual to domain-wall solutions that interpolate between two AdS critical points of the scalar potential 4 . If the AdS points preserve some amount of supersymmetry the domain-wall is called BPS [33]. Then it can be constructed by solving a set of first-order flow-equations defined in terms of a superpotential W (φ i ) with asymptotic behaviour ∂ φ i W z→±∞ = 0, 3 The ω-dependent family of maximal supergravities with a different G = SO (4,4) gauging was also investigated in ref. [15] and found to contain SO(4)-invariant unstable de Sitter critical points with arbitrary light tachyons controlled by ω. This intriguing phenomenon of tachyon amelioration was connected to a jump of gaugings involving an AdS/Mkw/dS transition in ref. [16]. 4 We refer the reader to refs [32,33] as well as section 9 in ref. [34] for general reading and also refs [20,21,[35][36][37] for domain-walls in the SU(3)-invariant sector of the electric SO(8) supergravity. Other domain-walls were explored in refs [38,39] for different (electric) gaugings.

5
where z is the coordinate along the direction transverse to the domain-wall. The flowequations for the set {φ i (z)} of scalar fields are schematically given by [37] where A(z) is the scale factor in the domain-wall metric Ansatz and K ij is the (inverse) Kähler metric accounting for non-canonically normalised kinetic terms for the scalars. In ref. [20], the exact form of the superpotential W was extracted from the fermi mass terms L fermi in the case of a purely electric SO (8) gauging. Using that ω = 0 superpotential, various BPS domain-walls were constructed in the literature [20,21,[35][36][37]. In the dual field theory picture they correspond to three-dimensional RG flows connecting an UV fixed point at z → ∞ to an IR fixed point at z → −∞ (the scalars flow to a constant value for BPS domain-walls). Furthermore, higher-dimensional embeddings as reductions of 11d supergravity on AdS 4 × S 7 (with a round, squashed, stretched or warped seven-sphere) with a 4-form flux were found and connected to the theory of multiple M2-branes [37,[40][41][42][43][44][45][46][47].
The outline of the paper is as follows. Section 2 collects standard results in N = 8 gauged supergravity in the framework of the embedding tensor which will be extensively used in this work (readers being familiar with the subject may go directly to section 2.4). In section 3 we carry out a supergravity derivation of the scalar Lagrangian, the superpotential(s) and the fermion mass terms for the SU(3)-invariant sector of N = 8 supergravity with a dyonic SO(8) gauging. In section 4 we make use of these results to study the stability of nonsupersymmetric AdS critical points and to obtain BPS domain-walls at ω = 0. We then discuss the results and make some final remarks. More technical computations and lengthy expressions are put into the appendices.

Crash introduction to maximal supergravity
After the previous discussion on R-symmetries and U(1)'s, we now summarise general results on N = 8 gauged supergravity in four dimensions (mostly from refs [8,48]) and elaborate more on the idea of dyonic gaugings [49].

Gaugings and scalar potential
The bosonic field content of the supergravity multiplet in maximal supergravity consists of the metric g µν , 56 vector fields A M µ (28 electric and 28 magnetic) and 70 complex scalars 6 Σ IJKL satisfying the self-duality condition with Σ IJKL = (Σ IJKL ) * , and which serve as coordinates in a coset space E 7(7) /SU (8). The index M = 1, ..., 56 will refer to the fundamental representation of the (global) duality group E 7(7) , whereas I = 1, .., 8 to that of the (local) R-symmetry group SU (8).
In the ungauged (non-interacting) case, the choice of an Sp(56, R) frame for the vector fields will not affect Physics. Thus, as an example of symplectic transformation, electricmagnetic duality will leave any observable invariant. However this picture changes dramatically once a set of charges X MN P is turned on and the theory becomes gauged (interacting).
In this case, symplectic transformations are no longer symmetries and these get reduced to the duality group E 7(7) ⊂ Sp(56, R).
The first sign that the theory has been gauged is that part of the duality group has been promoted to a non-abelian gauge theory G ⊂ E 7 (7) . The ordinary derivative is then replaced where Ω MN is the Sp(56, R) invariant matrix (skew-symmetric) satisfying Ω MP Ω NP = δ N M . These two sets of constraints guarantee the consistency of the gauged supergravity.
The second sign is that a non-trivial scalar potential V (Σ) is generated for the scalars spanning the E 7(7) /SU (8) Using the mixed E 7(7) /SU(8) coset representative V M N (Σ), one can build the scalar-dependent in terms of which the non-trivial potential induced by the charges X MN P of the gauged supergravity reads This potential is invariant under the linear action of E 7 (7) transformations and corresponds to the V appearing in (1.6).

