Brane-jet stability of non-supersymmetric AdS vacua

We classify the non-supersymmetric, and perturbatively stable within $D=4$, AdS vacua of maximal $D=4$ supergravity with a dyonic ISO(7) gauging in a large sector of the supergravity. Seven such vacua are established within this sector, all of them giving rise to non-supersymmetric $\textrm{AdS}_{4} \times \textrm{S}^{6}$ type IIA backgrounds with and without non-trivial warpings and with internal fluxes. Then, we analyse the dynamics of various probe D$p$-branes in these backgrounds searching for potential brane-jet instabilities. In all these cases, such instabilities are absent. Finally, an alternative decay channel through tunnelling is investigated, focusing on one of the seven backgrounds. We do not find instabilities either, but the analysis remains inconclusive.

D Geometric structures on S 5 and S 6 34 1 Motivation The issue of the non-perturbative (in)stability of non-supersymmetric vacua in string/Mtheory has gained a renewed interest in light of the weak gravity and swampland conjectures [1,2]. Focusing, for definiteness, in the class of anti-de Sitter (AdS) vacua that uplift from maximal supergravities in lower dimensions, all known non-supersymmetric AdS 5 × S 5 backgrounds of type IIB string theory that uplift [3,4,5] from extrema of D = 5 N = 8 SO(6)-gauged supergravity [6] have instabilites already at the perturbative level. This follows from [7,8], where the extrema of that gauged supergravity were scanned: all the non-supersymmetric extrema have Kaluza-Klein (KK) excitations, contained within the N = 8 supergravity, with mass below the Breitenlohner-Freedman (BF) bound [9]. Similar classification results exist [10] for AdS 4 ×S 7 vacua of M-theory that uplift [11] from extrema of D = 4 N = 8 SO(8)-gauged supergravity [12]. In this case, and in contrast to type IIB, there is one, and only one [10], non-supersymmetric critical point whose KK spectrum does not contain BF-unstable modes, at least within the slice of the KK towers  Table 1: All known supersymmetric, and non-supersymmetric but BF-stable within D = 4, AdS4 × S 6 solutions of massive IIA supergravity that uplift from critical points of D = 4 N = 8 dyonic ISO (7) supergravity. The reference marked with * contains only partial results on the IIA uplift. Non-available data are denoted with ?
type IIA solution, possibly recovering solutions like [31,33] first found by other methods. Perturbative and non-perturbative stability is guaranteed for the supersymmetric solutions but, notwithstanding the arguments of [1,2], stability of the non-supersymmetric solutions should be addressed on a case-by-case basis. Perturbative instabilities should manifest themselves as KK modes about the AdS 4 ×S 6 solutions with mass below the BF bound on AdS 4 . Two sectors of these KK towers are easily accessible: the slice containing the KK modes with all spins s ≤ 2 that are also contained in D = 4 N = 8 ISO(7) supergravity, and the massive s = 2 KK gravitons. The latter have been computed in the cases indicated in the table and, by gauge symmetry, are not expected to induce instabilities. The scalar, s = 0, and vector, s = 1, KK spectra in the D = 4 N = 8 slice have been computed in the references indicated in the table or in this paper if noted as [here], see appendix A. Out of all the D = 4 vacua that we find, only those supersymmetric and non-supersymmetric but free from BF-instabilities in the D = 4 N = 8 slice have been reported in table 1. Obtaining the full KK spectra about these AdS 4 × S 6 solutions and determining whether the non-supersymmetric ones contain BF-unstable modes outside the D = 4 N = 8 slice remains an open problem.
The main objective of the paper is to test the non-supersymmetric vacua summarised in table 1 for non-perturbative BJ instabilities, along the lines of [16,17]. For this purpose, in section 2 we place spacetime-filling D2-brane probes on each of these non-supersymmetric AdS 4 ×S 6 vacua. Remarkably, we find no BJ instabilities. This is perhaps not so surprising for the non-supersymmetric G 2 -invariant AdS critical point [30], as its AdS 4 ×S 6 uplift has a trivial warp factor. However, all the other solutions do involve non-trivial warp factors and yet they are BJ-stable under spacetime-filling D2-brane probes. BJ instabilities might still occur associated to other Dp-branes, with p = 4, 6, 8, wrapped around (contractible) cycles of the internal S 6 . These instabilities could be expected on the grounds that all of these solutions are supported by internal fluxes. In section 3 we address this question for the simplest of these solutions, the one with G 2 invariance. Again, rather surprisingly, we find no BJ instabilities.
Of course, the absence of BJ instabilities does not contradict the statements of [1,2]: these non-supersymmetric solutions might still decay in some other way, for instance tunnelling into a stable vacuum. In the supergravity, this decay would be signalled by the existence of a domain-wall connecting this solution to a different one, possibly supersymmetric. In section 4 we search for this type of domain-walls in the effective D = 4 N = 8 supergravity, focusing again on the simplest solution: the one with G 2 symmetry. We find no conclusive evidence for the existence of such domain-walls, suggesting that the non-supersymmetric G 2 solution is also stable against this decay channel. However, we do not claim comprehensiveness of this analysis, which deserves further future investigation.
