 Εντάξει, είμαι very pleased and very happy and honored, actually, to be here for the celebration of my good friend Samson's 60th. Before going to the, you know, I don't know when I really met face to face Samson. Again, he has been part of the landscape, I think, forever, but as usually in our field, you meet people through the archives before you actually see their faces, or since these were pre-archived dates through pre-prints. I think I was at CERN when I first started, you know, looking at papers written by Samson and company, and they looked terribly uninteresting to me at the time. So, you know, quantizing co-adjoint orbits of the Virasoro algebra, Eric talked about it. I didn't know what they were doing, why they were doing it. It was totally off the mark, and then, you know, it got worse before it got better. Here is, for instance, the next bunch of papers of Samson and company. If you just look at the title, Vesumino written model as a theory of free fields, multi-loop calculations, this makes no sense, right? What kind of free fields with multi-loop calculations are you talking about? Actually, you would notice also that not only the understanding of the Vesumino written model, but probably also English grammar was regressing, because for reasons that I never understood, the first title had Vesumino written or in capital. Then the second paper, just a week later, Zumino and Wittem were relegated to lowercase letters, but everything else was a capital. Now, in any case, the first paper I really read was the one that Eric talked about on a background independent open string field theory. That was a great paper, a very inspiring one. By that time I had also met Samson, so I knew that he was a great guy with lots of nice ideas, and that he was also a very gifted person in all sorts of ways. Indeed, trying to think a little during these days about what else Samson could have done. You know, there is a landscape of Samson, and of course there is our universe where he is a great physicist, but apparently he could equally well have been a champion of tennis, a champion of chess. I'm told a champion, a top violin player, even some less recommended options like this one, but I was wondering whether there is also Samson's swampland. Now there is no Samson. I couldn't find anything. Samson is not going to tell you that he won't be able to do something at the top level, so I was looking at all possible sources. The most reliable source was Nina, so I had to ask her. So she found one entry in a swampland, which is apparently singing, so apparently Samson cannot sing even though he's a violin player. I had a less reliable also option, namely my conjecture was that Samson cannot cook, because I have never seen him eat anything but red meat. But this may actually be a wrong entry in the swampland, so I encourage you to all try to find entries in this, because it's the other way around from quantum gravity. Quantum gravity of the swampland is huge, and there are few entries in the landscape here. It seems to be the opposite. So Samson, happy birthday, and I will now move to my talk, which is, so it's a subject that has interested me for a number of years now, multi-metric or multi-gravity theories, and let me simply tell you what the story is. There would be a talk by Gregory Gabbard, that's later in this conference, who will talk much more about multi-metric theories and massive gravity, but basically the question is the following. You are given two metric universes, so you have two metrics. I'll use the notation of lower and uppercase letters for the two universes. Now the manifolds are diffeomorphics, so there is also some mapping of the coordinates from here to there. You start with two decoupled guys, so they have independent Einstein actions, independent Planck scales, independent cosmological terms, and more generally matters, coupling to little g or capital G. But now if you have this diffeomorphism, you can of course also write down potentially new couplings, but you can also include g mu nu at little x and the pullback of the capital g mu nu g m n back to this manifold. So with these two metrics, you can make different variant couplings, and we are interested in a situation where the coefficient of this kind of couplings is very small in a way to be made precise. So the question is, does this make sense, or can you actually compute things with it? Now let me say very briefly since I think Gregory will talk much more about it, just a few comments about what was known and what is known. I think the first people to talk about these possibilities, namely a theory which has two differing variances, and it was called FG gravity, and the name has stuck even today, because I some Salaman strategy about 50 years ago now. And what happens is that for generic interaction Lagrangian, the spectrum includes a massless graviton and a second massive spin-to-field, so the two metrics, the two metric fluctuations, they mix like in a double well potential. One of them stays massless, the other becomes massive. The mass is typically of the order of this parameter ν², I'm assuming the coupling here is non-derivative in the matrix, by the way, for reasons that I'm not explaining here. So nu has dimensions of mass, the mass of the graviton is of this form, and in general there is a ghost when you do this. A quick way to understand why is that this field, which will become the Stuckelberg field, Xm of little x, this field that restores reparameterization symmetries on both sides, has the three missing polarizations of the massive graviton plus a fourth one, and the fourth one is typically a ghost. And very nice progress on this question has been made after many years of efforts in these papers of Derangava d'Agetoli and Hassan Rosen, who showed that there is a special three-parameter family of couplings of interactions in which, at least classically, the ghost drops out. Now, two more remarks. First of all, if you think about by gravity, you also think automatically about massive gravity. There is a very simple limit. You just take one of the Planck scales to infinity, the fluctuations of one metric freeze out the couple, and then you are just left with a massive graviton. So massive gravity is just a special limit of this biometric theory. We