Tachyonic Kaluza-Klein modes and the AdS swampland conjecture

We compute the Kaluza-Klein spectrum of the non-supersymmetric SO(3)$\,\times\,$SO(3)-invariant AdS$_4$ vacuum of 11-dimensional supergravity, whose lowest-lying Kaluza-Klein modes belong to a consistent truncation to 4-dimensional ${\cal N}=8$ supergravity and are stable. We show that, nonetheless, the higher Kaluza-Klein modes become tachyonic so that this non-supersymmetric AdS$_4$ vacuum is perturbatively unstable within 11-dimensional supergravity. This represents the first example of unstable higher Kaluza-Klein modes and provides further evidence for the AdS swampland conjecture, which states that there are no stable non-supersymmetric AdS vacua within string theory. We also find 27 moduli amongst the Kaluza-Klein modes, which hints at the existence of a family of non-supersymmetric AdS$_4$ vacua.


Introduction
The stability of anti-de Sitter (AdS) spacetimes has been a long-standing question in theoretical physics. The question is particularly interesting in the case of non-supersymmetric AdS spacetimes, which are not protected by supersymmetry arguments and corresponding positive mass theorems [1]. In the context of string theory, the fate of non-supersymmetric AdS vacua is especially important. For example, non-supersymmetric AdS vacua provide one of the most explicit ways to apply the AdS/CFT correspondence to QCD or condensed matter systems [2]. Moreover, non-supersymmetric AdS compactifications provide a simpler class of non-supersymmetric string solutions than de Sitter vacua, which are time-dependent. Therefore, non-supersymmetric AdS solutions can be seen as a natural stepping stone to understanding de Sitter vacua in string theory.
This lack of non-supersymmetric but stable AdS vacua, together with arguments based on a sharpened version of the Weak Gravity Conjecture (WGC) [22], has recently led to the AdS Swampland Conjecture [23], which states that all non-supersymmetric AdS vacua in string theory are unstable. The most prominent possible counterexample is the SO(3) × SO(3)-invariant nonsupersymmetric AdS 4 vacuum of the N = 8 SO (8) gauged supergravity in four dimensions [3,4].
This vacuum is perturbatively stable within the full four-dimensional N = 8 supergravity [19], which has long prompted hope that it may also be stable when uplifted to the full 11-dimensional supergravity.
Indeed, a perturbative instability would require the masses of the higher Kaluza-Klein modes to drop below those of lowest-lying modes which make up the consistent truncation to N = 8 supergravity and are above the BF bound [19]. Such behaviour of Kaluza-Klein towers has never previously been observed. Nonetheless, such a perturbative instability, therefore, would provide very concrete evidence for the AdS Swampland Conjecture, whose original arguments suggest a non-perturbative instability mechanism. However, since calculating the Kaluza-Klein Recently, a "brane-jet instability" was found for the SO(3) × SO(3) vacuum [24]. There it is argued that probe branes feel a net repulsive force in certain areas of the compactification manifold due to the varying warp factor of the 11-dimensional solution. This causes the probe branes to be expelled, hence signaling an instability. In [24] it is argued that because this instability is localised in the compactification manifold and the Coulomb branch of the probe branes may be reflected in scalars of 4-dimensional supergravity, this brane-jet result might indicate an instability triggered by higher Kaluza-Klein modes.
In this paper, we show that indeed the higher Kaluza-Klein modes of the SO(3) × SO(3)invariant AdS 4 vacuum are unstable. We do this by employing the recently developed method [25,26], which allow us to compute the Kaluza-Klein spectrum of any vacuum of N = 8 gauged supergravity obtained by a consistent truncation from 10-/11-dimensional supergravity, by tracking how the spectrum changes as the vacuum is deformed from the maximally-symmetric one.
Here this allows us to follow the spectrum as the round S 7 , corresponding to the maximally Interestingly, this instability is already present at the level of 11-dimensional supergravity and does not require non-perturbative effects of M-theory.