Fermi mass terms and SUSY transformations
The fermionic field content of the supergravity multiplet in maximal supergravity consists of 8 gravitini ψ I µ and 56 dilatini χ IJK = χ [IJK] . The fermion mass terms in the Lagrangian are given by where A IJK,LM N ≡ 8 The fermion mass terms can be used to compute mass spectra at maximally symmetric solutions (AdS, Minkowski or dS). When evaluated at a critical point of the potential, the scalar spectrum can be computed from and, as usual in supergravity theories, stability 5 is defined with respect to the normalised mass matrix m 2 L 2 = 3 |V 0 | (mass 2 ) , where V 0 denotes the value of the energy in the solution and L 2 = 3/|V 0 |. The masses of the vectors are obtained after diagonalising the mass matrix This matrix has (at least) 28 null eigenvalues associated to the unphysical linear combinations of vectors.
Finally, the counterparts of the last term in the modified SUSY transformation (1.4) are given by where the dots stand for terms already present in the ungauged case [8]. The number of supersymmetries preserved in a solution corresponds with the number of Killing spinors J satisfying Therefore, the fermion mass terms can be used to thoroughly explore maximally symmetric solutions of supergravity as well as the issues of stability and supersymmetry breaking.

The T -tensor
The fermion mass terms in (2.7) can be obtained from the so-called T -tensor. This tensor is related to the embedding tensor X MN P in (2.2) via a change of basis (2.14) discussed in the previous section. This is the route we will follow in order to compute the fermion mass terms, scalar/vector masses, etc. later on in the paper.

Dyonic gaugings
The possibility to embed a gauging dyonically inside the Sp(56, R) electromagnetic group of maximal supergravity was pointed out in refs [8,49,51,52] and made more concrete in ref. [13].
After this, various gauged supergravities models with G ⊂ SL (8) have been explored in the literature. This set-up is compatible with choosing electric charges ( , X as well as magnetic charges ( where the index A is now restricted to run over the 63 generators of SL(8) ⊂ E 7 (7) . The symmetric matrices θ and ξ specify the gauging as a function of the number of positive, negative and vanishing eigenvalues. The set of quadratic constraints in (2.3) take the form and allows for a parameter c interpolating between a purely electric gauging at c = 0 and a purely magnetic one at c = ∞. Most of the time it will be more convenient to move to a phase-like parameterisation such that purely electric gaugings (c = 0) correspond to ω = 0, purely magnetic (c = ∞) to ω = π 2 and dyonic gaugings to ω ∈ (0, π 2 ). In the present paper we will take a second look to the renowned SO(8) gauged supergravity, i.e. θ = ξ = diag(+1, ... ,+1), but will open the door for ω = 0 orientations of the gauging inside the electromagnetic Sp(56, R) group. This selects dyonic combinations of vector fields to span the SO(8) gauge symmetry. As mentioned in the introduction, there are the equivalence relations ω → −ω and ω → ω + π 4 for the choice of the electromagnetic phase, hence reducing its relevant range to ω ∈ [0, π 8 ].

N = 2 truncation
The dynamics of maximal supergravity results intractable if considering the entire set of fields in the theory. For that reason, it is customary to restrict the field content to a simpler subset invariant under the action of a certain subgroup of the R-symmetry group. We will consider here an SU(3)-invariant sector of the theory whose precise embedding inside the R-symmetry group is given by Sp (4). The gauging in the truncated theory is simply the U(1)×U(1) commuting with SU (3) inside SO (8). The decomposition of the eight gravitini in maximal supergravity features two singlets revealing the N = 2 supersymmetry preserved by the truncation [22].
The 70 complex scalars in (2.1) split into self-dual (SD) and anti-self-dual (ASD) irreducible representations (irreps) of SO (8). Schematically, Fields in the 35 s are proper scalars whereas those in the 35 c are pseudo-scalars. In the oxidation of the electric (ω = 0) SO(8) gauged supergravity to 11d supergravity, the former are related to deformations of the S 7 metric whereas the latter descend from the antisymmetric 3-form in the theory. On the other hand, the corresponding operators in the dual field theory are the traceless bosonic and fermionic bilinears, respectively. Given its relevance in this work, we will describe in detail the truncation of the scalar sector.

