A swamp of non-SUSY vacua

We consider known examples of non-supersymmetric AdS$_7$ and AdS$_4$ solutions arising from compactifications of massive type IIA supergravity and study their stability, taking into account the coupling between closed- and open-string sector excitations. Generically, open strings are found to develop modes with masses below the Breitenlohner-Freedman (BF) bound. We comment on the relation with the Weak Gravity Conjecture, and how this analysis may play an important role in examining the validity of non-supersymmetric constructions in string theory.


Introduction
String theory as a candidate theory of quantum gravity requires UV supersymmetry to be realized for its own consistency. Therefore, the most important challenge that arises in this context is that of understanding viable dynamical processes that cause supersymmetry breaking at lower energies. This would allow us to incorporate effective descriptions such as cosmic acceleration as well as the standard model physics at the electroweak symmetry breaking scale, within a UV-complete theory.
While it was originally argued in [1] that non-supersymmetric vacua are statistically abundant within the so-called "string landscape", the seminal work of [2] has caused a major paradigm shift in our way of analyzing the intimate relation between UV and IR physics. In particular, the weak gravity conjecture (WGC) proposed by the authors of [2] may be viewed as a bottom-up criterion that singles out those effective low energy models that can be successfully UV-completed in string theory thanks to their consistent spectrum of excitations. Only such models should be referred to as parts of the landscape.
Concretely, the WGC invokes the existence of microscopic states in the quantum spectrum of a system, whose mass is smaller than, or equal to, the charge. Recently in [3], a stronger version of the WGC was proposed according to which the mass/charge inequality may only be saturated by BPS objects in a supersymmetric vacuum. The main consequence thereof is that each and every non-supersymmetric AdS vacuum must be non-perturbatively unstable w.r.t. the emission of microscopic charged objects (see also [4] for a similar analysis). Further checks and explict realizations/counterexamples of the above idea in a stringy setup may be found e.g in [5][6][7].
Still inspired by the idea of studying universal instabilities of non-supersymmetric AdS vacua in string theory, but from a somewhat complementary viewpoint, a mechanism was proposed in [8] that generically predicts tachyons for all non-supersymmetric vacua admitting a so-called bran e picture [9]. These tachyons are associated with the extra matter fields living on the spacetime-filling branes participating in the above brane picture. These instabilities crucially rely on the coupling between closed-and open-string modes. Therefore, the above claim should not be viewed to be in conflict with the recent analysis in [10] (and in particular used, e.g. in [11], for holographic purposes), where some classes of non-supersymmetric AdS extrema in massive type IIA compactifications are found to be even non-perturbatively stable. This analysis is restricted to the closed-string sector, whereas the main take-home message in [8] is that one should worry more about open-string modes when it comes to stability of non-supersymmetric vacua. The aim of this paper is precisely that of further elaborating this proposal by giving concrete examples where the presence of such tachyons can be explicitly shown. The models chosen here are exactly those of [10], which were argued to be fully stable within the closed-string sector.
The paper is organized as follows. In section 2, we review the N = 1, D = 7 gauged supergravity description of the (non-)supersymmetric AdS 7 solutions of [12][13][14] found in the context of massive type IIA supergravity. We will start by studying the theory that accounts for all closed-string zero modes and subsequently move to deriving the D6-probe effective potential directly from the 10D perspective. Finally, we will consider the 7D coupling with extra vector multiplets describing the position moduli of spacetime-filling D6-branes and match the resulting mass spectra. In section 3, we start by reviewing the N = 4, D = 4 gauged supergravity description of the (non-)supersymmetric AdS 4 solutions of [15] obtained from massive IIA reductions on twisted tori with fluxes. The above supergravity model accounts for all closed-string zero mode excitations. Subsequently, we will propose an extended 4D theory obtained by the coupling with extra vector multiplets in order to describe position and YM moduli for the spacetime-filling D6-branes. In both the 7D and the 4D case, we will encounter tachyonic modes within the open-string sector. Possible end-points of these instabilities, and their relevance for a consistent theory of quantum gravity, will be discussed in section 4.
2 Massive type IIA on AdS 7 × S 3 The goal of this section is to analyze the different behavior of the open-string sector in the supersymmetric and the non-supersymmetric AdS 7 solutions found in the context of massive type IIA supergravity on S 3 . To this end, we will first review the form of the 10D solutions and subsequently we will move to their gauged 7D supergravity description. The closedstring excitations will be accounted for by considering the gravity multiplet coupled to three vector multiplets. Then we will move to the study of open-string excitations by calculating the effective potential of spacetime-filling D6 probes. In the non-supersymmetric extremum the position of the D6-branes will turn out to be tachyonic, while it will be above the BF bound [16] in the supersymmetric solution. Finally, we will be able to recover exactly the same result by computing the mass matrix of the corresponding 7D gauged supergravity theory when coupled to N extra vector multiplets.