Kaluza-Klein spectroscopy
We begin by reviewing the result of [25,26], which we will use to compute the Kaluza-Klein spectrum of the SO(3) × SO(3)-invariant AdS 4 vacuum. In [25,26], Exceptional Field Theory (ExFT) [27] was used to derive mass formulae for the Kaluza-Klein spectrum of any vacuum obtained by a consistent truncation to a lower-dimensional maximally supersymmetric supergravity.
ExFT provides a convenient reformulation of 10-/11-dimensional supergravity, which unifies gravitational and flux degrees of freedom in a way that makes a E 7(7) symmetry manifest. In particular, the bosonic sector consists of a four-dimensional metric g µν , a generalised metric M M N parameterising the coset space E 7(7) /SU (8), and a four-dimensional gauge field A µ M transforming in the 56 of E 7(7) , with µ = 0, . . . , 3 and M = 1, . . . , 56. As shown in [25,26], in ExFT the Kaluza-Klein fluctuations of any vacuum of a given N = 8 gauged supergravity, whose uplift to 10-/11-dimensional supergravity is known, can be conveniently expressed as a product of the lowest-lying modes, which make up the consistent truncation to the N = 8 supergravity, with the scalar harmonics of the compactification manifold, Y Σ , at the maximally symmetric point. For example, for all vacua within the four-dimensional SO(8) gauged supergravity, Y Σ would be the scalar harmonics of the round S 7 .
In this manner, the scalar fluctuations are parameterised by a tensor product of an e 7(7) su(8)-valued matrix with the scalar harmonics, which we label as above by Σ. Thus we represent the scalar fluctuations by j A B Σ , where A, B = 1, . . . , 56, and which, for fixed Σ, is valued in D is the projector onto the adjoint of E 7 (7) . From here onwards, we will freely raise/lower all A, B and Σ, Ω indices by δ AB and δ ΣΩ , respectively.
The method of [25,26] gives • the four-dimensional scalar matrix V A M ∈ E 7(7) /SU(8) of the vacuum we want to study.
The conventions for the four-dimensional scalar manifold are such that V A M = δ A M corresponds to the maximally symmetric point, e.g. the round S 7 for the SO(8) gauged supergravity.
We also need the following higher-dimensional data: • the scalar harmonics, Y Σ , of the compactification, • the linear action of the Killing vector fields of the compactification on the scalar harmonics, which we denote by T M Σ Ω . 4 To be explicit, T M Σ Ω is defined as where K M are the Killing vectors of the compactification 5 and L denotes the Lie derivative.
Therefore, the matrices T M Σ Ω correspond to the generators of the gauge group (generated by the Killing vectors K M ) in the representation of the scalar harmonics Y Σ . These are normalised relative to the four-dimensional embedding tensor, X M N P , such that We emphasise once more that we only require the higher-dimensional information for the maximally-symmetric compactification, e.g. the round S 7 for the SO(8) gauged supergravity, 4 For non-compact gaugings, we need to know the action of a different set of vector fields, which are not all Killing vectors. The full details are given in [25,26] but are not important here since we are working with the SO(8) gauged supergravity. 5 With appropriate modification for non-compact gaugings as discussed in [25,26].
irrespective of the four-dimensional vacuum that we are studying. The power of the method developed in [25,26] is that the effect of the deformation away from the maximally symmetric point can be captured simply by dressing the four-dimensional embedding tensor X M N P and the generators T M Σ Ω by the four-dimensional scalar matrix V A M .
In particular, the scalar mass matrix, M ABΣ,CDΩ for the scalar fluctuations, j AB,Σ , is given by where Kaluza-Klein modes can be found in [25,26].
Note that not all the fluctuations in j AB,Σ are physical. Some of these modes are Goldstone bosons eaten by massive vector fields, and some are eaten by the massive gravitons. We can remove these unphysical modes by fixing the gauge appropriately, as usual when computing Kaluza-Klein spectra.
Since the dressed embedding tensor X AB C is frequently given in SU (8)  The 36 and 420 representations of the embedding tensor are known as the fermion shift matrices, Explicitly, the relationship between the embedding tensor and the fermion shift matrices is [28] X ij,kl with the other components related by complex conjugation to the above.
Similarly, the dressed T A contain the SU(8) representations 28 ⊕ 28 as follows Finally, since the scalar j AB,Σ , for fixed Σ, parameterise the coset e 7(7) − su (8), their only non-zero components are given by wherej ijkl,Σ is the complex conjugate of j ijkl,Σ , and ijklmnpq is the eight-dimensional alternating symbol.
With the above conventions, the mass matrix (2.4) in SU(8)-covariant notation, is given by (2.11) The first three lines above correspond to the mass matrix of the four-dimensional N = 8 supergravity [29], while the last four lines provide corrections of the mass for the higher Kaluza-Klein levels.