SU(3)-invariant scalars
Let us denote the components of a real vector x ∈ 8 v by x = (x 1 , ..., x 4 , x1, ..., x4) and introduce complex variables These transform as 4 and4 of SU(4) ⊂ SO(8) and have a further "1 + 3" splitting where σ + , σ − ∈ C and σ R , σ I ∈ R. The basis of invariant forms in (3.5) is built using the These are the two real two-forms and the two complex four-forms together with the conjugates Σ * + and Σ * − . Inserting (3.6) and (3.7) into (3.5) and plugging the complex variables in (3.3), one can read off the components of Σ using the original coordinates (x i , xˆi).
The scalar fields σ R , σ I ∈ R and σ + , σ − ∈ C in the SU(3)-truncation of maximal super- (1) . It contains two factors which are respectively the special Kähler (SK) and quaternionic Kähler (QK) manifolds in the N = 2 truncated theory. The two supersymmetries are associated to the ψ 1 µ and ψ1 µ gravitini which are singlets under the SU(3) action. It becomes very convenient to introduce a set of new variables which amounts to an alternative expansion (3.9) Using this expansion, Re( ), Re( 1 ) and Re( 2 ) correspond to scalars in the 35 s whereas Im( ), Im( 1 ) and Im( 2 ) correspond to pseudo-scalars in the 35 c , in agreement with the splitting (3.2).
The main advantage of this parameterisation is that the U(1) × U(1) gauge symmetry in the truncated theory can be used to gauge-fixing θ = ψ = 0 [18,20,22]. This translates into 1 = λ e iφ and 2 = 0, so that we are left with a theory containing four real scalars (λ, α) and (λ , φ). Furthermore, this gauge choice implies that there are no four-forms with an odd number of hatted (unhatted) indices in the expansion (3.9), e.g. Σ1 234 , Σ 1234 , etc., since they only appear through Im(Σ + ) and Im(Σ − ). In the absence of these "odd" forms, the truncated theory admits an intermediate N = 4 formulationà la Schön&Weidner [53] that makes a connection to generalised type II flux compactifications feasible [11]. We would like to look into this in the future.

The scalar Lagrangian
The Lagrangian for the scalar sector of maximal supergravity is given by where M ≡ M MN is the scalar-dependent matrix in (2.5) built from the mixed vielbein V M P (λ, α, λ , φ, θ, ψ) . At this point we are not performing any gauge-fixing yet, so we deal with a six real fields problem. The construction of the vielbein depends on the specific choice of basis for the E 7(7) generators and other related issues. In order to keep this section alive, we have put all the details aside in the appendix A.
The covariant derivative induced by the gauging is totally encoded inside the ω-dependent embedding tensor in (2.15)-(2.16) and reads In this work we will consider vanishing vector fields A M µ = 0 compatible with maximally symmetric solutions of the theory and also with BPS domain-wall configurations interpolating between two of such solutions. As a consequence D µ → ∂ µ and the scalar Lagrangian 13 takes the form accounts for the kinetic energy associated to the fields (θ, ψ) which, as discussed before, can be gauged away.
The computation of the scalar potential V (λ, α, λ , φ, θ, ψ) for a dyonic gauging turns out to be rather cumbersome mostly due to the cubic term XXMMM in (2.6). To carry it out, it is helpful to use the parameter c in (2.17) instead of its compact version ω in (2.18). We set the normalisation with an overall factor 1/(1 + c 2 ). After a straightforward but tedious computation, the c-dependent scalar potential in (3.12) reads 14) The dyonic potential does not depend on the fields (θ, ψ) which can be gauged-away at any value of c , in analogy to the purely electric gauging c = 0 studied in ref. [21]. The reason is that the c parameter encodes an Sp(56) rotation that does not modify the embedding SU(3) ⊂ SO (8) . On the other hand, the above scalar potential is invariant under the transformations φ → φ + π and φ → −φ. The latter will be connected later to the existence of two different superpotentials in the N = 2 truncated theory. 14

The fermi mass terms
In this section we compute the fermi mass terms A IJ and A I JKL in (2.7) as a function of the scalars (after gauge-fixing θ = ψ = 0) and the electromagnetic phase ω in (2.18). To do so, we follow the prescription described in section 2.3 : first we build the T -tensor in (2.13) using the explicit form of the mixed vielbein V P P (λ, α, λ , φ) and then extract A IJ and A I JKL by taking the traces in (2.14).