Massive type IIA solutions in AdS
The AdS 7 × S 3 solutions of interest can be described in 10D massive IIA supergravity with a background including the RR 1-form C (1) , the Romans' mass F (0) , the NSNS B (2) field, the dilaton Φ and the metric g. We write these as [14] ds 2 3) It is useful to change from the coordinate r to a coordinate y via dr = 9 16 which allows one to analytically describe a family of solutions in terms of a function β = β(y). In terms of the y coordinate, the north pole of S 3 is located at y = −2. The case of our interest is the AdS vacuum supported by a stack of D6-branes located at y = −2, which corresponds to the following choice Note that the above background is a complete solution to the set of 10D field equations in appendix A, provided that the scalar X satisfies which holds for X = 1 (SUSY extremum), and X = 2 −1/5 (non-SUSY extremum).
Since the above solutions are supported by spacetime-filling D6-branes, they require the inclusion of a source term on the rhs of one of the BI in (A.8), to yield something of the form of where j (3) denotes a 3-form current. Such D6-branes would then fill AdS 7 and be fully localized at y = −2 inside S 3 .
In the next subsection we will review the 7D effective description of the above AdS vacua within N = 1 gauged supergravity coupled to three vector multiplets. This will immediately allow us to identify those with the critical points of the ISO(3)-gauged theory found in [17].

7D gauged supergravity description
Warped compactifications of massive type IIA supergravity on a squashed S 3 with spacetimefilling O6/D6 sources are known to admit a gauged N = 1, D = 7 supergravity description.
The theory that captures all of the closed-string zero modes is the one obtained through the coupling of the gravity multiplet with three extra vector multiplets. Such a supergravity model enjoys as a global symmetry, and its 64 bosonic degrees of freedom are arranged into the metric, six vector fields, one two-form potential and ten scalars. Such 7D degrees of freedom encode the full amount of information concerning the spectrum of zero modes within the orientifold-even sector of closed-string excitations. The explicit dictionary is given in table 1.
IIA fields Z 2 -even components 7D fields The scalar fields span the following coset geometry The set of consistent embedding tensor deformations that we are interested in comprizes a three-form f ABC parametrizing the gauging of a subgroup of SO (3,3), and a Stückelberg-like mass deformation for the two-form, which we denote by θ.   [17].
be written as The scalar potential induced by the embedding tensor reads where η AB denotes the SO(3, 3)-invariant metric, M AB is the inverse of M AB , and While the above results suggest full perturbative stability for both solutions, regardless of supersymmetry, our next step will be that of following the lines of [10] and look for perturbative instabilities within the open-string sector. We remind the reader that such a sector cannot be consistently disregarded since D6-branes crucially support the solutions, and hence they naturally carry open strings attached whose dynamics has to be taken into

