Massive Anti-de Sitter Gravity from String Theory

We study top-down embeddings of massive Anti-de Sitter (AdS) gravity in type-IIB string theory. The supergravity solutions have a AdS$_4$ fiber warped over a manifold M$_6$ whose shape resembles that of scottish bagpipes: The `bag' is a conventional AdS$_4$-compactification manifold, while the `pipes' are highly-curved semi-infinite Janus throats. Besides streamlining previous discussions of the problem, our main new result is a formula for the graviton mass which only depends on the effective gravitational coupling of the bag, and on the D3-brane charges and dilaton jumps of the Janus throats. We compare these embeddings to the Karch-Randall model and other bottom-up proposals for massive-AdS-gravity, and we comment on their holographic interpretation. This is a companion paper to [1], where some closely-related bimetric models with pure AdS$_5\times$S$^5$ throats were analyzed.


Introduction
Efforts to endow the graviton with a tiny mass have a long history, going back to the work of Fierz and Pauli [2] and continuing unabated today -for reviews and references see [3][4] [5].The problem is of obvious theoretical interest, and could have far-reaching implications for cosmology.In one recent development it has been argued that a key obstruction to the graviton mass -the appearance of a Boulware-Deser ghost [6], can be removed in certain classical non-linear extensions of the Fierz-Pauli action [7] [8](see also [9]).Questions, however, remain both in what concerns the consistency of such classical theories, and with regards to their range of validity if viewed as effective field theories around a given classical background. 1ne may hope to answer such questions by embedding massive gravity in an ultraviolet-complete theory like string theory.In this paper we will consider four-dimensional Anti-de Sitter (AdS) solutions of IIB string theory in which the lowest spin-2 mode has a tiny mass m g .In discussions of massive gravity, AdS is known to be an 'easier case' since it does not suffer from some of the difficulties encountered in the Mikowski and de Sitter backgrounds.There is, in particular, no van Dam-Veltman-Zakharov discontinuity [17] [18], and hence no need for the strong non-linearities known as Vainshtein screening [19].It remains therefore to be seen whether our embeddings of massive AdS gravity carry any lessons for these other backgrounds.
The general idea behind the embeddings, due to Karch and Randall [20][21], is to 'locally localize' the graviton on an AdS 4 brane living in AdS 5 bulk.Using a thin-brane approximation these authors showed that if the ratio of AdS radii is small, L 5 /L 4 1, the lightest graviton mode acquires a tiny mass.The proper string-theory realization of the idea had to wait for the derivation of exact solutions describing intersecting D3 , D5 and NS5-branes [22] - [27] .One key departure from the Karch-Randall model is the failure of the thin-brane approximation for the localizing brane, which is a D3-D5-NS5 bound state.As shown in [24] the AdS radius of this composite brane cannot be made parametrically larger than its thickness.As a result the Kaluza-Klein scale (beyond which any 4d description must break down) is L 4 , and not L 5 as in ref. [21].This is related to the familiar scale separation problem of AdS flux vacua, for a discussion see [28][29] [30]. 2he purpose of the present note is to derive an (almost universal) formula for the graviton mass in these string-theory embeddings.This is a follow-up paper to ref. [1] which analyzed closely related embeddings of bigravity models.Apart from the change in emphasis compared to [1], we will here also extend the results of this reference by allowing the dilaton to vary in the AdS 5 bulk which is deformed to the supersymmetric Janus background [22].This modifies the graviton mass by a multiplicative factor that we will compute.Our formula for the graviton mass is derived on the gravity side.It is an interesting open problem to match it with a computation of the anomalous dimension of the energy-momentum tensor on the CFT side.
There have been two other proposals in the literature for realizing massive AdS gravity in string theory.They relied either on transparent boundary conditions in AdS [31] [32], or on multi-trace deformations in CFT [33][34] [35].In these proposals the graviton mass is a quantum, one-loop effect.Although our embeddings could be possibly rephrased in these other frameworks by integrating out messenger degrees of freedom, they have the advantage of relying on proper classical solutions of 10d supergravity.They do not therefore suffer from some difficulties of the above proposals, namely non-locality of the worldsheet theory, or hard-to-control renormalization group flows [36][37] [38].The 'price to pay' is that the graviton mass is quantized and cannot be tuned continuously to zero.We will return to this point towards the end.
This paper is organized as follows: In section 2 we recall why defect or interface CFT [40] - [43] is the appropriate holographic setup for Higgsing the AdS graviton.Holographic duality is not crucial to our later analysis, but it provides useful insights on the underlying mechanism.Section 3 explains the group theory of the Higgsing, i.e. the recombination of representations of the N = 4 superconformal algebra osp(4|4) which is the symmetry of the relevant background solutions.This section can be skipped without affecting the flow of the paper.
Section 4 describes the qualitative characteristics of the supergravity solutions that lead to a small graviton mass.These solutions consist of AdS 4 fibers warped over six-dimensional manifolds with the shape of scottish bagpipes.The 'bag' describes a standard AdS 4 compactification, while the noncompact 'pipes' are highly-curved Janus throats.In section 5 we calculate, following [1], the graviton mass to leading order in the throat-to-bag size, and show that it only depends on few parameters of the solutions: the radius and dilaton variation in each throat, and the effective gravitational coupling of the bag solution.This is the main technical result of the paper.To extract its physical significance we reexpress it in three different ways.In section 6 we comment on the relation to bimetric and multi-trace models, while section 7 contains some concluding remarks.Explicit expressions for the metric and dilaton of the 'bagpipes' solutions are collected in the appendix.

Mass as holographic leakage
We begin our discussion of massive AdS gravity from the dual CFT side.This sheds instructive light on the Higgsing mechanism and motivates the construction of the dual supergravity solutions.Recall that holographic duality associates to any AdS 4 vacuum of string theory a three-dimensional conformal field theory (CFT 3 ).The AdS 4 graviton is mapped to the energy-momentum tensor T ab of the CFT 3 , and the mass (m g ) of the former to the scaling dimension (∆ g ) of the latter via [39] where L 4 is the AdS 4 radius. 3The operator T ab and its tower of derivatives arrange themselves in a spin-2 highest-weight representation of the conformal algebra so (2,3).Usually the energy-momentum tensor is conserved, ∂ b T ab = 0, so this representation must be short since it has three null descendant states.A simple algebraic computation then shows that T ab must have canonical scaling dimension ∆ g = 3, and the dual AdS 4 graviton is hence massless.
