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Weak gravity versus scale separation

We give evidence that fully supersymmetric Anti-de Sitter vacua of extended supergravity with a residual gauge group containing an abelian factor cannot be scale separated as a consequence of the weak gravity conjecture. We prove this for N=2 and N=8 supergravity and we explain how our argument applies also to vacua with partially broken, but extended residual supersymmetry. We finally discuss possible loopholes and especially how certain N=1 models can evade our reasoning. Our results suggest that N=0,1 supersymmetry at the lagrangian level might be the most promising chances to obtain a truly four-dimensional effective description of quantum gravity.


Introduction
From the point of view of a low energy observer our universe looks four-dimensional. However, the existence of extra dimensions is not ruled out experimentally and it is actually required by string theory for consistency. One has thus to explain why the typical size of the four observed spacetime dimensions is much larger than that of the yet unobserved additional ones. The puzzling existence of this hierarchy of scales goes commonly under the name of scale separation problem.
Aiming at investigating realistic scenarios, one should address such a problem within setups with positive background energy density, such as de Sitter vacua or quintessence. This would however introduce further complications, related to the unavoidable breaking of supersymmetry.
In this work, we thus take a modest attitude and concentrate on the scale separation problem for effective theories with Anti-de Sitter vacua and a certain amount of preserved supercharges.
Low energy effective theories coupled to quantum gravity should obey consistency criteria known as swampland conjectures (see e.g. [1,2] for reviews). Among these, some have consequences for the fate of scale separation. More specifically, the Anti-de Sitter distance conjecture [3] implies that the size of the Anti-de Sitter spacetime and that of the internal manifold cannot be parametrically decoupled. Further refinements have been proposed in [4], introducing logarithmic corrections, and in [5], considering the effect of discrete symmetries. These statements are supported by several examples already discussed in the corresponding articles, (see also [6] for a recent analysis of two-dimensional Anti-de Sitter solutions). On the other hand, certain classes of string theory compactifications [7], based on earlier works [8][9][10], are known to be in tension with these conjectures. 1 Such models have been revisited recently and their understanding has been improved beyond the smeared approximation for the involved sources [14][15][16][17].
At present, it is not clear if something is missing in the analysis in such a way that the Anti-de Sitter distance conjecture is eventually restored. The scale separation problem is thus still an open, fundamental question even for Anti-de Sitter vacua.
While this is clearly far from a phenomenologically acceptable setup, we should note that most of the scenarios that try to generate a positive cosmological constant in string theory start from an Anti-de Sitter vacuum and then add small, typically non-perturbative, effects to uplift the vacuum energy [18,19]. Our analysis is therefore certainly relevant also for this situation.
In this work, we do not assume any of the aforementioned swampland criteria directly against scale separation. Instead, we show that the absence of scale separation follows as a consequence of yet another conjecture, the (magnetic) weak gravity conjecture [20], which is among the oldest and arguably most tested ones. 2 In the spirit of the swampland program, we will employ a bottom-up perspective and require that the magnetic weak gravity conjecture applies to any 1 A systematic search for scale separation in large classes of four-dimensional anti-de Sitter vacua from a tendimensional perspective has been initiated in [11]. See instead [12], for three-dimensional scale-separated Anti-de Sitter flux vacua recently constructed in type IIA. See also [13], for more comments on various properties of these three-and four-dimensional Anti-de Sitter type IIA vacua in relation to swampland conjectures. 2 Notably, an argument connecting weak gravity to scale separation appeared already in the original paper [20]. effective theory, while remaining agnostic on the possible string theory origin. We will consider four-dimensional extended supergravity theories admitting supersymmetric Anti-de Sitter vacua with an unbroken gauge group containing an abelian factor. For these models, we will show that the cosmological constant is of the order of the ultraviolet cutoff dictated by the weak gravity conjecture, namely the gauge coupling We will additionally assume the charge q to be non-vanishing and quantized, but the precise details of charge quantisation will not be important. What matters for us is that in the expression of the vacuum energy there is no parameter which can be (parametrically) sent to zero.
Taking the Kaluza-Klein scale as a proxy for the ultraviolet cutoff, Λ KK ∼ Λ U V , the absence of (parametrical) scale separation for vacua such that (1.1) follows, because We will give an explicit proof that N = 2 and N = 8 Anti-de Sitter vacua with generic gaugings do obey (1.1). We will explain how our argument should apply also to N -extended partial supersymmetry breaking vacua of N ′ > N supergravity theories, explicitly showing that this is the case for partially broken N = 2 vacua of N = 8 supergravity. Finally, we discuss how N = 0, 1 models, such as [7], can evade our argument if they do not arise as consistent truncations of a higher supersymmetric theory.
Our methodology is analogous to the one employed in [21,22] to exclude de Sitter vacua in extended supergravity. This suggests that difficulties in constructing de Sitter vacua are somehow related to difficulties in obtain scale separation (see [23] for a similar reasoning from a different perspective).
This work is organised as follows. In section 2, we present our general argument in N = 2 supergravity and we apply it to a concrete example of Anti-de Sitter vacua arising from Mtheory compactified on Sasaki-Einstein manifolds. In section 3, we repeat the argument in N = 8 supergravity, which is the maximal theory in four-dimensions. In section 4, we discuss generic patterns of partially broken N = 2 vacua of N = 8 supergravity and show that the argument applies also in those cases. This gives support to the fact that it should indeed apply to general Anti-de Sitter vacua with extended supersymmetry, or at least to those obtainable as partial breaking within a higher supersymmetric theory. Finally, we discuss possible loopholes and emphasise that N = 0, 1 vacua seem to be the most promising candidates to reach scale separation. We work in Planck units.