Gravitino-gravitino terms
The computation of the gravitino-gravitino couplings A IJ (λ, α, λ , φ) reveals an splitting of the the ω-dependence of the form Recalling the index decomposition I → 1 ⊕ a ⊕1 ⊕â , the mass terms for the two gravitini which are singlets under SU(3) and therefore survive the truncation to the N = 2 theory read A 11 together with The remaining six non-singlet gravitini which are projected out in the truncated theory acquire a mass term together with In order to shorten the above expressions, as well as some forthcoming ones, we have introduced the functions f 1 (λ , φ) = cosh 4 (λ ) + e 4iφ sinh 4 (λ ) , g 1 (λ ) = 3 cosh(4λ ) + 1 . (3.20) As a check of consistency, the expressions in refs [20,21] for the pure electric SO(8) gauging are exactly recovered 7 by setting ω = 0.

N = 2 superpotentials
Due to the N = 2 supersymmetry preserved by the SU(3)-truncation, there exist two superpotentials, we will denote by W 1 and W1, from which the scalar potential in (3.14) can be derived. The W 1 and W1 superpotentials are identified with the A 11 and A11 mass terms of the two SU(3)-singlet gravitini in (3.15) [21]. As a consequence, they depend on the fields (λ, α, λ , φ) as well as on the electromagnetic parameter ω, namely, Looking at the form of (3.16), it is easy to see that both superpotentials remain invariant under the shift φ → φ + π and, by virtue of (3.17), are exchanged by the reflection φ → −φ.
Using any of the two complex superpotentials above, the scalar potential can be derived as 8 (3.26) In going from the first line to the second in (3.26) we write W = |W |e iArg(W ) and use the computed from (2.6). The real and ω-dependent function |W (λ, α, λ , φ)| will become the relevant one when looking at BPS domain-wall configurations in the next section.
(3.28) 8 The different coefficients with respect to refs [20,21] stem from a different normalisation: Using the form of the gravitino-gravitino mass terms in (3.16), and after some algebra manipulations, the W 1 superpotential in (3.24) takes the form (3.29) The above superpotential represents the generalisation to arbitrary values of ω of the one derived in ref. [22], which now we know corresponds to ω = 0. A conjectured ω-dependent superpotential was first presented in ref. [19]. Even though the generalisation hinged on symmetry arguments 9 involving the periodicity of ω, a full-fledged supergravity derivation of the ω-dependent superpotential was missing. Here we have provided such a derivation using the framework of the embedding tensor, finding that the conjectured superpotential in ref. [19] was correct up to an overall phase that could not be determined by symmetry arguments therein.

Scalar dynamics and BPS domain-walls
The dynamics of the SU(3)-invariant scalar sector is encoded in the action where we have collectively denoted Σ i = (λ, α, λ , φ). The field-space metric K ij can be read off from (3.12) finding and the scalar potential V (Σ i ) was given in (3.14) (alternatively (3.26)). We will make a domain-wall Ansatz for the space-time metric where z ∈ (−∞, ∞) is the coordinate transverse to the domain-wall and A(z) is the scale factor. 9 It was based on the invariant classifiers computed in ref. [13].
The non-vanishing components of the Einstein equations G µν = T µν obtained from the action (4.1) read

(4.4)
These two equations can be combined to obtain the simple monotonicity relation so that ∂ z A will decrease along the domain-wall solution. The Euler-Lagrange equations for the scalars with Γ µ νρ and Γ i jk denoting Christoffel symbols in space-time and field-space, give rise to the following equations of motion : We will obtain AdS solutions to the above system of equations as well as BPS domain-wall configurations which additionally satisfy first-order flow equations.

AdS solutions
Maximally symmetric solutions are characterised by scalar fields getting a constant vacuum expectation value (VEV), i.e. ∂ µ Σ i = 0. The equations in (4.7) boil down to extremisation and the Einstein equations reduce to G µν + V 0 g µν = 0. The cosmological constant V 0 is just the scalar potential evaluated at the critical point. The space-time metric then becomes that of Anti-de Sitter (AdS), Minkowski (Mkw) or de Sitter (dS) space for V 0 < 0, V 0 = 0 and V 0 > 0, respectively. In the case of AdS, which is the relevant in this paper, the solution to where L 2 = −3/V 0 is the AdS radius. By applying the radial coordinate redefinition r = e −z/L , the most familiar form of the AdS metric ds 2 = L 2 r 2 (η αβ dx α dx β + dr 2 ) is recast. The AdS boundary (z → ∞) is mapped to r = 0 and the deep interior (z → −∞) to r = ∞.