To obtain a massive graviton we must therefore allow 3d energy-momentum to 'leak out'. 4There are two possibilities that are consistent with so(2, 3) symmetry: (i) Couple the original theory to another 3d theory so that conformal symmetry is preserved.The coupling could be a double-trace deformation [33] [34], or it could be mediated by messenger degrees of freedom [1].If it is weak the dual low-energy string theory is a bimetric theory, with one graviton massless and the other obtaining a small mass; (ii) Consider the original theory as a defect or boundary of some higher-dimensional theory, in the simplest case a CFT 4 .The 3d energy-momentum can now leak out in the extra dimension There is therefore now no shortening condition, and T ab acquires an anomalous dimension [44], = ∆ g − 3 > 0 (unitarity requires that it be non-negative).
These two options are related -we will here focus on option (ii) which can be obtained as a limit of option (i).Since the graviton mass is proportional to the 3d energy-momentum leakage, we want the latter to be weak. 5In principle this could be achieved by fine tuning a (nearly or exactly) marginal bulk-boundary coupling, but this is not the mechanism at work here.Weak leakage will be instead ensured by the scarcity of the bulk CFT 4 degrees of freedom, as compared to those of the boundary CFT 3 .A consequence of this is that the Higgsing will not be a continuous process in these models, even though the graviton mass can be arbitrarily small.Let us be now specific about the defect CFT.The natural candidate for the bulk CFT 4 is N = 4 super Yang-Mills with gauge group SU (n) and coupling g YM .Its half -BPS superconformal boundaries and interfaces have been analyzed by Gaiotto and Witten [45].Half-maximal supersymmetry guarantees the stability of the solutions, and gives extra technical control, but it is not otherwise essential.The graviton mass, in particular, is not a protected quantity as we will see in a minute.Weak leakage of 3d energy-momentum could be achieved in the decoupling limit g YM → 0, but this limit is singular.A better alternative is to insist that there are much fewer degrees of freedom in the bulk than on the boundary.We will indeed show in section 5 that the anomalous dimension of T ab scales like ∼ n 2 / F3 , where F3 is the free energy on the 3-sphere which measures the boundary degrees of freedom [46].

Recombination of representations
Before moving to geometry, let us discuss the Higgsing from the point of view of representation theory.Let D(∆, s) denote a unitary highest-weight representation of so(2, 3) with conformal primary of spin s and scaling dimension ∆.Massive gravitons belong to long representations of the algebra.The decomposition of a long spin-s representation at the unitarity threshold reads [31] Thus the AdS 4 graviton (s = 2) acquires a mass by eating a massive Goldstone vector.In the 10d supergravity this vector must be the combination of off-diagonal components of the metric and tensor fields that is dual to the CFT operator T a4 .Since we will here deal with N = 4 backgrounds, fields and dual operators fit in representations of the larger superconformal algebra osp(4|4).These have been all classified under mild assumptions [47] [48].In the notation of [48] (slightly retouched in [49]) the supersymmetric extension of the above decomposition reads where L denotes a long representation, A i (B i ) a short representation that is marginally (absolutely) protected, and [s] (j;j ) ∆ denotes a superconformal primary with spin s, scaling dimension ∆ and so(4) R-symmetry quantum numbers (j; j ).The above decomposition (or recombination) describes the Higgsing of the N = 4 graviton multiplet in AdS 4 .That this is at all possible is not automatic.For instance N = 4 supersymmetry forbids the Higgsing of ordinary gauge symmetries because conserved vector currents transform in absolutely protected representations of osp(4|4) [50] [51].
The bosonic field content of the above N = 4 multiplets is as follows: The supergraviton multiplet A 2 has in addition to the spin-2 boson, six vectors and two scalar fields, making a total of 16 physical states. 6The eaten Goldstone multiplet B 1 contains 112 physical bosonic states and as many fermions.These latter include massive spin-3/2 states which are not part of the spectrum of gauged 4d supergravity [49].Higgsing with that much supersymmetry is thus necessarily a higher dimensional process.

Scottish bagpipes
We turn now to the gravity side of the Higgsing.The local form of all solutions of type-IIB supergravity with osp(4|4) symmetry has been derived by D'Hoker et al [22] [23] (see also [52] [53] for earlier work).Global solutions and the detailed holographic dictionary have been worked out in [26][27] [49].All solutions are warped products of AdS 4 over a base manifold M 6 , where ds 2 AdS4 is the metric of the unit-radius Anti-de Sitter spacetime, {y i } are the coordinates of M 6 with metric g ij , and L 4 (y) is the local radius of the AdS 4 fiber at a point y.The base manifold M 6 is itself the warped product of two 2-spheres over a Riemann surface.The complete AdS 4 ×S 2 × Ŝ2 fiber realizes the so(2, 3) × so(4) ⊂ osp(4|4) bosonic symmetry of the backgrounds.
For the lightest 4d graviton to acquire mass, M 6 should not be a compact manifold.The manifolds that lead to a small graviton mass actually resemble six-dimensional Scottish bagpipes: they have one or more semi-infinite throats (the 'pipes') attached to a large central core (the 'bag') as illustrated in figure 1 .The full 10d geometry of the pipes is AdS 5 ×S 5 , or its Janus generalization [23] in which the dilaton is also allowed to vary.A crucial technical remark [25] [26] is that under certain mild conditions (existence of both NS5-brane and D5-brane charges in the bag) pipes can be shrank smoothly away   leaving behind simple coordinate singularities.Doing this reduces the bag to a compact manifold M 6 , and AdS 4 × w M 6 becomes a standard AdS 4 vacuum with a massless 4d graviton.Since we want the graviton to acquire a small mass, we keep the throat radii finite but much smaller than the characteristic bag size.
The exact metric and dilaton backgrounds of the solutions are summarized in the appendix. 8The bag depends on a set of integer D5-, NS5-and D3-brane charges which can be arranged in two Young tableaux [26].Most of these will play however no role here.The only relevant bag parameters are (i) an overall measure of its size ∼ L bag to be defined below, and (ii) the values of the dilaton at the entries of the throats.Note that the bag is a sort of 'composite Karch-Randall brane'.Tuning the available parameters to make it flat, as in ref. [20], makes it so thick that it ends up occupying (figuratively, not literally) most of space [24].As stated above this is a facet of the scale non-separation problem of AdS vacua: L 4 is parametrically tied to the characteristic size of M 6 .
The spectral problem for spin-2 excitations around any AdS 4 supergravity solution was set up in ref. [24] (generalizing mildly [54]).Interestingly this problem only depends on the Einstein-frame metric, and not on the scalar and flux background fields.Mass eigenstates factorize as ψ(y) χ µν , where χ µν is an eigenfunction of the wave operator in AdS 4 , i.e.L AdS (2) χ µν = λχ µν where L AdS (2) is the (Lichnerowicz-Laplace) operator that acts on spin-2 (transverse-traceless) excitations, and the eigenvalue λ is related to the mass via Note that both the mass and the AdS 4 radius may vary as functions of the coordinates y i , but their product is constant.It is this invariant squared mass that replaces the left-hand side of eq. ( 1) in warped (as opposed to direct-product) solutions.The Kaluza-Klein mass spectrum is determined by the elliptic operator acting on the wavefunctions ψ(y) on M 6 [24] For direct-product solutions with constant L 4 , M 2 is simply the Laplace-Beltrami operator on M 6 .