don't have to think about it independently. But the second interesting fact is that this effective theory, obviously it's an effective theory, breaks down much before the Planck scale. Actually it breaks down at an intermediate scale called lambda 3 in this literature, and which is basically some geometric mean of the mass of the graviton or the Planck scale. And this was first pointed out in this very nice paper of Arkanihame Detalle, and there is much more work later. Okay, now, of course, before going into what I will tell you about, the real important question for a physicist is whether such effective field theory is compatible with the real-world physics. In particular, tests of GR and early cosmology, and this includes understanding the Weinstein mechanism. I won't say anything about this in this talk. Let me simply say that LIGO and Virgo quote now an upper bound on the mass of the graviton, which is about one kilo light years, if I remember correctly. And, of course, if the mass is smaller than the horizon scale, then it's hard to imagine its impact on phenomenology. I have nothing to say about this, so I won't talk about this. This talk will be just restricted to the formal question. Can multigravity arise from a UV-complete theory, strength theory? Is it part of the landscape or the swampland? And I will answer this in the positive, namely this part of the landscape, but with some twists. Indeed, the subject was revived, actually, from bottom-up approaches to embed brain-world models in higher dimensions. So it was two influential papers where the Randall-Schultrum and the DGP gravity. There is this paper that I will mention a little later by Thibault and Kogan. The problem, one problem with all these bottom-up approaches was very simple. The brains had to be heavy enough to back-react and change the spectrum of the graviton. But when they are heavy enough to back-react, they are never thin. So the approximations were not valid, even though the generic ideas were interesting. Inspiring. So what I will tell you about is work with my student, Ioannis Labdas, my former student. Now he's a post-doc who is in the room, in the audience. As you will see, there are no brains in the end of the story and one should really talk about weakly coupled flux universes. So that's in the way of introduction and now is there any question before I go on? So I'll change gear now and just talk about the same problem from the dual holographic side. So we know that now we can think about ADS gravity as CFT. So what's the question on the CFT side? Now the question is very simple to phrase. So the energy momentum tensor of the conformal field theory is the operator dual to the graviton. Its dimension is related to the mass of the graviton through this formula, where d is the dimension of the CFT. I will go to d equals to 3 later in the talk so as to talk about four-dimensional gravity. There is one other quantity which one can call the central charge even though it doesn't play the same role in higher dimensions and which is the normalization of the two-point function of the energy momentum tensor, c, and the dual quantity is simply the Planck scale in units of the ADS radius to some power. Okay, that's the dictionary and we also know that if the energy momentum tensor is conserved this means that its scale dimension is canonical, it's d and automatically this gives a massive graviton so to get anything like a massless graviton so to get a massive graviton you need somehow energy momentum tensor to leak out of the CFT. Okay, there's no way you can get it if the energy momentum tensor is conserved. This equation must have the right-hand side which is a vector and this vector is precisely the Stuckelberg whose role was played remember by the change of coordinates between the two universes. Okay, so the question then we want to ask is the following suppose we have two conformal field theories they have semi-classical gravity duals they are decoupled, they have independent Lagrangians now we want to add some coupling delta L and that will be of course the question which will make them coupled weekly in the sense that the graviton will just obtain a very tiny mass. Okay, that's the question. So as I said if delta L is zero then the two energy momentum tensors are separately conserved there are two massless gravitons, two decoupled universes if the coupling is weak but in a way that has to be qualified then you look at the two linear combinations there is the total energy momentum tensor it's still conserved and I have here normalized it so that the two point function is one the orthogonal combination which can acquire some anomalous scaling dimension and which is not conserved this means that there is exchange of energy momentum tensor between the two theories So what I mean by weak is that this anomalous scaling dimension epsilon has to be much much less than one actually parametrically smaller than one it has to be able to go to zero and at the same time all others spin two operators have dimensions that are separated from this by a gap Okay, otherwise there is no point we shouldn't even call this multi-gravity I want a situation where there is only one light graviton a gap and then eventually call it a Klein or other or multi-particle bound states So that's the definition of what we mean by weakly coupled CFTs Now you notice that I told you before that massive gravity is a special case you see it immediately in this formula if you take the big Planck scale or the big central charge to infinity formally this operator goes to zero whereas this operator goes formally simply to the little tij and which now has an anomalous dimension Okay, so this is simple Now when you think about coupling the two CFTs there are two ways you can do it The first was considered in the mid-2000s by Kiritsis and the Haroni-Clarkan car and they did the obvious thing they said well let's take an operator from theory one multiply it by an operator from theory two if we are lucky this could be a marginal coupling so lambda can be arbitrarily small So that's a way to try to do it There are many buds in this The first one is that this being a double trace operator is downed by powers of 1 over n in the larger expansion This means essentially that it's a quantum effect