For later reference, we conclude by collecting some relevant expressions related to the Dp-brane action. Following the Einstein frame conventions of [35], an extended Dp-brane (p = 0, 2, 4, 6, 8) with tension T p and charge Q p in the presence of non-vanishing Romans massF 0 is described by the action [35,36] (see also [37]) (1.1) The first piece here is the Dirac-Born-Infeld (DBI) term involving the pull-backs of the type II metricĝ ij , dilatonφ, and B-fieldB ij , as well as the field strength F ij of the Born-Infeld field A i defined on the worldvolume. The latter is parameterised by the (p + 1) coordinates ξ i . The second piece is the Wess-Zumino (WZ) term involving the Ramond-Ramond (RR) potentialsĈ n . Note that a non-vanishingB 2 induces a set of (p − 2r) RR charges with r = 1, . . . , p 2 on a Dp-brane via the e −B 2 factor in the WZ term. The last piece is the Chern-Simons form contribution to the WZ term forF 0 [36]. In order for the configuration to be supersymmetric, the tension-charge relation T p = ∓Q p must hold, with the upper sign for the Dp-brane and the lower sign for the anti-Dp-brane in our conventions. We will restrict our study to the case of unmagnetised Dp-branes whereby (1. 2) The first two terms in (1.1) simplify accordingly and the third term vanishes.
2 D2-brane-jet stability 2.1 Spacetime-filling D2-brane probes on AdS 4 × S 6 The non-supersymmetric type IIA AdS 4 × S 6 solutions of interest have been given on a case-by-case basis in the references indicated in table 1. As argued in appendix B, all the non-supersymmetric solutions in the table, as well as the supersymmetric ones with the exception of the two N = 1 U(1) R -invariant solutions, are encompassed by the formalism of [24,34]. Specifically, all the relevant AdS 4 × S 6 solutions can be parameterised in terms of eight real constants ϕ, χ, φ i , b i , i = 1, 2, 3, corresponding to D = 4 supergravity scalars: see appendices A and B for details. The specific values that these constants attain at each of the specific solutions of table 1, namely, the corresponding D = 4 scalar vacuum expectation values (vevs), can be found in table 3 of appendix B. Along with these constants, the solutions also depend on the R 7 coordinates µ I , I = 1, . . . , 7, constrained to lie on the S 6 locus δ IJ µ I µ J = 1 .
(2.1) These backgrounds are created by a stack of D2-branes in the presence of other Dp-branes. The ten-dimensional Einstein frame metric, the dilaton and the RR three-form potential for all these solutions are given by The quantities ∆ 1 , ∆ 2 and C here depend both on the constants ϕ, χ, φ i , b i , and on the S 6 angles µ I . Their explicit expressions [24,34] are reviewed in appendix B. In (2.2), we take ds 2 AdS 4 to be the line element of radius L AdS 4 in the Poincaré patch, so that here and in (2.2), A(r) = r L and x α , α = 0, 1, 2, are the Poincaré coordinates. The AdS 4 radius is specified in terms of the function V (the D = 4 scalar potential) of ϕ, χ, φ i , b i given in (B.8) of appendix B. Finally, g in (2.2) is an additional constant (the D = 4 N = 8 electric coupling) that sets an overall scale, and ds 2 S 6 in (2.2) is a family of metrics on a topological S 6 that depend on the D = 4 scalars ϕ, χ, φ i , b i and on the S 6 coordinates µ I . The explicit expression of this metric can be found in [24], but it will not be needed for the analysis of this section. The legs of the RR three-formĈ 3 along the internal S 6 (denoted with ellipses in (2.2)), the RR one-form and the B-field will not be needed either.
We now move on to place a probe D2-brane on this family of backgrounds. We choose to put the probe parallel to the AdS 4 boundary, i.e. along R 1,2 , so that the worldvolume coordinates are ξ i = x i , i = 0, 1, 2 . In this case, and with the simplifying assumption (1.2), the D2-brane action that follows from (1.1) reads the action (2.4) simplifies into where we have introduced the usual shorthand notation for the metric warp factor in (2.2), The action (2.6) implies an effective radial potential density 1 experienced by the D2-brane probe in the class of backgrounds (2.2). The force exerted by the background on the probe is therefore computed from This force is thus attractive or repulsive depending on the sign of the quantity analogue to that introduced in [16] in an M2-brane context (with the same relative sign once a different sign convention on the Freund-Rubin term is taken into account). For each solution in table 1 with the D = 4 scalar vevs ϕ, χ, φ i , b i fixed as in table 3, Θ is a function of the S 6 angles µ I . If Θ ≥ 0 everywhere on S 6 , the resulting force attracts the probe D2-brane towards the stack of branes located at r → −∞ that creates the background geometry. In this case, the IIA background is stable with respect to this decay channel. On the contrary, if Θ < 0 on certain directions along the S 6 , the resulting force pushes the probe D2-brane towards r → ∞ along those directions, and the massive IIA background suffers from a BJ-instability analogue to [16,19]. Let us now determine the behaviour of (2.10) on a case-by-case basis.