To define the spectral problem we need also to provide a norm.With canonically-normalized fields in ten dimensions 9 the Kaluza-Klein reduction of the inner product reads [24] The mass-squared operator (8) is thus hermitean and non-negative, as expected.
To summarize this section, we are interested in the smallest eigenvalue of the above mass operator, M 2 , for manifolds consisting of a large compact bag (M 6 ) attached to one or more thin semi-infinite Janus throats.It actually turns out that the solutions studied here, whose CFT duals are N = 4 linear-quiver gauge theories, admit at most two semi-infinite throats.But more general backgrounds based on star-like quivers could have more throats.As will be clear, each throat makes a separate contribution to the squared mass of the graviton in the L 5 /L bag 1 limit.

Mass from Janus throats
The general spectral problem defined in ( 8) and ( 9) is a difficult one.But we are only interested in the smallest eigenvalue given by the equivalent minimization problem Here λ 0 + 2 = ∆ g (∆ g − 3) is the lowest eigenvalue of M 2 , and the expression in square brackets is ψ|M 2 |ψ after an integration by parts.If M 6 were replaced by the compact bag M 6 (obtained by truncating the pipes) the minimum would have been the constant wavefunction corresponding to a massless graviton.But the infinite pipes make the constant ψ non-normalizable.Indeed, √ g L 4 reaches a minimum value L 8 5 inside the pipes, then blows up at infinity where the 10d geometry asymptotes to that of (half) the boundary of AdS 5 .This is explained in the appendix and illustrated in figure 2 .Normalizable wavefunctions should therefore go to zero inside the pipes.Furthermore, it is clear from eq. ( 10) that in order to minimize the mass ψ should go to zero in the region of minimal √ gL 4  4 where gradients have lower cost, as shown in fig. 2 .To make the argument precise, separate the manifold M 6 in three parts: (I) the bag, (II) the infinite throats, and (III) the matching regions where throats are attached to the bag.Minimizing M 2 in region (I) sets ψ to a constant, so the bag does not contribute to the graviton mass.On the other < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 7 q 3 M O y j r P y g Z r t T S k Y U 9 D S p e 3 g = " > A

e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " M j r 3 Y e Z S O A j e m D A c q O p p P N A q 8 o o = " > A A A C x n i c j V H L T s J A F D 3 U F + I L d e m m k Z i 4 a l q w
l t a X l l d y 6 8 X N j a 3 t n e K u 3 u d J E q 5 y 9 p u 5 E e 8 5 9 g J 8 l t a X l l d y 6 8 X N j a 3 t n e K u 3 u d J E q 5 y 9 p u 5 E e 8 5 9 g J 8 l t a X l l d y 6 8 X N j a 3 t n e K u 3 u d J E q 5 y 9 p u 5 E e 8 5 9 g J 8 l t a X l l d y 6 8 X N j a 3 t n e K u 3 u d J E q 5 y 9 p u 5 E e 8 5 9 g J 8 ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " l r q C 2 L s M 6 j i + / O / 6 I + V y h L 2 x A r s = " > A A A C x 3 i c j V H L T s M w E B z C q 5 R X g S O X i A q J U 5 S m p e 2 x g g v c i k Q f E i C U G A M W e S l 2 K q q K A z / A F f 4 M 8 Q f w F 6 x N K s E B g a M k 6 9 m Z s X c 3 S E M h l e u + z V i z c / M L i 6 W l 8 v L K 6 t p 6 Z W O z L 5 M 8 Y 7 z H k j D J h o E v e S h i 3 l N C h X y Y Z t y P g p A P g r t D n R + M e C Z F E p + q c c o v I v 8 m F t e C + U p D 5 1 J E l 5 W q 6 3 h e v V 3 f t 1 2 n 0 W w 1 v Q Y F X r 3 l t V t 2 z X H N q q J Y 3 a T y i n N c I Q F D j g g c M R T F I X x I e s 5 Q g 4 u U s A t M C M s o E i b P 8 Y A y a X N i c W L 4 h N 7 R 9 4 Z 2 Z w U a 0 1 5 7 S q N m d E p I b 0 Z K G 7 u k S Y i X U a x P s 0 0 + N 8 4 a / c 1 7 Y j z 1 3 c b 0 D w q v i F C F W 0 L / 0 k 2 Z / 9 X p W h S u 0 T Y 1 C K o p N Y i u j h U u u e m K v r n 9 r S p F D i l h O r 6 i f E Y x M 8 p p n 2 2 j k a Z 2 3 V v f 5 N 8 N U 6 N 6 z w p u j g 9 9 S x r w d I r 2 7 0 H f c 2 q u U z t p V D s H x a h L 2 M Y O 9 m i e L X R w h C 5 6 5 H 2 L J z z j x T q 2 E m t k 3 X 9 R r Z l C s 4 U f y 3 r 8 B G R r k R k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l r q C 2 L s M 6 j i + / O / 6     as functions of the coordinate x of the Janus throat.The radius reaches a minimum value of L 5 inside the throat, and grows as L 5 cosh x on either side where the geometry asymptotes to AdS 5 /Z 2 ×S 5 .The left half of the throat is cut-off by the 'bag' at a characteristic radius ∼ L bag L 5 .The graviton wavefunction approaches a constant in this region, and vanishes exponentially at infinity.Since the matching region (III) contributes neither to the norm nor to the mass, it can be shrunk for our purposes to a point.hand, at leading order in L 5 /L bag only the bag contributes to the norm of ψ.The reason is that √ gL 2  4 decreases exponentially fast in the matching region, and ψ vanishes exponentially fast in the throat as will be shown in a minute.This fixes the constant value ψ 0 ψ bag in region (I).
The region of minimal √ gL 4  4 , on the other hand, where ψ 0 can vary with minimal mass cost, lies deep inside the throat regions.Only the throats will therefore contribute to the graviton mass at this leading order, an assumption whose validity will be again verified a posteriori.
In summary, the leading-order contribution to the norm comes from the bag, while the leadingorder contribution to the mass comes from the bottom of the throat where √ gL 4 4 is minimal.The matching region (III) contributes to neither and can be neglected.We may thus reformulate the problem as a variational problem in the Janus geometry: The only residual dependence on M 6 (viz.on the composite NS5-D5-D3 brane) is via the boundary value ψ bag whose physical meaning will soon be made clear.