Glossary of AdS critical points at ω = 0
The structure of SU(3)-invariant critical points of the purely electric SO(8) gauged supergravity at ω = 0 was classified thirty years ago by Warner in ref. [18]. In this case, the theory is known to contain an AdS solution at the origin preserving N = 8 supersymmetry and G 0 = SO(8) residual symmetry as well as other five types of AdS critical points preserving smaller (super)symmetry. The relevant data for these points 10 is summarised in Table 1. 10 An exact form is known for the numbers in Table 1 (see appendix A in ref. [22]). We typed the numerical values in order to compare with other tables in the text for which only numerical values are available.  On the other hand, the issues of perturbative stability and higher-dimensional origin of these critical points have also been thoroughly investigated (see ref. [27] for a list of references). The analysis of the SU(3)-invariant sector at ω = 0 showed that, whenever supersymmetry did not protect solutions to have instabilities, these showed up somewhere in the full N = 8 spectrum. However, counterexamples to this were found soon after by analysing the SO(4)-invariant sector of the theory still with ω = 0 [54,55] as well as within the G 2 -invariant sector with ω = 0 [14]. The scalar mass spectra at these points turned out [14,16] to be independent of ω.  Table 3: The genuine SU(3)-invariant critical points of the SO(8) gauged supergravity at ω = π 8 . These points have no counterpart at ω = 0. For those solutions preserving N = 1, the mark * singles out the superpotential (W 1 vs W1) with respect to which supersymmetry is preserved.

Glossary of critical points at ω = 0
The structure of SU(3)-invariant critical points at ω = 0 was explored in ref. [19] using a superpotential differing from (3.29) by an overall phase, as discussed in section 3.4. However it is clear from (3.26) that the scalar potential is not sensitive to overall phases, so the critical points associated to (3.29) coincide with those found in ref. [19]. Turning on ω was found to modify the location and energy of the critical points existing at ω = 0 as well as to create new ones with no counterpart at ω = 0. As a check of the scalar potential in (3.14), we have exhaustively verified the set of critical points found in ref. [19]. The entire set of AdS solutions can be divided into two categories : i) points which are shifted counterparts of those at ω = 0 : These points have the same normalised mass spectra as their counterparts at ω = 0, hence inheriting their stability properties [13,14,19]. The list of these points at ω = π 8 is shown in Table 2.
ii) points with no counterpart at ω = 0 : These points are genuinely associated to dyonic SO(8) gaugings. There are novel N = 1 AdS solutions with either G 2 or SU(3) residual symmetry as well as non-supersymmetric critical points preserving either G 2 or SU (3) too. The set of these points at ω = π 8 is summarised in Table 3.
Perturbative stability of the non-supersymmetric point preserving G 2 was checked in refs [14,19]. However, the lack of a derivation from scratch of the ω-dependent supergravity quantities, concretely of the fermi mass terms, made an analysis of stability for the novel non-supersymmetric and SU(3)-preserving point impossible. Now we are at the position to perform such an analysis here.

Stability of the new
This AdS solution was shown to have ω-dependent mass spectra in ref. [19]. Furthermore, the scalar masses for the SU(3)-singlets were computed at ω = π 8 and found to satisfy the B.F. bound, but the stability of the full scalar spectrum remained an open question. Plugging the VEVs of the scalars displayed in Table 3  where one identifies the eight massless vectors associated to the SU(3) residual symmetry.
The first two masses correspond to the SU(3)-singlets and reflect the complete breaking of the U(1) × U(1) gauging in the truncated theory.
We want to highlight that (up to our knowledge) this is the first example of a nonsupersymmetric and nevertheless fully stable critical point in new maximal supergravity with a scalar mass spectrum being sensitive to the electromagnetic phase ω. Previous stable cases were insensitive [13,14] and those being sensitive were unstable [15,16].