Solutions with G 2 symmetry
As explained in appendix B.2, the coefficient Θ becomes constant (independent of the S 6 angles), for the solutions with G 2 -invariance. Specifically, we obtain positive in both cases (recall that c ≡ m/g > 0). Both G 2 -invariant configurations are thus BJ-stable. This was expected for the supersymmetric solution, but comes a bit as a surprise for the non-supersymmetric one. In retrospect, perhaps the N = 0, G 2 result is not so surprising either, as the warping becomes constant and in the previous examples [16,19] the BJ instabilities tend to come associated with non-trivial warpings (not always, though: the D = 11 SO(7) − solution is unwarped and BJ-stable [16]; however, it is BFunstable). We have also evaluated (2.10) for the BF-unstable SO(7) point [38], which also lies in the sector with at least G 2 -invariance and thus also leads to trivial warping in IIA (see (4.8) of [32]). It turns out to also be BJ-unstable with Θ = 3 −1 · 5 . More importantly, all other non-supersymmetric solutions in table 1 involve non-trivial warpings and, as we will now next see, they are also free from D2-BJ-instabilities.

Solutions with at least SU(3) symmetry
The IIA backgrounds with at least SU(3) symmetry but not G 2 , are cohomogeneity-one, and the coefficient Θ develops a dependence in the S 6 angle α described in appendix B.3. Bringing (B.3)-(B.8) with (B.10), (B.11) to the expression (2.10), and further particularising to the D = 4 scalar vevs contained in table 3, we compute: 10 + 2 cos 2α , (2.14) The last two cases correspond to the relevant solutions in the order they appear in table 1.
The functions (2.13) and (2.14) are non-zero in the entire domain (B.12) of α, leading to stability of these supersymmetric solutions against D2 BJs. This is analogue to the M2 BJ-stability of the supersymmetric solution discussed in [17]. More surprisingly, the functions (2.15) and (2.16) are also non-negative for all α (see figure 1), leading as well to D2-BJ stability of the corresponding warped, non-supersymmetric AdS 4 × S 6 solutions. This is in contrast to the non-supersymmetric warped AdS 4 × S 7 solutions analysed in [16,19], which are BJ unstable w.r.t. the corresponding spacetime-filling probe branes. Equations (2.13)- (2.16) show that all solutions in the SU(3) sector that are BF-stable, are also D2-BJ-stable. For completeness, we have tested the D2-BJ-stability of the SO(6) solution ((4.7) of [32]), which also belongs to the class of solutions with at least SU(3) symmetry but is known to be BF-unstable [38]. This solution is also BJ-unstable, as it has Θ(α) = 2 − 13 12 · 3 −1 c − 1 6 9 cos 2α + 7 , which dips below zero in a subinterval of (B.12).

Solutions with an explicit factor of SO(3) R
Finally, we turn to assess the D2-BJ-stability of the solutions in table 1 whose residual symmetry exhibits an explicit factor of SO(3) R as defined in equation (B.1). These solutions also involve non-trivial warpings and, as reviewed in appendix B.4, they are co-homogeneity one, two or three depending on whether their symmetry is enhanced to include SO ( Both solutions are thus BJ-stable with respect to spacetime-filling probe D2-branes. The N = 3 case is again analogue to the supersymmetric cases considered in [17] and above. The non-supersymmetric case is also BJ-stable, like the cases above and unlike [16,19]. We will omit the explicit form of Θ for the higher cohomogeneity cases, as the resulting expressions are not particularly enlightening, and simply refer to the plots in figure 2. The behaviour of the function Θ(β,θ) is qualitatively very similar for both non-supersymmetric solutions in table 1 with U(1) d × SO(3) R symmetry. In both cases, β = π 2 defines a line of global minima independent ofθ, with Θ( π 2 ,θ) ≈ 0.3162 and Θ( π 2 ,θ) ≈ 0.2965 for the solutions with cosmological constants V = −23.4561 g 2 c − 1 3 and V = −23.4588 g 2 c − 1 3 , respectively. In both cases the function is positive at its global minimum, leading to D2-BJ stability for these solutions. Similarly, for the non-supersymmetric SO(3) R -invariant solution, β = π 2 defines a plane of global minima of Θ(β,θ,φ) independent ofθ andφ. This minimum is positive, Θ( π 2 ,θ,φ) ≈ 0.3195, leading to D2-BJ stability for the SO(3) R solution as well.
We have also tested the BJ-stability of the remaining non-supersymmetric solution of ISO(7) supergravity with residual U(1) d × SO(3) R symmetry: the solution labelled ii) in [34]. This solution is BF-unstable (see appendix A below) yet, curiously, it is D2-BJinstability-free. Finally, note that the two N = 1 solutions in table 1 with symmetry U(1) R are excluded from our analysis. Being supersymmetric, they are also expected to be BJ-stable.

Dp-brane-jet stability of the G 2 -invariant vacua
The presence of internal fluxes in the non-supersymmetric solutions listed in table 1 might lead to BJ instabilities associated to Dp-brane probes with p > 0 wrapped on (contractible) cycles of the internal S 6 . In this section, we systematically compute the effective potential density V (r) for such D4, D6 and D8 brane probes, focusing in the simplest backgrounds: the two of them with G 2 symmetry. We find that the N = 1 vacuum is stable under these class of perturbations, as expected. More interestingly, the N = 0 solutions is also stable.
Our starting point is the class of G 2 -invariant backgrounds of massive IIA supergravity presented in [22], parameterised by the two scalars (ϕ, χ) in (B.9), and obtained upon uplift of the G 2 -invariant sector of the ISO(7) maximal supergravity [23]. These backgrounds are given by in terms of the SU(3)-invariant two-form J and three-form Ω specifying the homogeneous nearly-Kähler structure on S 6 (see appendix D). The scalar-dependent quantities ∆ , B and C take the expressions ∆ = e − 3 4 ϕ 1 + e 2ϕ χ 2 − 3 4 , B = g −2 e 2ϕ χ 1 + e 2ϕ χ 2 −1 , C = − L 3 g e ϕ 1 + e 2ϕ χ 2 2 5 − 7e 2ϕ χ 2 + m e 7ϕ χ 3 , before being evaluated at the scalar vevs for the N = 1 and N = 0 G 2 -invariant vacua of table 3. Once such vevs are inserted into (3.1)-(3.2), the ten-dimensional G 2 -invariant backgrounds, as expressed in [32], are obtained. We will now consider higher-dimensional Dp-branes (p > 2) wrapping specific (contractible) cycles within the S 6 . To this end, it is convenient to describe the round S 6 as the sine-cone over S 5 , and refer its homogeneous nearly-Kähler structure to the Sasaki-Einstein structure on S 5 , see appendix D. In (3.3), α is the S 6 angle with range (B.12) introduced in (B.11) that also appears in section 2.3 above. The volume form associated to (3.3) is Here and in (3.3), η ≡ dψ + A 1 is a one-form along the S 5 Hopf fibre, with ψ an angle ranging as ψ ∈ [0, 2π] and A 1 a local one-form potential for the Kähler form J on CP 2 , normalised as dA 1 = 2 J . See appendix D for further details.