The N = 4 supersymmetric Janus solution [23] depends on two parameters, the radius L 5 and the dilaton variation δφ.Like all other solutions in this class it has the fibered form The scale factors L 4 , f, f , ρ depend on the complex coordinate z that parametrizes the infinite strip.We write z = x + iτ with τ ∈ [0, π/2].The metric factors and dilaton are given in the appendix.Here we only need the combination that enters in the square brackets in (12).Things simplify actually further because the spin-2 eigenfunctions in the Janus geometry factorize into spherical harmonics on the 2-spheres, and separate functions of x and τ , and the lightest mode is only function of x [24].
Integrating over the 2-spheres and τ gives10 where the function G(x), computed in the appendix, reads We have cutoff the integral at some large negative value x c , at the boundary of the matching region.The value of x c will drop out and could be replaced by −∞, its only role is to remind us that ψ would have been a non-normalizable mode in the complete Janus geometry.
The variational problem ( 14) can be easily solved, where c 1 , c 2 are integration constants.We can perform the integral analytically with the result We here set cosh δφ = a and chose the lower integration limit so that I is an odd function of x.Fixing c 1 ,c 2 so as to satisfy the boundary conditions (14) gives finally the graviton wavefunction in the throat Note that I approaches its limiting values exponentially, so ψ 0 (x c ) ψ bag up to exponentially small corrections.Furthermore at x → +∞, ψ 0 = O(e −2x ) as required for the norm to be finite.The reader can now check that this contribution to the norm is parametrically smaller than that of the bag, and can be neglected as claimed earlier.
Plugging the above wavefunction in the expression ( 14) leads to the graviton mass.Note that ψ 0 obeys eq. ( 16), so the integrand is a total derivative and one finds Since G(x)dψ 0 /dx = −ψ bag /2I(∞, a) and [ψ 0 ] ∞ xc = −ψ bag , we finally get where we have introduced the Janus correction factor J(a), As a = cosh δφ ranges from 1 to ∞, J(a) decreases monotonically from 1 to 0, see figure 3 .The function is normalized so as to drop out for AdS 5 ×S 5 throats, while more generally it has the effect of reducing the graviton mass.
This can be understood intuitively as follows: δφ is the difference between the value of the dilaton at the entry of the throat and its value at infinity.The former is fixed by the bag (see the appendix).The latter is a free parameter that determines the coupling constant g YM of the dual 4d, N = 4 super Yang-Mills theory.Taking g YM to zero (and hence |δφ| → ∞) decouples the bulk CFT from the defect, restores conservation of T ab and sends the graviton mass to zero.The same is true, by S-duality, if g YM is taken to infinity -it is the bulk magnetic theory now that decouples manifestly.These limits are however singular.Not only does supergravity break eventually down, but also the spectrum in the Janus throat becomes quasi-continuous [24] invalidating any effective 4d description.
g 1 u / i 2 j I c P J L 2 T t g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 g 1 u / i 2 j I c P J L 2 T t g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 g 1 u / i 2 j I c P J L 2 T t g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6

Bimetric and double-trace models
In ref. [1] we analyzed solutions with a highly-curved AdS 5 ×S 5 throat13 capped-off at both of its ends by bags of much larger size.The low-energy theory is in this case a two-graviton theory.This is a concrete realization of the idea of 'Weakly Coupled Worlds' [64] in which two or more Universes endowed with separate metrics are coupled through the mixing of their gravitational fields.It is well known that massive gravity can be obtained from bigravity in a decoupling limit, and this is also true for our solutions.Before exhibiting this decoupling limit, we will first generalize the analysis of [1] from AdS 5 ×S 5 to Janus throats.
The manifold M 6 now consists of a Janus throat capped-off on both sides by two bags, M 6 and M 6 .For economy of notation we introduce the parameters and v := .
Note that v is just a short-hand for the parameter V 6 L 2 4 bag = ψ −1/2 bag of the previous section.Using the inner product ψ 1 |ψ 2 = M6 √ g L 2 4 ψ * 1 ψ 2 one finds easily two orthogonal, low-lying spin-2 states.A massless state with constant wavefunction throughout M 6 (which is normalizable because M 6 is now compact), and a massive state whose wavefunction is approximately constant in the bags, Since the throat makes a subleading contribution to the inner product, the above wavefunction is clearly orthogonal to the constant one, i.e. to the wavefunction of the massless graviton.This second mode is necessarily massive because ψ 0 is forced to vary inside the Janus throat in order to extrapolate between the above values at the exits.
One can now repeat almost verbatim the calculation of the previous section.The wavefunction in the Janus throat with the new boundary conditions reads where I(x, a) has been defined in eq. ( 17).Inserting the above wavefunction in (19), and reexpressing v and v in terms of radii and effective couplings gives This agrees with the result derived in [1] for pure AdS 5 ×S 5 throats for which J = 1. 14It also reduces to our formula of the previous section in the decoupling limit v → ∞, i.e. κ 4 → 0 or equivalently L 2 4 bag → ∞.In this limit the massless graviton has vanishing wavefunction and decouples, whereas ψ 0 is concentrated entirely in the (unprimed) bag M 6 and in the throat.
From the perspective of the dual field theory, these bigravity solutions are not 4d defect CFTs, but rather 3d CFTs of a special kind.They are superconformal gauge theories based on linear quivers with a low-rank 'weak' node [1].Removing this node breaks the quiver into two disjoint quivers.One could in principle integrate out the scarce messenger fields, thereby generating multitrace couplings between disjoint theories in the spirit of [33][34].In contrast with these references, the couplings are however non-local (they are generated by massless messengers) and exactly scale invariant (the AdS 4 symmetry is manifest).Conversely, integrating back in the messenger fields restores the interpretation of the multitrace couplings in terms of a classical supergravity background, and resolves the conflicts with string-theory locality discussed in refs.[36][37].
Similar comments apply to the relation of our models with the transparent boundary conditions of [31][32].These could conceivably mimic the effects of the semi-infinite throats, but they are obscuring the issues of locality and scale invariance.It is nevertheless interesting that they lead to the same parametric dependence of m g on the effective gravitational coupling κ 4 .

Final remarks
As these top-down embeddings demonstrate, massive AdS 4 gravity is part of the string-theory landscape.String theory is believed to be a consistent theory, so we expect the effective 4d low-energy gravity to be free of any pathologies.We have seen that the effective theory must break down at the AdS radius L 4 , which is comparable to the Kaluza-Klein scale, a feature that is related to the scale non-separation problem and could be generic.This still leaves a range of energies, m g < E < L −1 4 , in which to try to compare with effective actions such as those of refs.[7][9].A technical complication is that string theory is rarely minimal -the low-energy theory would have extra fields in addition to the massive graviton.