BPS domain-walls
We now move to study BPS domain-wall configurations where the scalars develop a profile Σ i (z) and the scale factor A(z) is no longer linear in z. By plugging the domain-wall Ansatz (4.3) into the action (4.1) one finds where a is the area transverse to the domain-wall direction. The energy per unit of transverse area is then given by [33]  The fact that V can be obtained from a superpotential as (3.26) allows the energy density (4.13) to be writtenà la Bogomol'nyi (completing squares) by using the relations in (3.27) [20]. Then it is extremised by BPS domain-wall solutions for which gravitational stability is guaranteed [33]. These solutions are found to satisfy the first-order set of equations 11 (4.14) 11 As shown in ref. [21], the gauge choice θ = ψ = 0 holds along the flow such that the kinetic function T (λ , φ, θ, ψ) in (3.13) does not contribute to the energy density. and connect two supersymmetric AdS points at z = ±∞ along a steepest descent path 12 of |W |. At the two end points, one has ∂ z Σ i ∝ ∂ Σ i |W | = 0 and, using the AdS/CFT correspondence, the dual field theory is conjectured to flow from an UV fixed point at the boundary of AdS (z → +∞) to an IR fixed point at the deep interior (z → −∞). When approaching these asymptotic regions, the scale factor behaves as A(z) ∼ L −1 z | z→±∞ with Using the (inverse) metric K ij in (4.2), the flow equations in (4.14) can be written in the more compact form Near the asymptotic regions at z → ±∞, the non-linear flow is well approximated by a linear one satisfying where Σ i 0 denote the VEVs of the scalars at one of the asymptotic AdS points, L 0 is the AdS radius and the matrix ∆ i j is also evaluated at that point. The eigenvalues of ∆ i j encode the masses of the fields and therefore also the conformal dimension of the dual operators.
The aim is to solve the set (4.15) of differential equations numerically using the ωdependent superpotential |W | in section 3.4. From now on, we will take W = W 1 in (3.24) without entailing a loss of generality 13 and investigate two types of BPS domainwalls: i) domain-walls flowing between two supersymmetric points in Table 2, so they can be understood as ω-deformations of others existing at ω = 0 ii) domain-walls which have no counterpart at ω = 0 as they flow towards some of the genuine supersymmetric points in Table 3.  [35][36][37]. In addition, the connection to BLG theory and deformations thereof by adding mass terms was put forward in refs [37,45,46]. Here we will numerically solve the first-order equations in (4.15) to determine how the flows get modified when turning on the electromagnetic phase ω. 12 The actual flow occurs in the opposite direction as V | z→±∞ ∼ −6g 2 |W | 2 and it runs from higher to lower values of the potential. 13 To be consistent with this choice, one has to select the ( * -marked) AdS solutions preserving W 1 and not W1 for those domain-walls flowing towards N = 1 points in Tables 2 and 3. In order to plot the flows of the four-field superpotential W (λ, α, λ , φ), it is necessary to take a two-dimensional slice. We will take with sections α * (λ ) and φ * (λ ) of the form The above choice of slice is then guaranteed to catch pairs of critical points located at Σ (1) and Σ (2) . Let us emphasise that, irrespective of the slicing, we are solving the actual system of first-order equations in (4.15) and not any projected version of it.
Setting ω = 0 , the N = 8 & SO (8) Table 3 also exists. (4.22) 14 In order to avoid confusion between the N = 1 points preserving G 2 in Table 2 and in Table 3, we have attached the labels G 2 and G * 2 respectively.  [56]. Remarkably, the very convenient parameterisation we used to perform the supergravity computations in the previous sections, is also adequate to capture all the types of flows at ω = π 8 .

Domain
15 Upon submission of version 1 of this manuscript, we became aware of the preprint [56] where an exhaustive study of domain-walls and RG flows at ω = 0 has been carried out in terms of the field variables (z, ζ 12 ).
Therein, the sets of flows in Figures 3 and 4