D4-brane wrapping internal two-cycles
We start by considering a probe D4-brane parallel to the AdS 4 boundary, i.e. along R 1,2 , and wrapping S 2 ⊂ CP 2 ⊂ S 6 . 2 From (3.1) and (3.3), combined with (D.5) and (D.9), the worldvolume of the D4-brane readŝ One therefore encounters non-trivial WZ terms of the formĈ 3 ∧B 2 andĈ 5 (see (C.7) and (C.8)) when pulling back the background (3.1) into the D4-brane worldvolume (3.5). Inserting (C.7) and (C.8) into (3.6), and computing the DBI term using (3.1), the effective action for the probe D4-brane takes the form with eφ and the quantities ∆ and B given in (3.1) and (3.2). The D4-brane wrapped over S 2 ⊂ CP 2 then then experiences an effective potential density (3.9) This function depends on the coordinate α , and therefore BJ instabilities could occur at certain values of this angle. However, an explicit evaluation of (3.9) shows that namely, the net force is always attractive towards r = 0 . As a result, there is no BJ instability in this probe D4-brane channel. This is a remarkable fact as theB 2 potential in (3.1) has an internal component on S 2 , as can be seen from (D.11) and (D.10), that could have triggered such an instability.
Together with the probe D4-brane wrapping S 2 ⊂ CP 2 ⊂ S 6 above, there are other D4-branes that also require an analysis of BJ instabilities. The reason is that the WZ term in (3.6) has non-vanishing components along J as required by the G 2 invariance of the backgrounds (3.1). Then, by virtue of (D.11) and (D.8), there are seven independent such WZ components which stand a chance of inducing BJ instabilities on the probe D4-branes wrapping the respective two-cycles. The computation of the effective potential density for the six new D4-branes parallels the one performed above so we omit all the details here. The final outcome is summarised as follows: As a result, we find no BJ instabilities for any of the seven D4-branes when placed in the class of G 2 -invariant backgrounds in (3.1).

D6-brane wrapping internal four-cycles
We consider now a probe D6-brane parallel to the AdS 4 boundary, i.e. along R 1,2 , and wrapping CP 2 ⊂ S 6 . The D6-brane coordinates are identified with ξ i = x i ( i = 0, 1, 2 ) together with those on CP 2 ⊂ S 6 . 3 From (3.1) and (3.3), the D6-brane worldvolume is given byv 12) and the action (1.1)-(1.2) reduces to Three non-trivial WZ terms (see (C.9) and (C.10)) contribute now to the D6-brane action (3.13) when pulling back the background (3.1) into the D6-brane worldvolume (3.12). Using (3.1), a straightforward computation of the DBI and WZ terms yields an action for a probe D6-brane of the form with eφ , and ∆ and B , given in (3.1) and (3.2). As a result, the effective potential density for the probe D6-brane simplifies into Note that Θ in (3.16) does not depend on the S 6 angles. A direct evaluation of this coefficient at the two G 2 -invariant vacua gives Therefore, a probe D6-brane wrapping CP 2 ⊂ S 6 does not suffer from a BJ instability despite the fact that theF 4 flux in (C.4) has an internal component on CP 2 . Finally, there are other D6-branes that require a careful analysis of BJ instabilities. Again, the reason is that the WZ term in (3.13) has non-vanishing components along J ∧J due to the G 2 invariance of the backgrounds (3.1). Combining (D.11) and (D.8), one finds seven independent such WZ components which can potentially induce BJ instabilities on the probe D6-branes wrapping the respective four-cycles. Omitting the details on the computation of the effective potential densities, the final outcome is summarised as follows: • D6-branes on four-cycles • D6-brane on the four-cycle CP 2 ⇒ Θ = cst > 0 (see (3.16)) . Therefore, we do not observe BJ instabilities for any of the seven D6-branes when placed in the class of G 2 -invariant backgrounds in (3.1).
The computation of the WZ terms, together with the gravitational DBI term using (3.1), proceed uneventfully and yields a probe D8-brane action with eφ and the quantities ∆ and B given in (3.1) and (3.2). The effective potential density for the probe D8-brane then takes the form Once again Θ turns out to be independent of the S 6 angles. Finally, a direct evaluation of the coefficient (3.23) shows that Therefore, as in all the previous cases, a probe D8-brane wrapping the internal S 6 is also free from a BJ instability.