Massive Minkowski gravity is harder to embedd and could possibly lie in swampland.One way to see the difficulty is as follows: a key feature of the Karch-Randall model is the existence of a local minimum of the AdS scale factor L 4 .The existence of a minimum seems however to be in tension with the holographic c-theorem when the AdS 4 fiber is replaced by Minkowski [65][66] [20].It is an interesting question whether this obstruction can be somehow relaxed.
In a different direction one can look for massive-gravity and bimetric models in other dimensions and/or with different amounts of supersymmetry.Many exact AdS D solutions with D > 4 and halfmaximal supersymmetry are known by now, for instance [67] for AdS 7 , [68][69] [70] for AdS 6 , and [71] for AdS 5 (for a review and more references see also [72]).Some cases can be a priori excluded.A prime example is 6d N = 1, where the stress tensor belongs to a protected B-series multiplet [48] and cannot acquire an anomalous dimension.Thus massive AdS 7 supergravity is a priori excluded. 15The stress tensor multiplet is also absolutely protected for N = 1 in 5d, and for more-than-half-maximal supersymmetry (N > 2 in 4d and N > 4 in 3d).This is consistent with the fact that there exist no candidate defect CFTs with so many unbroken supersymmetries. 16A situation with no protection is N = 2 AdS 5 .It would be interesting to search for embeddings of massive AdS supergravity or bigravity in this case.It would be even more interesting to search for non-supersymmetric embeddings that allow a Kaluza-Klein cutoff L 4 along the lines of [28].
Aknowledgement: We have benefited from discussions or email exchanges with Laura Bernard, Ali Chamseddine, Cedric Deffayet, Eric D'Hoker, Gregory Gabadadze, Slava Mukhanov, Alessandro Tomassielo and Christoph Uhlemann.C.B. aknowledges the hospitality of the Mani L. Bhaumik Institute of UCLA during the last stage of this work.

Figure 1 :
Figure 1: The 'bagpipes' manifold M 6 consists of semiinfinite pipes with cross-sectional radius L 5 , attached to a compact bag of typical size ∼ L bag L 5 .The dark curves on the bag depict 5-brane singularities.The AdS 4 scale factor diverges at infinity in the pipes so that the full 10d geometry asymptotes to (AdS 5 /Z 2 )×S 5 .

AdS 4 ⇥
M6 < l a t e x i t s h a 1 _ b a s e 6 4 = " + i t 4 6 I T R 8 v a r J 3 V o B u 9 0 O 8 u R 1 M k = " > A A A C 5 H i c j V H L S s N A F D 3 G V 3 1 H X b o w W A R X J Z G i L n 1 s 3 A g V r R Z s C U k 6 r U P z Y j I R p H T p z p 2 4 9 Q f c 6 r e I f 6 B / 4 Z 0 x g g 9 E J y Q 5 c + 4 9 Z + b e 6 6 c h z 6 R t P w 8 Z w y O j Y + O l i c m p 6 Z n Z O X N + 4 S R L c h G w e p C E i W j 4 X s Z C H r O 6 5 D J k j V Q w L / J D d u r 3 9 l T 8 9 I K J j C f x s b x M W S v y u j H v 8 M C T R L n m c r 8 p I m u n f T R w q 0 3 J I 5 Z Z T d 8 T m j 0 Y u B u u W b Y r t l 7 W T + A U o I x i 1 R L z C U 2 0 k S B A j g g M M S T h E B 4 y e s 7 g w E Z K X A t 9 4 g Q h r u M M A 0 y S N q c s R h k e s T 3 6 d m l 3 V r A x 7 Z V n p t U B n R L S K 0 h p Y Z U 0 C e U J w u o 0 S 8 d z 7 a z Y 3 7 z 7 2 l P d 7 Z L + f u E V E S t x T u x f u o / M / + p U L R I d b O k a O N W U a k Z V Fx Q u u e 6 K u r n 1 q S p J D i l x C r c p L g g H W v n R Z 0 t r M l 2 7 6 q 2 n 4 y 8 6 U 7 F q H x S 5 O V 7 V L W n A z v d x / g Q n 6 x X H r j i H 1 f L 2 b j H q E p a w g j W a 5 y a 2 s Y 8 a 6 u R 9 h X s 8 4 N H o G N f G j X H 7 n m o M F Z p F f F n G 3 R u 7 h Z u k < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + i t 4 6I T R 8 v a r J 3 V o B u 9 0 O 8 u R 1 M k = " > A A A C 5 H i c j V H L S s N A F D 3 G V 3 1 H X b o w W A R X J Z G i L n 1 s 3 A g V r R Z s C U k 6r U P z Y j I R p H T p z p 2 4 9 Q f c 6 r e I f 6 B / 4 Z 0 x g g 9 E J y Q 5 c + 4 9 Z + b e 6 6 c h z 6 R t P w 8 Z w y O j Y + O l i c m p 6 Z n Z O X N + 4 S R L c h G w e p C E i W j 4 X s Z C H r O 6 5 D J k j V Q w L / J D d u r 3 9 l T 8 9 I K J j C f x s b x M W S v y u j H v 8 M C T R L n m c r 8 p I m u n f T R w q 0 3 J I 5 Z Z T d 8 T m j 0 Y u B u u W b Y r t l 7 W T + A U o I x i 1 R L z C U 2 0 k S B A j g g M M S T h E B 4 y e s 7 g w E Z K X A t 9 4 g Q h r u M M A 0 y S N q c s R h k e s T 3 6 d m l 3 V r A x 7 Z V n p t U B n R L S K 0 h p Y Z U 0 C e U J w u o 0 S 8 d z 7 a z Y 3 7 z 7 2 l P d 7 Z L + f u E V E S t x T u x f u o / M / + p U L R I d b O k a O N W U a k Z V F x Q u u e 6 K u r n 1 q S p J D i l x C r c p L g g H W v n R Z 0 t r M l 2 7 6 q 2 n 4 y 