in gravity So you have to really go to quantum gravity to see it But more seriously there is a second place where this idea doesn't fly Namely double trace couplings What do you make of them in strength theory We know that from Wheaton's prescription that in gravity they correspond to modified boundary conditions in ADS or an alternative quantization but there is no strength theory rule for what it means to make this coupling to turn on this coupling So in a sense there is from the very start no real strength theory embedding but they are at best doing supergravity Up to now maybe this will change and I will have a comment on this later on Be it as it is a Harony et al did do a calculation They said well we can just do conformal perturbation theory assuming lambda is marginal they did a very nice calculation and obtained epsilon as a number that I didn't even write down lambda squared times 1 over the central charge of theory one plus 1 over the central charge of theory two and you will see that this same type of formula will arise but definitely in our problem What we did was something different but which turns out to be much more controllable Well instead of putting a double trace operator So here are the two CFTs Now I write for those of you that are not in the subject The quiver is simply some way to encode the information of a gauge theory So circles are gauge groups Squares are fundamental matter representations Never mind this is some normal gauge field theory This is the same one I assume that they have a common global symmetry which then you gauge So in a sense by gauging a common global symmetry you insert a mediator, a messenger which makes the two theories communicate Now if this messenger were weakly coupled or maybe as you will see if the rank of this gauge group is very small there is a chance that the energy leaking between the two is small and that's what we'll calculate actually the leaking of energy and the corresponding anomalous scaling dimension will be small They live in the same space or they touch at the boundary? They live in the same space here A flat space, right? Here the CFTs are just in flat space There's no gravity in the CFTs That's just CFTs Everything back from the second copy to the first copy Is that what you've done? That was done in the gravity side So in the gravity I need to take one manifold to the other to write down a local coupling Was this one if you had two times as well as two operators or two sets of operators? Well, you have two deformorphic manifolds but in the end of the day think of living in just one So the other guy is somewhere else it's some hidden sector but not hidden in the usual sense of the word hidden also from gravity I mean it's somewhere else Now as you will see what this allows us to do is called to treat things classically in gravity no quantum effects and to embed things in strength theory in a very controllable manner Okay, so that's what I will show to you but before even starting one can do some very simple you know some almost free launch representation theory Can this be possible? How can the graviton obtain a mass given that we also will add some supersymmetry to make as usual in strength theory things controllable Well, let me So let me tell you a few things about some kind of algebraic constraints or classification or a NAM in the same way that NAM classified conformal field theories First of all you know particle states and in this sheet there are representations of the conformal group of course the conformal group has the rotation group and the u1 or r in the covering space which plays the role of scaling dimension or energy and the representations that enter are always highest weight so in three dimensions we need to specify basically the spin s and the scaling dimension delta under the conformal group for the lowest weight states and the corresponding infinite unitary representation I will denote by s delta so spin and scaling dimension The massless particles for spin bigger than a half zero and a half are special have scaling dimension s plus one so for s equals two the graviton this is delta equals two three remember and this is then a short representation basically the conservation law of the corresponding current truncates gives null states or truncates the representation Now Poratti pointed out in 2001 and 3 that the higging of a gauge symmetry in gravity is the same as the breaking of a global symmetry in the conformal field theory and the way this works is simply by recombining short representations into a long one multiplet together with a short Stuckelberg multiplet can recombine into a long graviton multiplet So for spin two this means that you take the massless spin two graviton a massive vector which plays the role of a Stuckelberg and together they give a massive graviton That's simply recombination Now if you have supersymmetry you just super everything So the group is super conformal In three dimensions S0 to 3 becomes OSP4n where n is now the number of supersymmetries And now we are lucky that all unitary highest weight representations of all super groups have been very nicely compiled in this work of Cordova-Dumitrescu and the interrogator based on a lot of previous works so many special cases were known before but they have put everything very conveniently and nicely in a nice article So we just look at these representations In particular the graviton would be now a spin two super multiplet the Stuckelberg fields a Stuckelberg super multiplet which always includes spin three half states actually and it should be possible for the two to combine into a long spin two representation You inspect the tables and you see that in some cases the energy momentum tensor representation is just absolutely protected never appears in the right hand side of the decomposition of a long representation this happens when the dimension of the CFT is bigger than four and when the number of supersymmetries is more than half maximal actually So in these cases massive gravity is just excluded but it's just kinematics So these rules out certain cases not always obvious here is a table but for instance there is no massive gravity with an equal five or higher supersymmetries in ADS-4 just for kinematic reasons Now what's allowed in principle is therefore