Summary & final remarks
In this paper we have revised the SU(3)-invariant sector of the one-parameter family of SO (8) gauged supergravities discovered in ref. [13]. Using the powerful framework of the embedding tensor, we performed a supergravity derivation of the scalar Lagrangian (section 3.2), the fermion mass terms (section 3.3 + appendix B) and the N = 2 superpotential(s) (section 3.4) as a function of the electromagnetic phase ω and the six real scalars in the theory.
The precise knowledge of the fermi mass terms allowed us to check the stability of a nonsupersymmetric AdS critical point preserving SU(3) symmetry which only exists for ω = 0, hence being genuinely dyonic. We find that this AdS solution is fully stable under scalar fluctuations and has a mass spectrum that is sensitive to the electromagnetic phase. As mentioned in the main text, this is the only example (up to our knowledge) of such a critical point in new maximal supergravity.
In the second part of the paper, we presented some first results on BPS domain-walls for ω = 0. Making use of the ω-dependent superpotential(s) in (3.24), we derived the firstorder flow equations in (4.15) and solved them numerically at ω = π 8 . In this way we 30  Tables 2 and   3 (see Figure 6). Some of them have a purely electric counterpart and behave in a similar way, e.g. steepest descents smoothly lie inside the bounding cone. The others will not have such a smooth behaviour as they flow towards or pass nearby an AdS point which simply does not exist at ω = 0. In these cases, it would be very interesting to explore how the bounding cones blow up when taking the limit ω → 0 in which the N = 1 & G * 2 and N = 1 & SU(3) points run off to infinity in field space [19]. A dedicated study of domain-walls in dyonic gauged supergravities will be presented somewhere else.
We would like to finish by commenting on potential applications of our results and also future directions. The first one concerns the search for a reduction Ansatz of 11d supergravity that could accommodate the electromagnetic phase ω. To this end, if it is at all possible, the knowledge of the T -tensor and the fermion mass terms could play a central role [27]. We have derived these quantities as a function of the phase ω and the scalars in the SU(3)-invariant sector. This sector of the theory already encompasses many of the AdS points for which an 11d lifting could be figured out in the case of an electric SO(8) gauging (ω = 0). For this reason, we believe that the ω-dependent expressions obtained here might help in getting some insights in this direction. A second remark concerns the conjectured three-dimensional RG flows that the BPS domain-walls at ω = π 8 would be dual to. In the case of ω = 0, these were connected to deformations of the BLG theory of M2-branes by a mass term of the form [37,45,46] ∆W BLG = 1 2 m 1 Φ 2 1 + 1 2 m1 Φ 2 1 , (5.1) and the bounding cone for the steepest descents in Figure 2 was related to the (m 1 , m1) mass parameters. Therefore, a possible generalisation to ω = 0 again demands the role of the electromagnetic phase to be better understood in the context of 11d supergravity.
That goes beyond the scope of this work. Here, our aim was to construct flows between supersymmetric AdS points in new maximal supergravity. Nevertheless, the types of flows that we obtained for a dyonic SO(8) gauging could help in this task. We hope to come back to these issues in the future.

A Unitary gauge & E 7(7) /SU(8) parameterisation
The N = 8 supergravity multiplet in four dimensions contains 70 real scalars which parameterise an element V of the coset space E 7 (7) SU (8) . The SU (8) in the denominator represents the maximal compact subgroup, so, in the unitary gauge, the physical scalars are associated to the non-compact generators of E 7 (7) . In order to build the 56 × 56 coset representa- SU (8) with both indices in the SU(8) basis, we will make extensive use of the Γ-matrices of SO(8) ⊂ SU(8) we discuss now.
Majorana SO(8) spinors will be defined with an index down χ µ . For the Γ-matrices and the charge conjugation matrix C we adopt the conventions in ref. [57] [ with definite symmetry properties. The γ a building blocks are the 8 × 8 matrices and are used to split the 70 real scalars in the theory into self-dual (SD) and anti-self-dual where the expression for the tensors [S m ] abcd and [C m ] abcd in terms of (A.6) can again be found in ref. [58]. 16 We use normalised δ αβγδ IJKL Kronecker symbols with weight one such that With all the above ingredients, the prescription to build the mixed coset representative V M N entering the scalar matrix (2.5) for a given G 0 -invariant sector of maximal supergravity is as follows: 1) The precise embedding of G 0 inside the R-symmetry group SU(8) specifies the set of G 0 -invariant four-forms and therefore the set of components in Σ IJKL which are compatible with the residual symmetry.
2) After identifying the G 0 -invariant components inside Σ IJKL , it is immediate to read off which fields ϕ

B Gravitino-dilatino mass terms
In this appendix we present the explicit form of the A I JKL tensor corresponding to the gravitino-dilatino mass terms in (3.22).
• Couplings involving the ψâ µ gravitini The couplings to these gravitini match those already found for their counterparts ψ a µ . They are given by and complete the set of gravitino-dilatino couplings of the SU(3)-truncated theory.