8 6 U 7 F q H x S 5 O V 7 V L W n A z v d x / g Q n 6 x X H r j i H 1 f L 2 b j H q E p a w g j W a 5 y a 2 s Y 8 a 6 u R 9 h X s 8 4 N H o G N f G j X H 7 n m o M F Z p F f F n G 3 R u 7 h Z u k < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + i t 4 6 I T R 8 v a r J 3 V o B u 9 0 O 8 u R 1 M k = " > A A A C 5 H i c j V H L S s N A F D 3 G V 3 1 H X b o w W A R X J Z G i L n 1 s 3 A g V r R Z s C U k 6 r U P z Y j I R p H T p z p 2 4 9 Q f c 6 r e I f 6 B / 4 Z 0 x g g 9 E J y Q 5 c + 4 9 Z + b e 6 6 c h z 6 R t P w 8 Z w y O j Y + O l i c m p 6 Z n Z O X N + 4 S R L c h G w e p C E i W j 4 X s Z C H r O 6 5 D J k j V Q w L / J D d u r 3 9 l T 8 9 I K J j C f x s b x M W S v y u j H v 8 M C T R L n m c r 8 p I m u n f T R w q 0 3 J I 5 Z Z T d 8 T m j 0 Y u B u u W b Y r t l 7 W T + A U o I x i 1 R L z C U 2 0 k S B A j g g M M S T h E B 4 y e s 7 g w E Z K X A t 9 4 g Q h r u M M A 0 y S N q c s R h k e s T 3 6 d m l 3 V r A x 7 Z V n p t U B n R L S K 0 h p Y Z U 0 C e U J w u o 0 S 8 d z 7 a z Y 3 7 z 7 2 l P d 7 Z L + f u E V E S t x T u x f u o / M / + p U L R I d b O k a O N W U a k Z V F x Q u u e 6 K u r n 1 q S p J D i l x C r c p L g g H W v n R Z 0 t r M l 2 7 6 q 2 n 4 y 8 6 U 7 F q H x S 5 O V 7 V L W n A z v d x / g Q n 6 x X H r j i H 1 f L 2 b j H q E p a w g j W a 5 y a 2 s Y 8 a 6 u R 9 h X s 8 4 N H o G N f G j X H 7 n m o M F Z p F f F n G 3 R u 7 h Z u k < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + i t 4 6 I T R 8 v a r J 3 V o B u 9 0 O 8 u R 1 M k = " > A A A C 5 H i c j V H L S s N A F D 3 G V 3 1 H X b o w W A R X J Z G i L n 1 s 3 A g V r R Z s C U k 6 r U P z Y j I R p H T p z p 2 4 9 Q f c 6 r e I f 6 B / 4 Z 0 x g g 9 E J y Q 5 c + 4 9 Z + b e 6 6 c h z 6 R t P w 8 Z w y O j Y + O l i c m p 6 Z n Z O X N + 4 S R L c h G w e p C E i W j 4 X s Z C H r O 6 5 D J k j V Q w L / J D d u r 3 9 l T 8 9 I K J j C f x s b x M W S v y u j H v 8 M C T R L n m c r 8 p I m u n f T R w q 0 3 J I 5 Z Z T d 8 T m j 0 Y u B u u W b Y r t l 7 W T + A U o I x i 1 R L z C U 2 0 k S B A j g g M M S T h E B 4 y e s 7 g w E Z K X A t 9 4 g Q h r u M M A 0 y S N q c s R h k e s T 3 6 d m l 3 V r A x 7 Z V n p t U B n R L S K 0 h p Y Z U 0 C e U J w u o 0 S 8 d z 7 a z Y 3 7 z 7 2 l P d 7 Z L k j P n 3 n N m 7 r 1 B G o p M u e 7 L g D U 4 N D w y O j Y + M T k 1 P T N r z 8 0 f Z U k u G a + x J E z k S e B n P B Q x r y m h Q n 6 S S u 5 H Q c i P g 8 6 O j h 9 f c p m J J D 5 U V y l v R H 4 7 F u e C + Y q o p r 3 U r c v I 2 W o d 9 J q V u h I R z x z H U A e 9 s 0 r T L r l l 1 y z n J / A K U E K x 9 h L 7 G X W 0 k I A h R w S O G I p w C B 8 Z P a f w 4 C I l r o E u c Z K Q M H G O H i Z I m 1 M W p w y f 2 A 5 9 2 7 Q 7 L d i Y 9 t o z M 2 p G p 4 T 0 S l I 6 W C F N Q n m S s D 7 N M f H c O G v 2 N + + u 8 d R 3 u 6 J / U H h F x C p c E P u X r p / 5 X 5 2 u R e E c m 6 Y G Q T W l h t H V s c I l N 1 3 R N 3 c + V a X I I S V O 4 x b F J W F m l P 0 + O 0 a T m d p 1 b 3 0 T f z W Z m t V 7 V u T m e N O 3 p A F 7 3 8 f 5 E x y t l T 2 3 7 O 2 v l 6 r b x a j H s I h l r N I 8 N 1 D F L v Z Q I + 9 r P O A R T x a z b q x b 6 + 4 j 1 R o o N A v 4 s q z 7 d 8 Y A m h o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 7 q 3 M O y j r P y g Z r t T S k Y U 9 D S p e 3 g = " k j P n 3 n N m 7 r 1 B G o p M u e 7 L g D U 4 N D w y O j Y + M T k 1 P T N r z 8 0 f Z U k u G a + x J E z k S e B n P B Q x r y m h Q n 6 S S u 5 H Q c i P g 8 6 O j h 9 f c p m J J D 5 U V y l v R H 4 7 F u e C + Y q o p r 3 U r c v I 2 W o d 9 J q V u h I R z x z H U A e 9 s 0 r T L r l l 1 y z n J / A K U E K x 9 h L 7 G X W 0 k I A h R w S O G I p w C B 8 Z P a f w 4 C I l r o E u c Z K Q M H G O H i Z I m 1 M W p w y f 2 A 5 9 2 7 Q 7 L d i Y 9 t o z M 2 p G p 4 T 0 S l I 6 W C F N Q n m S s D 7 N M f H c O G v 2 N + + u 8 d R 3 u 6 J / U H h F x C p c E P u X r p / 5 X 5 2 u R e E c m 6 Y G Q T W l h t H V s c I l N 1 3 R N 3 c + V a X I I S V O 4 x b F J W F m l P 0 + O 0 a T m d p 1 b 3 0 T f z W Z m t V 7 V u T m e N O 3 p A F 7 3 8 f 5 E x y t l T 2 3 7 O 2 v l 6 r b x a j H s I h l r N I 8 N 1 D F L v Z Q I + 9 r P O A R T x a z b q x b 6 + 4 j 1 R o o N A v 4 s q z 7 d 8 Y A m h o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 7 q 3 M O y j r P y g Z r t T S k Y U 9 D S p e 3 g = " k j P n 3 n N m 7 r 1 B G o p M u e 7 L g D U 4 N D w y O j Y + M T k 1 P T N r z 8 0 f Z U k u G a + x J E z k S e B n P B Q x r y m h Q n 6 S S u 5 H Q c i P g 8 6 O j h 9 f c p m J J D 5 U V y l v R H 4 7 