(Meta)stability and decay through domain-walls
In previous sections we have established the absence of BJ instabilities for spacetime-filling D2-branes in all non-supersymmetric, yet perturbatively BF stable, vacua listed in table 1, and of more general BJ instabilities for the vacuum with G 2 symmetry. However, generic swampland arguments [1,2] suggest that non-supersymmetric AdS 4 vacua should present some type of instability. Thus, we dedicate this section to the exploration of an alternative decay channel for the non-supersymmetric vacuum with G 2 symmetry in table 1. More concretely, we look at its potential quantum tunnelling into a different vacuum with strictly lower potential energy. An unstable vacuum in a bulk theory of gravity can decay to a true, stable vacuum by bubble nucleation under certain conditions [39]. The nucleation is quantum in nature and has a decay rate (per unit volume per unit time) given in the semi-classical approximation by the expression where S false is the Euclidean on-shell action evaluated at the non-supersymmetric AdS 4 vacuum, and S DW refers to the same quantity evaluated at the domain-wall (DW) solution that interpolates between the non-supersymmetric and supersymmetric AdS 4 vacua. The computation of the coefficient A in (4.1) involves the evaluation of a functional determinant that depends on the details of the model [40]. The expression (4.1) for the nucleation decay rate assumes the existence of a DW solution, with a (in this case Lorentzian) 4D metric is the radius of the unstable (stable) AdS 4 spacetime in the Poincaré patch, as in (2.3). We focus on DWs that asymptote to the N = 0 , G 2 vacuum at r → +∞ (ultraviolet, UV). Then, due to the condition V − < V + (with V ± being the gravitational potential evaluated at the scalars vevs that determine the AdS 4 vacua with radius L ± ), only a limited number of AdS 4 vacua can be asymptotically approached when r → −∞ (infrared, IR), in agreement with the holographic c-theorem of [41] and the F-theorem of [42]. Note that the N = 3 vacuum of table 1 is degenerate in energy with the N = 0 , G 2 vacuum, and it is thus excluded from our analysis. In this section we investigate DW solutions that asymptote to (some of) the AdS 4 vacua of table 1, in the IR, with a value of the potential V lower than the corresponding value at the N = 0 , G 2 vacuum. To keep the discussion simple, we will focus on the SU(3) invariant sector presented in [23], whose action we bring to (4.3) below. Our aim here is to qualitatively describe the procedure without entering technical details, since these have been extensively explained elsewhere. The interested reader can find detailed accounts in the early work [41], in [43] for numerical integrations in the SU(3) invariant model (4.3), or in [24] for calculations in other sectors of the ISO(7) maximal supergravity. We have not found a DW configuration connecting the N = 0 , G 2 vacuum in the ultraviolet (UV) to any of the SU(3) symmetric vacua of table 1, either with N = 0 or N = 1 supersymmetry, in the IR. However, it is worth emphasising that we have restricted our search to the SU(3) invariant sector of the theory.

SU(3) invariant sector: bosonic action
The bosonic sector of the SU(3) invariant model in [23] consists of four real scalars and the metric field. The dynamics is dictated by the Einstein-scalar action which includes a scalar potential (4.4) Here g and m are the electric and magnetic couplings of ISO(7) maximal supergravity, being related to the inverse S 6 radius and the Romans mass parameter, respectively. We will set g = m = 1 in this section without loss of generality. To compare with the parameterisation in table 3 of appendix B, one must identify ρ = 2 ζ in (B.10).
Contrary to the case in [23], where supersymmetric DW solutions of this model were investigated, we are interested here in solutions involving the N = 0 , G 2 vacuum. This requires that we consider the second order Euler-Lagrange equations of motion for the scalars and the metric. This gives rise to a coupled set of five second order differential equations: one for each scalar field and one for the DW function A(r) in the ansatz (4.2). This system is supplemented with a first order differential equation, given by the Hamiltonian constraint of the gravitational system. This implies that deviations around any AdS 4 vacuum will be parameterised by nine constants of integration, although one of them can always be reabsorbed in a shift of the radial coordinate r .

Numerical analysis
DW solutions that asymptote to the N = 0 , G 2 vacuum in the UV ( r → +∞ ) are described by deviations around such a solution (see table 3) that involve four different powers of the radius: each one being parameterised by two constants of integration, and with L = 3 3/4 2 −13/6 (recall that we are setting g = m = 1 ). Regularity in the UV immediately forbids the e − 3− √ 33 2 r L radial dependence, leading to a problem with six parameters. In a way, this parameter space can be thought of as the phase space that will give rise to the coefficient A in (4.1) upon evaluation of a functional determinant [40]. In order to find DW solutions in this parameter space one has to resort to numerical methods, which turn out to be plagued with technical difficulties, rendering this a daunting task. However, it is common lore that the numeric integration of the equations of motion is simpler if one constructs the DW starting from the IR end-point instead, which we will do in the following.