Probe potential
Our scope now is to compute the mass of the scalar modes corresponding to the D6-branes' position. To this end, we will have to evaluate the brane action within the 10D background defined by (2.10) in the probe approximation, up to quadratic order. A single Dp-brane interacts with a given background due to its tension and charge. Action-wise, these two contributions are given by so-called the Dirac-Born-Infeld (DBI) term and the Wess-Zumino (WZ) term, respectively describing the brane's gravitational and electromagnetic interaction with the background fields. At first order in α , these can be written in string frame as where ...| Σ denotes the pullback of the corresponding fields over the brane worldvolume Σ, τ p is the tension of the brane, F is the field strength of the worldvolume gauge field and C ≡ n C (n) is a polyform introducing the formal sum over all RR potentials.
Spacetime-filling D6-branes are localized at the "north pole" of the S 3 and fluctuations depend only on the motion along the fiber. We have parameterized this direction with the y coordinate, which will be now promoted to a field Y (x α ) = y + 2 describing the brane excitations around the critical point y = −2. The first thing we need to do then, is to expand the 10D fields around y = −2. This procedure yields which reveals a D6-singularity at y = −2. Secondly, we take all contributions up to second order in Y into account, both in the DBI and WZ action.
For the WZ term, there is only one contribution to C coming from the C (7) that satisfies (2) . On the other hand, the fundamental contribution to the DBI comes from the warp factor of the AdS 7 and the dilaton. Expanding

Equation (2.21) produces then DBI and WZ terms which both have linear contributions in
Y that cancel each other exactly. Note that this is crucial in order to verify that Y = 0 is a critical point of the action. Up to second order in Y , we find the effective Lagrangian to be andg αβ is the unwarped metric of AdS 7 . Interestingly, the mass term for the SUSY point comes solely from the WZ term while the non-SUSY point mass term comes from the DBI.
We want to evaluate the mass of the Y mode normalized to the cosmological constant, which is Λ = −60L −2 . The normalized masses, by also taking the kinetic metric into account, are then given by where the latter (non-SUSY) solution is tachyonic, since the BF bound in 7D is − 3 5 .

The coupling to (3 + N ) extra vector multiplets
Now we would like to reproduce the same result of the previous subsection by using gauged 7D supergravity as a tool. As we said earlier, the theory with three vector multiplets only contains information concerning closed-string excitations. Hence, if we want to capture the dynamics of the D6-brane position moduli, we have to consider an extension of the theory in subsection 2.2. Since each spacetime-filling D6-brane carries a 7D vector multiplet, the correct extended description should be the one where the gravity multiplet is coupled to The unique peculariaty of supergravity theories with 16 supercharges is that of having to coupling to vector multiplets completely determined by supersymmetry requirements, regardless of the amount of vectors that one wants to include. This means that our new theory will now have as a global symmetry, and the amount of scalar fields will be where the new 3N scalars are denoted by Y Ia I=1, ...,N and each of them parametrizes the position coordinates Y a of the the I-th D6-brane.
As we argued earlier, the form of the kinetic Lagrangian (2.17) and of the scalar potential (2.19) will be identical in this case, except for the fact that now all contractions will now be taken over (6 + N )-dimensional indices, i.e. promote where M ≡ A ⊕ I. Now just stick to the embedding tensor deformations of the previous , otherwise . (2.29) Then we adopt the following explicit parametrization of the SO(3,3+N ) where we have put together the moduli in the vector 33) and the non-normalized masses in the matrix In this presentation, we have made a change of basis for the φ a , picking instead the mass As it can be seen, the equations of motion are trivial except for the one for X which is given where ω a bc , ω a jk and ω i bk are constants.
The explicit 10D background reads [19] ds 2 10 = ds 2 where ρ, Φ 0 , m, f p , and h l are all suitable constants that satisfy some algebraic constraints in order for the above background to solve the field equations in appendix A. The above background describes both SUSY and non-SUSY AdS extrema, depending on the choices of the constants in the flux Ansatz. Upon using the dictionary in [20], these solutions may be interpreted as those in [21,22]  In the next subsection, we will present the effective 4D N = 4 description, where the full information regarding the closed-string zero modes consistent with the orientifold involution are contained in the theory coupled to six vector multiplets.

4D gauged supergravity description
Massive type IIA orientifold reductions on twisted tori with fluxes down to four dimensions are known to admit a 4D half-maximal gauged supergravity description [19,24]. As already mentioned above, one has to consider the coupling of the gravity multiplet with six extra vector multiplets in order to comprize all of the closed-string zero modes. This supergravity theory enjoys as a global symmetry, and its complete set of bosonic degrees of freedom includes the metric, 12 vector fields and 38 scalars. The dictionary between these 4D degrees of freedom and the 10D orientifold-even sector of closed-string excitations can be found in table 4.
IIA fields Z 2 -even components 4D fields The set of consistent embedding tensor deformations that we are interested in are all contained within an object in the (2,220), denoted by f αM N P , parametrizing the gauging of a subgroup of SO (6,6). By means of group-theoretical considerations, it is possible to single out the parts of f which describe all parameters giving rise to non-trivial contributions to the scalar potential, i.e. the internal curvature, the H (3) flux and the RR-fluxes. This results in the embedding tensor/fluxes dictionary spelled out in table 5.