The SO(3) × SO(3)-invariant AdS 4 vacuum
The SO(3) × SO(3)-invariant AdS 4 vacuum was first found as a solution of four-dimensional N = 8 SO(8)-gauged SUGRA [3,4]. Since the SO(8)-gauged SUGRA arises as a consistent truncation of 11-dimensional supergravity on S 7 [30], this non-supersymmetric AdS 4 vacuum can be uplifted to a solution of 11-dimensional supergravity. Concretely, this is done by making use of the known uplift formulas for the internal metric [31] and the internal components of the three-form [32]. For the SO(3) × SO(3)-invariant vacuum the resulting 11-dimensional solution has been worked out in [33]. As the relevant expressions are rather complicated and not needed here, we refer readers there for further details and explicit formulas.
Indeed, using the methods of [25,26], we only need the explicit form of the four-dimensional scalar matrix at the SO(3) × SO(3) stationary point [3] in order to compute the Kaluza-Klein spectrum. In four-dimensional N = 8 supergravity, the scalar matrix is an .
where after gauge fixing we no longer need to distinguish between SO(8) and SU (8) indices. In this gauge, the u and v matrices are expressed as infinite sums, viz.
To give the scalar matrix at the SO (3) where the two SO(3) subgroups act on the subspaces defined by I = 1, 2, 3 and I = 6, 7, 8, respectively. Note that tensors Y + and Z + are self-dual, while Y − and Z − are antiself-dual . These objects satisfy a number of elementary identities which have been listed in [33].
With these definitions, we can parametrize the SO(3) × SO(3) invariant scalar field config-urations as with two independent parameters λ and ω. To exponentiate the scalar expectation value it is, furthermore, useful to define the Hermitian projector 6 which satisfies so Π is a Hermitian projector. In particular, using identities from [33], we find that After these preparations it is straightforward to determine the u and v matrices For the above scalar field configuration, we get [2,3] A ij 1 (λ) = diag a, a, a, b, b, a, a, a , (3.12) 6 With the short-hand notation A B ≡ (A B)IJKL ≡ AIJMN BMNKL.
For A 2 i jkl (λ) we have [2] ( 14) The potential of the four-dimensional supergravity depends only on A ij 1 and A 2 i jkl and for the field configurations considered here is given by and does not depend on ω. As shown in [33], a rotation by an angle ω corresponds to a diffeomorphism in the internal dimensions, hence does not change the physical solution. The extremum is attained at so that

Scalar harmonics
To compute the Kaluza-Klein spectrum, we now only need the higher-dimensional information coming from the round S 7 . The scalar harmonics on S 7 can be expressed as symmetric traceless polynomials in the elementary harmonics Y a , with a = 1, . . . , 8, which satisfy Y a Y b δ ab = 1.
Thus,  Our summation convention for the harmonic indices Σ, Ω is such that  In order to match the notation of section 3, we must convert between the SL(8) basis used here and the SU(8) basis used there. We explain how to do this in appendix A. In the conventions of section 3, the round S 7 has A ij 1 = δ ij and A 2 i jkl = 0. Therefore, the corresponding embedding tensor in the SL(8) basis is given by (4.9)
We can now compute the masses of these Kaluza-Klein modes using the mass matrix (2.4).
We Tachyonic Kaluza-Klein modes The level 0 scalar fields, S 0 , i.e. those of N = 8 supergravity, have masses at and above the BF bound [19]. We find this is also true for S 1 , the scalars at Klein tower eventually has increasing masses with increasing level n. Therefore, we expect that at a high enough level, there will be no more tachyonic states and the total number of tachyons in the Kaluza-Klein spectrum is finite.
Indeed, we expect the total number of tachyonic modes to be finite due to the fact that the scalar mass operator on a compact manifold is a self-adjoint elliptic operator. More specifically, it is a generalised Lichnerowicz operator which in the case at hand is obtained by computing the gauge (see e.g. [35] for the corresponding analysis around the maximally supersymmetric SO (8) vacuum), this operator has the same principal symbol as the Laplace-Beltrami operator for the given metric, hence a discrete spectrum with no accumulation point [36]. What is remarkable, however, is that the mass spectrometer formula of [25,26] allows us to explicitly follow the "flow" It would be interesting to determine if these moduli can be integrated into finite deformations.
Since this is a non-supersymmetric vacuum, already the existence of infinitesimal moduli suggests some hidden structure. This same structure may also protect the moduli from obstructions at higher orders. If these modes can be integrated to finite moduli, the non-supersymmetric  dimensions.
Using the fluctuation Ansatz of [25,26] it is possible to determine the 11-dimensional fields corresponding to the tachyons and moduli we identify. This would allow us to study the possible endpoint of the instability and whether there are obstructions to integrating up the moduli to finite deformations.

Conclusions
In this paper, we computed the Kaluza-Klein spectrum of the non-supersymmetric SO(3) × SO(3)-invariant AdS 4 vacuum of 11-dimensional supergravity [33]. We showed that some of the  in the ISO(7) gauged theory which are stable within the four-dimensional gauged supergravity [6,7,10,37]. Moreover, using the consistent truncation of massive IIA supergravity on S 6 [38,39], these vacua can be uplifted to 10-dimensional solutions of string theory. Therefore, the method of [25,26] can be readily applied to address the stability of these non-supersymmetric AdS 4 vacua, which have also recently be shown to be brane-jet stable [37]. Moreover, there is a far richer variety of non-supersymmetric stationary points for maximal (and non-maximal) gauged supergravities in three dimensions than for higher dimensions D ≥ 4, whose stability remains to be investigated. For instance, the maximal gauged SO(8)×SO(8) gauged theory of [40] possesses more than 2700 critical points [41], among them at least one stable non-supersymmetric one [42].
However, a major unsolved problem here concerns the possible uplifts of these vacua to 10 or 11 dimensions, as these cannot correspond to standard Kaluza-Klein compactifications.
HN are supported by the ERC Advanced Grant "Exceptional Quantum Gravity" (Grant No.