Flowing to the N = 0 , SU(3) vacua in the IR
To present the logic in a simpler manner, we will focus momentarily on one specific candidate for the IR asymptotics of the DW: the N = 0 , SU (3) vacuum of table 1  (4.7) An exploratory analysis of solutions shows that the constant of integration α 1 must always be non-zero for non-trivial solutions to exist. Then, by virtue of a shift of the radial coordinate, it can be set to any convenient value, such that its magnitude does not carry physical significance. This allows us to set α 1 = 1 and leaves us with a twodimensional parameter space spanned by (α 2 , α 3 ) . At the origin of this two-dimensional parameter space, namely (α 2 , α 3 ) = (0, 0) , the solution at the UV does not approach an AdS 4 vacuum. Instead, the scalars and DW function A(r) approach asymptotically a scaling behavior dictated by the presence of D2-branes in the setup (see e.g. [43]), which are the objects that dominate generically the stress-energy tensor near the boundary. The presence of the Romans mass and the condition that the DW ends on an AdS 4 geometry in the IR imply that this scaling behavior is never a full solution to the equations of motion, but just an asymptotic solution. By exploring the space of parameters (α 2 , α 3 ) we find that there is a closed region where the UV asymptotics of the solutions are governed by the presence of D2-branes, denoted by the shaded area in figure 3.
For values of the parameters (α 2 , α 3 ) outside the shaded area in figure 3 we find that at least one scalar diverges at a finite value of the radius, rendering these integrations unphysical. Most importantly, in the highlighted parts of the border between DWs that approach the D2-brane behaviour in the UV and unphysical ones (the thick green line at the bottom of figure 3), new phenomenology appears. For this uni-parametric family of solutions the D2-branes are not the only dominant sources in the stress-energy tensor near the UV. Instead, all the branes present in the setup contribute and change the dynamics accordingly. Geometrically, the metric approaches an AdS 4 solution in the UV, with all scalars approaching constant values. By examining the radius of the geometry, and the magnitude of the scalar vevs, we conclude that the asymptotic UV solution corresponds to the N = 0 , SO(7) vacuum with V = −19.614907 of [38]. For values of the parameters close to, but not exactly, the ones in the one-parameter family of solutions that asymptote to an AdS 4 metric, the DWs asymptote to the D2-branes geometry, as generically in the shaded area of the figure. However there is a range of the radial coordinate r where the scalar fields and DW function A(r) are very well approximated by their values at the N = 0 , SO(7) vacuum, with very small gradients. The closer one gets to the critical values in parameter space, the longer (in radial coordinate) the N = 0 , SO(7) vacuum is realised before the DW continues towards the D2-brane behaviour.
Importantly, within the precision of our numerics, we have not found DW solutions other than the ones exposed above and summarised in figure 3. This suggests that, within the SU(3) invariant sector of the ISO (7)   four parameters are needed to describe the r → −∞ behaviour (4.9) An exploratory analysis shows this time that the parameter β 3 must be turned on for a regular DW to exist in the UV, and we can set β 3 = 1 without loss of generality. This leaves us with a large three-dimensional parameter space spanned by (β 1 , β 2 , β 4 ) to scan, and we have done it by repeating the procedure explained above for several slices along one of the parameters. We present three such slices in figure 4.
We observe a connected region in the parameter space where the DW generically approaches the behaviour dictated by dominating D2-branes in the UV. Outside this region there are diverging scalars that render the solutions unphysical. At the border between these two regions there exist lower-dimensional families of DWs that approach an AdS 4 vacuum in the UV, but none of them correspond with the N = 0 , G 2 vacuum we are after. Instead, we only find DWs that asymptote to the N = 1 , G 2 vacuum in the UV of the type discussed in [43]. The way these solutions are approached falls into two categories, denoted by two different highlighted areas in figure 4.
• Close to the critical region with AdS 4 UV asymptotics denoted by the green line in figure 4, the DW solutions approach the D2-brane behaviour by first spending an arbitrarily large (the closer one gets to the critical values) range of radial coordinate near the N = 1 , G 2 vacuum.
• For DW solutions close to the critical region with AdS 4 UV asymptotics denoted by the red line in figure 4, the solutions first spend an arbitrary large range of radial coordinate near the N = 1 , G 2 vacuum and then, closer to the UV, the radial profile spends an arbitrary large range of radial coordinate near the N = 0 , SO (7) vacuum. After that, the DWs approach the D2-brane behaviour in the far UV.
Therefore, due to the collected evidence against the existence of DWs connecting the N = 0 , G 2 vacuum in the UV with other AdS 4 vacua in the IR, we conclude that the probability of decaying by quantum tunnelling for the N = 0 , G 2 vacuum must be highly suppressed, at least withing the SU(3) invariant sector of the ISO(7) maximal supergravity.

Discussion
We have determined the non-supersymmetric, but BF-stable within D = 4, AdS vacua of maximal supergravity in four dimensions with a dyonic ISO(7) gauging [23] that lie within the seven N = 1 chiral subsector [24] of the supergravity. There are seven such solutions, summarised in table 1 of the introduction along the six known supersymmetric ones. By the results of [21,22], all these vacua give rise to AdS 4 × S 6 backgrounds of massive type IIA string theory. Then, we have tested these solutions for BJ-instabilities against spacetime-filling D2-branes, following the analogue spacetime-filling M2-brane analysis of [16,17]. All the type IIA solutions test negative for this type of BJ-instabilities. This is to be contrasted to the AdS 4 × S 7 M-theory case, where the only non-supersymmetric, but BF-stable within the D = 4 supergravity, solution [13,14] is BJ-unstable [16]. We have also performed a thorough stability test of the simplest of these non-supersymmetric vacua, the one with G 2 symmetry, against BJs of Dp-branes, p > 2, wrapped around different contractible cycles of the internal S 6 . We find no BJ instabilities either. It would be interesting to extend the wrapped-Dp-BJ-stability test to the other non-supersymmetric solutions in table 1.
Of course, these non-supersymmetric AdS 4 × S 6 solutions could still decay through other channels. In fact, one would expect that they did, based on the general arguments of [1,2]. One possibility is that their full KK spectrum does contain BF-unstable modes. After all, the D = 4 N = 8 supergravity captures only a small slice of the full KK towers, and these might still contain modes below the BF bound outside that slice. It would be possible to address this question with the technology put forth in [44]. Another possibility is that these vacua tunnelled into other ones, becoming eventually stable at the end of a decay chain. Such decay would be signalled by the existence of a DW solution connecting the non-supersymmetric AdS solution to a different one -an iterative process that would stop when a stable, supersymmetric vacuum be reached. In this paper we have partially assessed this possibility for the non-supersymmetric G 2 -invariant vacuum.
We have not found a DW connecting the N = 0 G 2 solution in the UV to any SU(3)symmetric one in the IR, within the SU(3) invariant sector of ISO(7) supergravity, and irrespectively of the amount of supersymmetry present in the latter. Our analysis is numeric in nature and is therefore subject to an inherent degree of uncertainty, but the evidences suggest that there is no such DW. Therefore no quantum tunnelling process seems possible for this solution. Some caveats of this interpretation are the following: 1. It is possible that the DW does after all exist in the SU(3) invariant sector (4.3) but our procedure was not able to find it. In this case, given the extreme amount of fine-tuning of parameters that such solution may require, we consider that the decay rate factor A in (4.1) would be highly suppressed in this channel, rendering the N = 0 G 2 solution metastable.