N = supergravity
We start by proving that four-dimensional N = 2 Anti-de Sitter vacua with an unbroken gauge group containing an abelian factor are of the form (1.1). Through this section, we follow the conventions of [24,25].

The argument
The vacuum energy of fully supersymmetric Anti-de Sitter vacua is given by the trace of the gravitino mass matrix where L Λ are the symplectic sections of the special Kähler manifold and P x Λ are the quaternionic moment maps. Fully supersymmetric solutions of four-dimensional N = 2 gauged supergravity have been discussed in detail in [26]. Among the vacuum conditions given there, we recall that where f Λ i ≡ D i L Λ is the Kähler covariant derivative of the symplectic sections. Squaring and using the special geometry identity where g i is the Kähler metric, (ImN ) ΛΣ the kinetic matrix of the vector fields and P 0 Λ the special Kähler moment map. This relation is central in our argument, because it expresses the gravitino mass in terms of the gauge couplings. Notice that we also used the special geometry relation [25,27], where k i Λ are special Kähler Killing vectors, and notice that this quantity vanishes on the vacuum [26]. Thus, we can express the vacuum energy entirely in terms of the moment maps, containing the gauge charges and couplings Since we want to show that this expression is related to the square of the charge of the gravitinos on the vacuum, it is convenient to define an SU(2) charge matrix such that the gravitino covariant derivative takes the form D µ ψ ν A = · · · + iA Λ µ Q Λ A B ψ ν B and the vacuum energy can be rewritten as where To recast (2.6) into the form (1.1), we have first to identify the abelian vector associated to the weak gravity conjecture and canonically normalise its kinetic term. To this purpose, we split the vector fields as where the projectors P Λ Σ , P ⊥ Λ Σ , defined as in [21] by split the vectors into the combination contributing to the U(1) factor and the orthogonal ones.
The vector gauging the weak gravity abelian factor is chosen to bẽ for some coefficients Θ Λ . Substituting these expressions into the vectors' kinetic term, we get from which we identify the weak gravity gauge coupling [28] The kinetic term being canonically normalised, we have now to identify the charge. To this purpose, we repeat a similar procedure as above. We split the charge matrix (2.5) as (2) indices are understood when this is not source of confusion). Using also (2.8), the gravitino covariant derivative becomes where the U(1) charge matrix is (2.14) as one can check by using implying Finally, we can look at the vacuum energy (2.6), conveniently rewritten as where the mixed term TrQ Λ Q ⊥ Σ vanishes due to the projectors' orthogonality. Since the kinetic matrix (Im N −1 ) ΛΣ is negative definite, we have Inserting (2.15) into this expression, we get eventually In absolute value, this means where in the last step we enforced the magnetic weak gravity conjecture. These vacua are therefore of the form (1.1) and cannot be scale separated. We stress that our proof applies to any N = 2 Anti-de Sitter vacuum with a generic (non-abelian) gauge group. The only working assumption we need is the presence of an unbroken abelian factor in the vacuum, which is required to apply the weak gravity conjecture.
While the argument above requires an abelian gauge group on the vacuum, we can argue that also a non-abelian gauge group would lead to the same conclusion, in a similar fashion to [22]. In fact, gauged extended supergravities contain the would-be goldstones of the non-abelian gauge symmetries and one could apply the argument above by looking at points in field space that are infinitesimallly close to the vacuum, but where the gauge group is broken to a subgroup containing an abelian factor. In this case (which can be achieved in various ways, according to [22]), the cut-off of the theories defined infinitesimally close to the vacuum must approach the cut-off of the non-abelian theory, while the U(1) gauge coupling will approach the non-abelian coupling. Hence the application of the magnetic weak gravity conjecture for the theories close to the vacuum will give the desired results as a limit.