F u e C + Y q o p r 3 U r c v I 2 W o d 9 J q V u h I R z x z H U A e 9 s 0 r T L r l l 1 y z n J / A K U E K x 9 h L 7 G X W 0 k I A h R w S O G I p w C B 8 Z P a f w 4 C I l r o E u c Z K Q M H G O H i Z I m 1 M W p w y f 2 A 5 9 2 7 Q 7 L d i Y 9 t o z M 2 p G p 4 T 0 S l I 6 W C F N Q n m S s D 7 N M f H c O G v 2 N + + u 8 d R 3 u 6 J / U H h F x C p c E P u X r p / 5 X 5 2 u R e E c m 6 Y G Q T W l h t H V s c I l N 1 3 R N 3 c + V a X I I S V O 4 x b F J W F m l P 0 + O 0 a T m d p 1 b 3 0 T f z W Z m t V 7 V u T m e N O 3 p A F 7 3 8 f 5 E x y t l T 2 3 7 O 2 v l 6 r b x a j H s I h l r N I 8 N 1 D F L v Z Q I + 9 r P O A R T x a z b q x b 6 + 4 j 1 R o o N A v 4 s q z 7 d 8 Y A m h o = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 7 q 3 M O y j r P y g Z r t T S k Y U 9 D S p e 3 g = " k j P n 3 n N m 7 r 1 B G o p M u e 7 L g D U 4 N D w y O j Y + M T k 1 P T N r z 8 0 f Z U k u G a + x J E z k S e B n P B Q x r y m h Q n 6 S S u 5 H Q c i P g 8 6 O j h 9 f c p m J J D 5 U V y l v R H 4 7 F u e C + Y q o p r 3 U r c v I 2 W o d 9 J q V u h I R z x z H U A e 9 s 0 r T L r l l 1 y z n J / A K U E K x 9 h L 7 G X W 0 k I A h R w S O G I p w C B 8 Z P a f w 4 C I l r o E u c Z K Q M H G O H i Z I m 1 M W p w y f 2 A 5 9 2 7 Q 7 L d i Y 9 t o z M 2 p G p 4 T 0 S l I 6 W C F N Q n m S s D 7 N M f H c O G v 2 N + + u 8 d R 3 u 6 J / U H h F x C p c E P u X r p / 5 X 5 2 u R e E c m 6 Y G Q T W l h t H V s c I l N 1 3 R N 3 c + V a X I I S V O 4 x b F J W F m l P 0 + O 0 a T m d p 1 b 3 0 T f z W Z m t V 7 V u T m e N O 3 p A F 7 3 8 f 5 E x y t l T 2 3 7 O 2 v l 6 r b x a j H s I h l r N I 8 N 1 D F L v Z Q I + 9 r P O A R T x a z b q x b 6 + 4 j 1 R o o N A v 4 s q z 7 d 8 Y A m h o = < / l a t e x i t > x x c bag < l < l a t e x i t s h a 1 _ b a s e 6 4 = " l e A + l 6 / c 3 F z 0 m 3 m 3 Y T Y c 6 G J w p Y k = " > A A A C y X i c j V H L T s J A F D 3 U F + I L d e m m k Z j g p m l B B H Z E N y a 6 w E Q e C R L S l k G r p a 3 t 1 I D E l T / g V n / M + A f 6 F 9 4 Z S 6 I L o t O 0 v X P u O W f m 3 m s F r h N x X X 9 P K X P z C 4 t L 6 e X M y u r a + k Z 2 c 6 s Z + X F o s 4 b t u 3 7 Y t s y I u Y 7 H G t z h L m s H I T O H l s t a 1 u 2 x y L f u W R g 5 v n f B x w H r D s 0 r z x k 4 t s k J a p 7 1 D v K j / V 4 2 p 2 s F v W h U S q q u F a v V Y q F M Q a F U K h 6 W V E P T 5 c o h W X U / + 4 Z L 9 O H D R o w h G D x w i l 2 Y i O j p w I C O g L A u J o S F F D k y z / C I D G l j Y j F i m I T e 0 v e K d p 0 E 9 W g v P C O p t u k U l 9 6 Q l C r 2 S O M T L 6 R Y n K b K f C y d B T r L e y I 9 x d 3 G 9 L c S r y G h H N e E / q W b M v + r E 7 V w D F C R N T h U U y A R U Z 2 d u M S y K + L m 6 o + q O D k E h I m 4 T / m Q Y l s q p 3 1 W p S a S t Y v e m j L / I Z k C F X s 7 4 c b 4 F L e k A U + n q M 4 O m g X N 0 D X j / C B X O 0 p G n c Y O d p G n e Z Z R w w n q a J D 3 D Z 7 x g l f l V L l T R s r D N 1 V J J Z p t / F r K 0 x e y H Z E t < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l e A + l 6 / c 3 F z 0 m 3 m 3 Y T Y c 6 G J w p Y k = " > A A A C y X i c j V H L T s J A F D 3 U F + I L d e m m k Z j g p m l B B H Z E N y a 6 w E Q e C R L S l k G r p a 3 t 1 I D E l T / g V n / M + A f 6 F 9 4 Z S 6 I L o t O 0 v X P u O W f m 3 m s F r h N x X X 9 P K X P z C 4 t L 6 e X M y u r a + k Z 2 c 6 s Z + X F o s 4 b t u 3 7 Y t s y I u Y 7 H G t z h L m s H I T O H l s t a 1 u 2 x y L f u W R g 5 v n f B x w H r D s 0 r z x k 4 t s k J a p 7 1 D v K j / V 4 2 p 2 s F v W h U S q q u F a v V Y q F M Q a F U K h 6 W V E P T 5 c o h W X U / + 4 Z L 9 O H D R o w h G D x w i l 2 Y i O j p w I C O g L A u J o S F F D k y z / C I D G l j Y j F i m I T e 0 v e K d p 0 E 9 W g v P C O p t u k U l 9 6 Q l C r 2 S O M T L 6 R Y n K b K f C y d B T r L e y I 9 x d 3 G 9 L c S r y G h H N e E / q W b M v + r E 7 V w D F C R N T h U U y A R U Z 2 d u M S y K + L m 6 o + q O D k E h I m 4 T / m Q Y l s q p 3 1 W p S a S t Y v e m j L / I Z k C F X s 7 4 c b 4 F L e k A U + n q M 4 O m g X N 0 D X j / C B X O 0 p G n c Y O d p G n e Z Z R w w n q a J D 3 D Z 7 x g l f l V L l T R s r D N 1 V J J Z p t / F r K 0 x e y H Z E t < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l e A + l 6 / c 3 F z 0 m 3 m 3 Y T Y c 6 G J w p Y k = " > A A A C y X i c j V H L T s J A F D 3 U F + I L d e m m k Z j g p m l B B H Z E N y a 6 w E Q e C R L S l k G r p a 3 t 1 I D E l T / g V n / M + A f 6 F 9 4 Z S 6 I L o t O 0 v X P u O W f m 3 m s F r h N x X X 9 P K X P z C 4 t L 6 e