2.
A second possibility is that the decay is protected in the presence of SU(3) invariance, such that a DW solution exists with lower, or none, symmetry. Considering for concreteness the model with seven-chiral fields introduced in [24], the three supersymmetric candidates present there would have 4-, 6-and 9-dimensional parameter spaces 5 to be explored, with each dimension corresponding respectively to the last three supersymmetric points of table 1. In the full N = 8 supergravity, the dimensions of the parameter spaces would be 35, 40 and 55 (see appendix A for the corresponding spectra). The search of this type of DW solutions in such high-dimensional parameter spaces is certainly beyond the purposes of this work.

3.
A third possibility is that there is indeed no DW connecting the N = 0 G 2 AdS 4 solution to any other vacuum of D = 4 N = 8 ISO(7) supergravity in the IR.
In any case, the swampland logic [1,2] would still require a decay channel to exist for the non-supersymmetric AdS 4 ×S 6 vacua of type IIA string theory that we have considered in this work. Such an explicit decay channel for these solutions remains to be found. A All known vacua of D = 4 N = 8 ISO (7) supergravity In appendix A of [24] we constructed an N = 1 discrete, (Z 2 ) 3 -invariant truncation of D = 4 N = 8 dyonic ISO (7) supergravity [23] that retains seven real scalars, seven pseudoscalars, and no vector fields. In N = 1 language, this truncation retains the largest number, seven, of N = 1 chiral fields contained in the N = 8 scalar manifold E 7(7) /SU (8), and is analogue to the (Z 2 ) 3 truncation constructed for other D = 4 N = 8 gaugings in [45,46,47,48]. The real Kähler potential K and the holomorphic superpotential W for this model can be found in equations (A.8) and (A.9) of [24]. The scalar potential V follows from K and W through the standard N = 1 formulae (2.7), (2.8) of [24].
In [24], we scanned the superpotential W of this model and found two previously unknown N = 1 points. For the present project, we have scanned the potential V in this sector for vacua, and we 1. recover all known supersymmetric points summarised in table 1 of the text; 2. recover the non-supersymmetric points previously known to be BF-stable within the full N = 8 ISO(7) theory: one point with G 2 symmetry [30], two points with SU(3) symmetry [23], and one point with SO(3) d × SO(3) R symmetry [23]; 3. recover the non-supersymmetric points labelled as ii)-v) in [34], which were found in that reference to be free of BF instabilities within the subsector considered therein; 4. recover the non-supersymmetric points previously known to have BF-instabilities: the SO (7) and SO(6) points of [38], and the U(1) × SO(3) R -invariant point labelled as i) in [34]; and 5. find 43 new critical points, all of them non-supersymmetric.
Next, we scan for BF instabilities for the critical points in items 3 and 5 above. In order to do this, we first compute the scalar masses around these points within the seven-chiral sector. This analysis is enough to find BF instabilities in point ii) of [34] and in all 43 new critical points (see table 2). Finally, we scan for BF instabilities for the remaining points in item 3 by computing their spectra within the full N = 8 theory using the mass formulae of [49]. The points iii)-v) in [34] are found to be BF-stable within the full N = 8 theory. Together with those in item 2, these are the critical points that have been brought to table 1 of the introduction: the points with symmetry U(1) d × SO(3) R are iii) and v) of [34],  Table 2: Scalar masses within the seven-chiral model for the 43 new non-supersymmetric AdS4 vacua found in this paper, labelled by the value of the potential. All of them present BF instabilities. and the point with symmetry SO(3) R is iv) of [34]. This analysis brings the knowledge of vacua of D = 4 N = 8 dyonic ISO(7) supergravity to the following state-of-the-art: there exist, at least, • 6 supersymmetric points, recorded in table 1, with bosonic spectra within D = 4 N = 8 ISO(7) supergravity reviewed below; • 7 non-supersymmetric points, but BF-stable within the D = 4 N = 8 supergravity, recorded in table 1; their bosonic spectra within the D = 4 supergravity is given below; and • 47 non-supersymmetric points, which are BF-unstable already within the D = 4 N = 8 supergravity; the spectra of the previously known ones can be found in the references above, while for the new ones the spectra within the seven-chiral model of [24] is given in table 2 above.
Further comments on the symmetries of the critical points in table 1 will be made in appendix B. We emphasise that the present classification is not exhaustive within the full N = 8 theory: a complete classification along the lines of [10,7,8] should still be made. We also note that the present perturbative analysis only sees a small cross section of the full KK towers about the corresponding type IIA AdS 4 × S 6 solutions: nothing excludes BF instabilites in KK modes not contained within D = 4 N = 8 supergravity.
For convenience, we now summarise the bosonic spectra within N = 8 supergravity of all the solutions of table 1. Please refer to the table for the original references. Together with a massless graviton, the vector and scalar mass spectra are given subsequently.
The set of normalised vector masses is given by whereas the normalised scalar masses read The set of normalised vector masses is given by whereas the normalised scalar masses read The set of normalised vector masses is given by whereas the normalised scalar masses read The set of normalised vector masses is given by M 2 L 2 = 6 (×1) , 28 9 (×6) , 25 9 (×6) , 2 (×1) , The set of normalised vector masses is given by The set of normalised vector masses is given by The set of normalised vector masses is given by The set of normalised vector masses is given by The set of normalised vector masses is given by  The sector of D = 4 N = 8 ISO (7) supergravity invariant under SO(3) R was constructed in [34]. As per (B.1), all the non-supersymmetric points of table 1 (as well as the supersymmetric ones with the exception of the N = 1 U(1) R points) are contained within the SO(3) R -invariant sector. In addition, as explained in appendix A, all these critical points enjoy the additional discrete symmetry (Z 2 ) 3 that truncates the N = 8 supergravity to its seven-chiral subsector. In conclusion, all relevant points in table 1 belong to the intersection of the SO(3) R and (Z 2 ) 3 invariant subsectors. This is the model described in section 2.1 of [24] containing four N = 1 chiral fields. These are respectively composed of four real scalars ϕ, φ i , i = 1, 2, 3, and four real pseudoscalars χ, b i , i = 1, 2, 3, parameterising the scalar manifold (SL(2)/SO(2)) 4 . These are the eight real scalar fields used in section 2 of the main text. In this language, the relevant critical points of table 1 are attained at the D = 4 scalar vevs recorded in table 3 of this appendix.
The uplift of the SO(3) R sector of the D = 4 supergravity into the type IIA metric and dilaton was presented in appendix D.1 of [24]. The external components of the RR three-form potential can also be easily reconstructed with the tensor hierarchy expressions of appendix B of [34] by employing (3.43) of [22]. These are all the ingredients needed for the D2 BJ-stability analysis of section 2. For convenience, we record here the relevant uplifting formulae following [24,34]. For all the solutions in table 1 except the N = 1 U(1) R -invariant ones, the relevant type IIA fields are given in equation (2.2) of the main text with the quantities ∆ 1 , ∆ 2 and C given in terms of the R 7 coordinates µ I constrained to the S 6 locus (2.1) and the D = 4 scalars fixed at the values recorded in table 3. It is SUSY bos. sym.  convenient to pack φ i and b i , i = 1, 2, 3, into the 3 × 3 matrices and to split µ I = (µ i , ν a ), i = 1, 2, 3, a = 4, 5, 6, 7, while also introducing the vector notation µ and ν for the components µ i and ν a . With these definitions, we have [24] and [24] where [34] and 6 [34] H 0 (4) = g 2 e ϕ + e and L = −6/V with [34] The lengthy expression for the internal metric ds 2 S 6 in (2.2) can be found in [24]. It is not needed for the analysis of section 2. While the Freund-Rubin term (B.5) generically depends on the S 6 embedding coordinates µ, ν, at an AdS 4 critical point it becomes independent of them and proportional to the scalar potential B.8: see (3.37) of [22].