IIA fluxes Θ components
f +ājk = f +ībk , f +abc Table 5: The embedding tensor/fluxes dictionary for the case of massive type IIA reductions on a twisted T 6 with gauge fluxes. The SO(6, 6) index M is further split into its light-cone directions (a, i,ā,ī). Adapted from [25].
The scalar potential induced by the embedding tensor reads [26] V = 1 64 (3.14) The By using half-maximal gauged supergravity as a tool, the spectrum of the closed-string excitations was computed in [27], and the result is summarized in table 6.
id  Just as in the previous seven-dimensional case, the analysis within the closed-string spectrum of excitations would let us conclude that there exists non-SUSY, but nevertheless stable, AdS vacua, such as the ones within the family '3' and '4'. Here, as opposed to the 7D case, one even has the option of interpreting the above vacua as solutions of N = 8 supergravity, since they exhibit all vanishing tadpoles for spacetime-filling sources. This approach was adopted in [25], where the full mass spectrum in N = 8 was computed for the various solutions. Interestingly, the ones within the family '4' were found to be completely tachyon-free. Even more interestingly, in [8], still within this effective description, they were even found to be non-perturbatively stable.
To understand what is going on, one should recall that the only descriptions of the above AdS 4 solutions that actually makes sense, have just the right amount of D6 branes parallel to the O6-planes introduced in (3.10), so as to produce a vanishing tadpole in all directions. However, given these UV realizations of our solutions, one would have to consider the dynamical degrees of freedom living on all D6's in order to claim stability at the full quantum level.
In the next section, we want to make use of the coupling of half-maximal supergravity with (  The first natural issue to be addressed concerns end-point of such an instability. Since in both of the cases analyzed here, the tachyons are related to D6-brane physics, we expect this to be realized through brane polarization via the Myers' effect [33]. More concretely, any other positions of the D6-branes in internal space than their original one, involves a wrapping of a 2-cycle of finite size where they have polarized into D8-branes [34]. This is in line with the supporting D8-brane developing a the near horizon AdS 7 × S 2 throat. We checked explicitly that there is no other real minimum of the effective potential when the closed-string moduli are held fixed. The most natural option is that there could be a new solution crucially involving non-zero F gauge flux representing dissolved D6-brane charge, in the spirit of the non-supersymmetric AdS 7 solutions found in [35]. On the other hand, these constructions heavily rely on the interplay between closed and open strings and hence they are genuinely stringy, since such competition may only occur at finite α . This makes the actual stability of those solutions an extremely delicate issue that we wish to come back to in the future. In fact, the application of the WGC advocated in [10] suggest that they indeed are unstable.
Finally, the other relevant issue that arises from our present analysis, concerns the impact of our results on non-supersymmetric holographic constructions. In [3], the instabilities were viewed as a direct consequence of the WGC, and argued to destroy the holographic dual by shrinking the lifetime of the boudary CFT to zero. Our analysis suggests that the issue is much more subtle since the instabilities are only visible when retaining the coupling between open and closed strings. As suggested in [10], the holographic correspondence might then work in a limit where these two sectors can be consistently decoupled from each other. This is not possible in string theory, which could imply that non-supersymmetric duality never is exact but breaks down at small scales. We therefore believe that the fate of nonsupersymmetric holography needs to be investigated further.

Acknowledgments
We would like to thank Thomas Van Riet and Suvendu Giri for very stimulating discussions.
The work of the authors is supported by the Swedish Research Council (VR).

A Massive type IIA supergravity
In this appendix we review our working conventions concerning massive type IIA supergravity in ten dimensions. The bosonic part of its string-frame Lagrangian is given by where ∇ M is the covariant derivative w.r.t. the Levi-Civita connection. The trace part of the Einstein equation, The (modified) BI instead are given by