An example: M-theory on Sasaki-Einstein manifolds
To give a better understanding of how the above argument works in practice, we now provide a simple example.
Compactifications of M-theory on seven-dimensional Sasaki-Einstein manifolds give rise to N=2 supergravity with abelian gauging [26,29]. Following [26], we take the prepotential and the symplectic sections to be The Kähler potential is then There is no gauging on the special Kähler manifold, thus k i Λ = 0 = P 0 Λ . The metric on the quaternionic manifold is where ρ and σ are real scalars, while ξ is complex. We gauge the isometries generated by the abelian killing vectors The associated quaternionic prepotentials are The theory admits a supersymmetric Anti-de Sitter vacuum at The field σ does not appear in the scalar potential and is thus a free parameter. The gauge kinetic matrix of the vectors is diagonal on the vacuum (2.30) The two killing vectors are gauging a U(1)×U(1) isometry of the quaternionic manifold, but on the vacuum the prepotentials reduce to Therefore, the vacuum contains an unbroken U(1) factor. In the language of our general argument, we can here identify i.e.
where q is the abelian charge. The gauge coupling is thus Eventually, we see that the vacuum energy (2.29) agrees with (2.19) and it is then of the general form (1.1).

N = 8 supergravity
The structure of the argument presented in section 2.1 is general, though the details depend on the specific N = 2 structures. To show that it can be easily extended to any gauged supergravity theory with N > 1 supersymmetries, we will now show how it works for N = 8 supergravity, which is the maximal theory in four dimensions. We follow the conventions of [30] and, assuming that one might be less familiar with N = 8 than with N = 2 supergravity, we recall few facts about the former.
The couplings of the N = 8 Lagrangian relevant for our analysis are the vectors' kinetic term, the scalar potential and the gravitino covariant derivative. The kinetic term is where the kinetic matrix is negative definite and given by

2)
L Λ ij being the coset representatives of the scalar manifold E 7(7) /SU (8). The scalar potential is and we recall the relations [30] 3 The quantities A 1 ij (gravitino mass) and A 2i jkl will also enter the supersymmetry transformations given below. The tensor P M ijkl is defined as where X M N P are the structure constants of the 56 representation of E 7 (7) . The gauge charges Finally, the gravitino covariant derivative contains the gauge connection term Since we want to study supersymmetric vacua, the supersymmetry transformations of the fermions are also relevant. Their form, up to three fermion terms, is

The argument
We have first to determine the form of the scalar potential in a maximally supersymmetric Anti-de Sitter vacuum. Given that H ± ij ρσ is the (anti-)self dual part of the improved vector field strength H ij ρσ (formula (2.51) of [30]) and that P µ ijkl = iΩ N P L M ij D µ L N kl contains derivatives of the scalars and a term with the gauge connection A µ M , on a maximally symmetric vacuum we set Furthermore, a maximally supersymmetric vacuum is such that all supersymmetry variations of the fermions vanish. Taking into account (3.12), the condition δχ ijk = 0 can only be solved on The vanishing of the gravitino variation leads instead to the killing spinor equation which allows for a non-vanishing gravitino mass and thus for an Anti-de Sitter vacuum. Inserting (3.13) into (3.6), we obtain which is the analogous of (2.3) and it enables us to express the gravitino mass in terms of the gauge charges and kinetic function. The vacuum energy becomes then In the last step we used (ImN −1 ) ΛΣ = −M ΛΣ , which can be derived from (3.2), (3.4) and the In the purely electric frame, the vacuum energy (3.16) becomes then This is the analogous of formula (2.6) of the N = 2 setup.
To recast the expression of the vacuum energy above in the form (1.1), we have now to identify the combination of vectors gauging the abelian factor associated to the weak gravity conjecture and canonically normalise the kinetic term. Given that we have already presented the manifestly covariant procedure in section 2.1, in the following we arrive at the same result by choosing a preferred direction among Λ = 0, . . . , 27 and thus avoid some level of complication.
We follow closely the steps of [22] (which were performed for the de Sitter case).
To canonically normalise the vectors, we introduce the vielbeins Θ A Λ , such that In these expressions, g is the abelian gauge coupling, which is in general function of the scalars analogous to (2.12). The canonically normalised vectors are then given by and the now hermitean charge matrix is such that the gravitino covariant derivative reduces to Finally, the vacuum energy (3.18) becomes Choosing for definiteness the direction of the weak gravity abelian group to be A = 1, we can write (q 1 ≡ q) which in absolute value means where in the last step we enforced the magnetic weak gravity conjecture. As for the N = 2 case, these vacua are precisely of the form (1.1) and therefore cannot be scale separated.
Also in this case we should stress that while our argument assumes the existence of an abelian factor on the vacuum, it can be easily extended to non-abelian gauge groups along the lines discussed at the end of section 2. This is especially important in this case because we know that fully supersymmetric vacua of N = 8 supergravity with a negative cosmological constant preserve an SO(8) gauge group [31].