B.2 Solutions with G 2 symmetry
The simplest AdS 4 ×S 6 solutions to analyse are those whose internal symmetry is enlarged to G 2 , as the resulting solutions become homogeneous. These solutions have the D = 4 scalars restricted as On (B.9), the dependence on the internal S 6 coordinates µ I = (µ i , ν a ) drop out from (B.3)-(B.5) by virtue of (2.1). The warp factor and dilaton in (2.2) become constant (i.e. independent of the S 6 coordinates), and (2.2) reduce to the expressions given in (4.3) of [22]. The individual AdS 4 × S 6 solutions are obtained by further fixing (B.9) to the corresponding vevs in table 3. In our conventions, the complete type IIA N = 1 G 2 solution can be found in (4.6) of [32], and the N = 0 G 2 solution, in (4.9) of that reference. For these solutions, the coefficient Θ in (2.10) for the force experience by spacetime-filling probe D2-branes becomes constant (i.e., independent of the µ I angles), see (2.11), (2.12).

C G 2 -invariant backgrounds: internal fluxes and WZ terms
The goal of this appendix is two-fold: on the one hand, we establish our conventions regarding Romans' massive IIA supergravity [20] and, on the other hand, we present the various WZ terms entering the effective action of the probe Dp-branes placed in the class of G 2 -invariant backgrounds of section 3.
Following the conventions of [22,50], the bosonic action for the massive IIA closed string sector reads where the ten-dimensional gravitational coupling constant is expressed in terms of the string length s = √ α as 2κ 2 = (2π) 7 8 s . All the AdS 4 × S 6 backgrounds of massive IIA supergravity discussed in this work solve the equations of motion that follow from the action (C.1). However, in order to evaluate the probe Dp-brane action (1.1), with p > 2 , it is convenient to use the democratic formulation of the RR sector of the theory [51].
Together with the field strengths entering the action (C.1), namely, where m is the Romans mass parameter [20], this (re)formulation introduces dual RR potentialsĈ 5 ,Ĉ 7 andĈ 9 , which do not carry an independent dynamics, and have associated field strengthŝ 3) Therefore, in order to obtain the dual potentialsĈ 5 ,Ĉ 7 andĈ 9 that enter the probe Dp-brane action (1.1), one must integrate the dual field strengthsF 6 ,F 8 andF 10 in (C.3) recursively.
For the class of G 2 -invariant AdS 4 × S 6 backgrounds in (3.1), the associated field strengths in (C.2) take the form [22]  with L being the radius of AdS 4 . The quantities B and C depend on the constants (ϕ, χ) in (B.9) and are explicitly given in (3.2). Finally, the two-form J and three-form Ω are SU(3)-invariant forms specifying the nearly-Kähler structure on S 6 (see appendix D). It is then a straightforward (but tedious) task to obtain the dual potentialsĈ 5 ,Ĉ 7 andĈ 9 from the duality relations (C.3) upon particularisation to the background fluxes in (C.4) and using (3.1). In terms of the various gauge potentials, the relevant WZ terms entering the probe Dp-brane effective actions in the class of G 2 -invariant backgrounds (3.1)-(3.2) analysed in section 3 are given subsequently.

WZ terms for the D0-brane
The class of G 2 -invariant backgrounds in (3.1) haŝ C 1 = 0 , (C. 5) so no WZ term is induced on the worldvolume of a probe D0-brane. The effective potential density that derives from the action (1.1)-(1.2) is purely gravitational and thus of attractive nature.