X M y u r a + k Z 2 c 6 s Z + X F o s 4 b t u 3 7 Y t s y I u Y 7 H G t z h L m s H I T O H l s t a 1 u 2 x y L f u W R g 5 v n f B x w H r D s 0 r z x k 4 t s k J a p 7 1 D v K j / V 4 2 p 2 s F v W h U S q q u F a v V Y q F M Q a F U K h 6 W V E P T 5 c o h W X U / + 4 Z L 9 O H D R o w h G D x w i l 2 Y i O j p w I C O g L A u J o S F F D k y z / C I D G l j Y j F i m I T e 0 v e K d p 0 E 9 W g v P C O p t u k U l 9 6 Q l C r 2 S O M T L 6 R Y n K b K f C y d B T r L e y I 9 x d 3 G 9 L c S r y G h H N e E / q W b M v + r E 7 V w D F C R N T h U U y A R U Z 2 d u M S y K + L m 6 o + q O D k E h I m 4 T / m Q Y l s q p 3 1 W p S a S t Y v e m j L / I Z k C F X s 7 4 c b 4 F L e k A U + n q M 4 O m g X N 0 D X j / C B X O 0 p G n c Y O d p G n e Z Z R w w n q a J D 3 D Z 7 x g l f l V L l T R s r D N 1 V J J Z p t / F r K 0 x e y H Z E t < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l e A + l 6 / c 3 F z 0 m 3 m 3 Y T Y c 6 G J w p Y k = " > A A A C y X i c j V H L T s J A F D 3 U F + I L d e m m k Z j g p m l B B H Z E N y a 6 w E Q e C R L S l k G r p a 3 t 1 I D E l T / g V n / M + A f 6 F 9 4 Z S 6 I L o t O 0 v X P u O W f m 3 m s F r h N x X X 9 P K X P z C 4 t L 6 e X M y u r a + k Z 2 c 6 s Z + X F o s 4 b t u 3 7 Y t s y I u Y 7 H G t z h L m s H I T O H l s t a 1 u 2 x y L f u W R g 5 v n f B x w H r D s 0 r z x k 4 t s k J a p 7 1 D v K j / V 4 2 p 2 s F v W h U S q q u F a v V Y q F M Q a F U K h 6 W V E P T 5 c o h W X U / + 4 Z L 9 O H D R o w h G D x w i l 2 Y i O j p w I C O g L A u J o S F F D k y z / C I D G l j Y j F i m I T e 0 v e K d p 0 E 9 W g v P C O p t u k U l 9 6 Q l C r 2 S O M T L 6 R Y n K b K f C y d B T r L e y I 9 x d 3 G 9 L c S r y G h H N e E / q W b M v + r E 7 V w D F C R N T h U U y A R U Z 2 d u M S y K + L m 6 o + q O D k E h I m 4 T / m Q Y l s q p 3 1 W p S a S t Y v e m j L / I Z k C F X s 7 4 c b 4 F L e k A U + n q M 4 O m g X N 0 D X j / C B X O 0 p G n c Y O d p G n e Z Z R w w n q a J D 3 D Z 7 x g l f l V L l T R s r D N 1 V J J Z p t / F r K 0 x e y H Z E t < / l a t e x i t > L 5 b t n X P P O T P 3 X j v y 3 I T r + n t O W V p e W V 3 L r x c 2 N r e 2 d 4 q 7 e 5 0 k T G O H t Z 3 Q C + O e b S X M c w P W 5 i 7 3 W C + K m e X b H u v a k 3 O R 7 9 6 z O H H D 4 J p P I z b w r X H g 3 r q b t n X P P O T P 3 X j v y 3 I T r + n t O W V p e W V 3 L r x c 2 N r e 2 d 4 q 7 e 5 0 k T G O H t Z 3 Q C + O e b S X M c w P W 5 i 7 3 W C + K m e X b H u v a k 3 O R 7 9 6 z O H H D 4 J p P I z b w r X H g 3 r q b t n X P P O T P 3 X j v y 3 I T r + n t O W V p e W V 3 L r x c 2 N r e 2 d 4 q 7 e 5 0 k T G O H t Z 3 Q C + O e b S X M c w P W 5 i 7 3 W C + K m e X b H u v a k 3 O R 7 9 6 z O H H D 4 J p P I z b w r X H g 3 r q b t n X P P O T P 3 X j v y 3 I T r + n t O W V p e W V 3 L r x c 2 N r e 2 d 4 q 7 e 5 0 k T G O H t Z 3 Q C + O e b S X M c w P W 5 i 7 3 W C + K m e X b H u v a k 3 O R 7 9 6 z O H H D 4 J p P I z b w r X H g 3 r q z s r b A q z 0 l D T j 4 P s 6 f o L 1 W C / x a s L t e r W + V o x 7 D I p a w Q v P c Q B 0 7 a K J F 3 l e 4 x w M e v a 5 3 7 d 1 4 t + + l 3 k C p W c C X 5 d 2 9 A S M 2 n C 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S O l 4 7 K 7 z s r b A q z 0 l D T j 4 P s 6 f o L 1 W C / x a s L t e r W + V o x 7 D I p a w Q v P c Q B 0 7 a K J F 3 l e 4 x w M e v a 5 3 7 d 1 4 t + + l 3 k C p W c C X 5 d 2 9 A S M 2 n C 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S O l 4 7 K 7 z s r b A q z 0 l D T j 4 P s 6 f o L 1 W C / x a s L t e r W + V o x 7 D I p a w Q v P c Q B 0 7 a K J F 3 l e 4 x w M e v a 5 3 7 d 1 4 t + + l 3 k C p W c C X 5 d 2 9 A S M 2 n C 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S O l 4 7 K 7 z s r b A q z 0 l D T j 4 P s 6 f o L 1 W C / x a s L t e r W + V o x 7 D I p a w Q v P c Q B 0 7 a K J F 3 l e 4 x w M e v a 5 3 7 d 1 4 t + + l 3 k C p W c C X 5 d 2 9 A S M 2 n C 0 = < / l a t e x i t >

Figure 2 :
Figure 2: Schematic drawing of the AdS 4 radius L 4 (green curve) and of the graviton wavefunction ψ 0 (blue curve) t e x i t s h a 1 _ b a s e 6 4 = " O / v C c 6 M y 6 F s L f 5 d B O G e L s v m j Z O H d F 3 d z 8 U p U k h 5 A 4 h T s U j w g z r Z z 0 2 d S a W N e u e u v o + J v O V K z a s z Q 3 w b u 6 J Q 3 Y / j n O a V D b K 9 l W y T 7 b L 5 S P 0 l F n s Y V t F G m e B y j j B B V U y f s K j 3 j C s 3 F u 3 B h 3 x v 1 n q p F J N Z v 4 t o y H D y a S l B s = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " O / v C c 6 M y 6 F s L f 5 d B O G e L s v m j Z O