Partially supersymmetric vacua and possible loopholes
So far, we focussed on vacua preserving all supersymmetries. One could therefore ask if we can employ a similar argument for vacua where supersymmetry is partially broken. To begin, let us state the transformation rules of the maximal theory at the 2-fermion level: where we recall that the covariant derivative of the supersymmetry parameters contain also the SU(8) composite connection, Q µ i j , i.e.
It is also useful to rewrite the scalar transformation as where we the employed the coset manifold properties of the scalar manifold.
If we have a vacuum preserving 2 supersymmetries, we can split the supersymmetry parameter accordingly: (4.5) where A = 1, 2 specifies the preserved transformations (rotating under the residual U(2) Rsymmetry) and a = 1, . . . , 6 counts the supersymmetries that are violated on the vacuum, which we assume for now to be transforming in the fundamental representation of SU (6). The truncation of the theory to the fields and transformations preserving the 2 residual supersymmetries described by ǫ A can be obtained by following the prescription in [27]. This effectively amounts to set ǫ a = 0 consistently in the transformation rules above. Also, since we want an ordinary gauged N = 2 supergravity as a result, we should truncate away the corresponding gravitini, ψ µ a = 0.
Once we discard the ψ µ a gravitini, we need to prove that its supersymmetry transformation under the residual supersymmetries consistently vanishes, i.e.
This requires that we truncate The first condition implies the factorisation of the scalar manifold. In detail, we can check the components of the scalar curvature and see that This in turn implies that either P ABab = 0 = P abcd , (4.9) or P Aabc = 0, (4.10) as one can see from (4.8) by employing the duality properties of the vielbein, which tells us that Let us analyse the two cases separately.
In the case we impose (4.10), consistency further requires that which means that we have to remove from the spectrum 20 fermions χ abc = 0. and the fermion shifts Finally, consistency of the fermion supersymmetry rules further constraints the fermion shifts: Altogether this results in the following residual supersymmetry variations:  1). Moreover, the fields we truncated away are also filling proper supersymmetric representations with respect to N = 2 supersymmetry, if we assume that the supersymmetry breaking vacuum has vanishing cosmological constant. In this case, we find that the 6 gravitini arrange themselves together with the other fields in 6 semilong gravitino multiplets and the remaining fields can be collected in 10 hypermultiplets.
In the case we decided to impose the other condition, namely (4.9), the same consistency relations would have required setting P ABab = P abcd = 0, We can now comment on possible loopholes of our general argument for vacua preserving N supersymmetries in N ′ -extended supergravity theories. The first one is that the abelian factor used in the original N ′ > N extended supergravity may not survive the truncation. This is certainly an element of concern, but since the N supersymmetric vacuum is still a vacuum whose cosmological constant is generated by a gauging procedure (assuming N > 1), its value must be related to some charges of the residual gauge group of the truncated theory and we should still be able to apply our argument, at least in the weaker form for non-abelian factors.
Another reason of concern is the use of the KK scale as a proxy for the ultraviolet cutoff,  [33], which suggests that the existence of a (string) scale parametrically smaller than Λ KK is pathological and indeed never realised in any known example [34].
Finally, yet another loophole is the possibility that the vacuum is generated in an N = 1 theory in a way that it cannot be related to any consistent truncation of an extended supergravity.
This can happen if the model contains sources that break supersymmetry and at the same time violate the Jacobi identities of any candidate N > 1 parent theory. The vacuum should be N = 1, otherwise our argument, which crucially uses gauged supergravity, would still apply. One could in fact build N = 1 models where the superpotential and D-terms cannot be expressed as truncations of an N > 1 theory. An example of how this can happen can be seen in [35], where N = 1 IIA vacua and effective actions were discussed. The origin of the N = 1 theory comes from the superposition of orbifolds and orientifolds that preserve different sets of the original supersymmetries. This means that while each system of branes and orientifold can preserve half of the original supersymmetry and be described by a N = 4 gauged supergravity, whose Jacobi identities correspond to the fluxes and localised sources, the simultaneous presence of different projections renders the final theory genuinely N = 1. Actually, the precise relation between orbifolds respecting the complex structure of the underlying manifold and gauged supergravities has been worked out in [37]. We should also stress that the presence of orientifolds as a necessary ingredient is somehow expected to obtain scale separated vacua, because they allow the violation of generic energy conditions respected by other sources.

Conclusion
In this work, we investigated the existence of scale separated Anti-de Sitter vacua of extended supergravity in four dimensions. We presented a general argument showing that such vacua are in tension with the magnetic gravity conjecture. For N = 2 and N = 8 vacua our analysis was explicit, and we explained how it can be extended to vacua with partial breaking of supersymmetry fitting in the between. A necessary working assumption is the presence of an unbroken abelian factor in the vacuum, needed to consistently apply the weak gravity conjecture. In case only non-abelian groups survive, we can still give a weaker version of our argument, along the lines of [22]. While one could envisage vacua where the residual gauge symmetry group is empty (see for instance [38] for maximal supergravity), these are clearly uninteresting for phenomenological applications and in any case they mostly arise from models which are ruled out by the presence of other vacua with residual abelian factors.
Our results indicate that, contrary to what one might expect, scale separation is not straightforward to obtain already at the effective field theory level, if the amount of preserved supersymmetry is not minimal or even absent. Indeed, the most realistic way we foresee to circumvent our argument is to consider (at most) N = 1 theories in four dimensions which cannot be obtained as consistent truncations of a parent theory with a higher amount of preserved supersymmetry. 5 This could explain why certain classes of string compactifications with residual N = 0, 1 supersymmetry in four dimensions, such as [7], seem indeed to be scale separated.
The work here presented can be extended along various directions. For example, our analysis in section 4 was meant to give a proof of principle that the argument should work more generally in any N -extended supergravity. It would be of interest, even in its own rights, to perform a detailed analysis of the possible patterns of partial supersymmetry breaking of the maximal theory and construct explicitly the corresponding actions. With respect to the scale separation problem, a major step forward would clearly be to understand precisely if and how N = 0, 1 Anti-de Sitter vacua do indeed evade our argument in the way we suggested above, or perhaps in some other manner. It would be also interesting to extend the argument to spacetime dimensions higher than four. Indeed, in d > 4 dimensions the scalar potential of gauged supergravity necessarily stems from a gauging procedure and thus we believe that our argument applies to such models as well. If true, this could exclude scale separation for anti-de Sitter vacua in d > 4, N > 0 theories. We leave an explicit verification of this statement for future work.  Table 1: Multiplet content. We listed the states by helicities. For massive multiplets, each massive vector multiplets will eat one of the scalar degrees of freedom and each gravitino will eat one of the spin 1/2 degrees of freedom.
We also provide tables for each of the 11 possible routes of partial supersymmetry breaking, giving the symmetry breaking pattern, the number of surviving hyper n H and short vector n V multiplets and the spectrum of the truncated states (as noted in [27], additional cases arise as further truncations of these "maximal" cases). These cases follow from the classification of the possible truncations in [27]. The novelty here is that we analysed if and how the matter fields in such truncations could be arranged in consistent representations of N = 2 supersymmetry, so that we could interpret them as the result of a spontaneous supersymmetry breaking scenario.
In detail, the 11 truncation patterns and their relation to the residual scalar manifolds are summarised in tables 2-9.     Table 9: Supersymmetry breaking patterns and truncation of the spectrum -case 11