four-dimensional gravity with less than four supersymmetries and five-dimensional gravity with less or equal than two supersymmetries and in both cases you can easily show that the gauge-mediator mechanism can be applied and in both cases therefore these bounds are saturated so there are existence proofs and here I will show you explicitly the upper example You have to do a little more work to see what the double trace couplings can do and it turns out that double trace couplings that have to be relevant or marginal of course otherwise they go away are only allowed with one quarter supersymmetries so they are more restricted than the weight gauging and this I won't describe this you can see it in this recent preprint of last year basically you can just show that with the tensor product of two super fields and you look for top components these would be supersymmetric deformations that are relevant or marginal there are none if there is more than one quarter supersymmetry so here then are the allowed cases in a sense massive ADS-5 gravity with half maximal supersymmetry and massive ADS-4 gravity with again half maximal supersymmetries are the two the two maximal situations by analogy with again nums 6 dimensional super conformal symmetry and those are the two I want to concentrate on it's a good point so I'll make a comment in the end on this let me simply show you very explicitly the hexing of the n equal 4 4 dimensional super graviton as I said you have the graviton it has 16 bosonic states it has 6 vectors 2 scalars and the graviton the Stuckelberg guy has 112 bosonic degrees of freedom it includes 4 gravitini actually and if you combine the two the same multiplet which has exactly the same content as the n equal 8 maximal supergravity so the long massive multiplet we are talking about seems to be in the same field space as the maximal supergravity of Kramer and Zoullia and we can show indeed that it can be obtained at threshold by gauging n equals 8 but only at threshold there is no obvious way we work on this with Sever and Lus of trying to go further and deform it to make this long multiplet massive that's a side remark now I want to show to you how you embed this into strength theory but are there any questions on this before this recombination is a continuous process right? yes but here you don't have a continuous parameter hold on so you will see so I will have a continuous parameter and I will also have a parametrically small parameter a rational number essentially which can be made but you can never make it continuous but it doesn't matter at the level of recombination of representations it doesn't matter the excluded cases are cases where there is a mass gap of one and there is no way with a parametrically small deformation that you can make ok so let me show you now the embedding in strength theory so let's say I will consider the two maximally supersymmetric allowed cases there is some whoever gauge theory on one side another one on the other side we gauge a common global symmetry and now the two cases were either d equal 4 and equal 2 or the CFT or d equal 3 and equal 4 now d equal 4 and equal 2 sounds like the simpler one and there is this total quiver theory it's a total quiver now exactly but somewhere in the middle there is a weak link that I want to break that quiver is also super controlled yes the entire thing I don't go away from anti-desitter yeah what does this mean by the weak link like the gauge coupling in the week so hold your horses so indeed in four dimensions one would say there is an obvious way to decouple things the gauge coupling can be marginal just take it continuously to zero the two things decouple but then you don't have an ADS dual but the problem is indeed that I don't have now an ADS dual a geometric dual so it's some quantum phenomenon which I cannot control now in three-dimensional field theory or four-dimensional gravity things are more fortunately more favorable now here you may you think you start with a less controllable situation the three-dimensional gauge coupling is of course massive flows to infinity in the infrared and yet as I will show to you the leaking of energy momentum tensor can be parametrically small because of two other parameters that would come in so this is the example that we know how to control this one we don't and there is work in progress but we don't on that one actually the four-dimensional one there is a bunch of gauge couplings yes you can play games yeah but the real game is here in a sense you have somewhere to cut the thing in two and stop the exchange of energy how this place is distinguished ok you are rushing ahead but in a sense if you think about what can happen how can this be distinguished A the gauge coupling could be weak but then we are away from semi-classical gravity B and that's what I will use the rank of the gauge group can be very small so there are very few degrees of freedom that are involved in the exchange compared to the semi-classical gravity now in four dimensions that's impossible you can easily show that the ranks of the gauge groups along the quiver are of x-function so it can never be weak in three it can and I will use it but then there is a third parameter that you cannot see from what I draw and which will come into here which allows us again to tune continuously actually the thing to zero so anyway this even though it looks the harder case is the controllable one this is still to be worked out sorry so the idea is to take the rank to be small I will take the rank to be small and the second parameter that I will explain now smaller than the gauge coupling is again weak it's the same as in 4D no it's three dimensional gauge but still the gauge coupling is going to be weak it's like that look with limit of large enough and three dimensional gauge theory with large enough they don't have a gravity tool ok so you will see that there is ok so again hold on I mean it's the ratio of ranks that enters so of course I want to stay with gravity dual ok now let me tell you about the geometry I have told you about field theory now the reason why you can control all this is because there is a huge class of N equal for holographic dual pairs that are pretty much well understood now on the CFT side they were conjectured to exist by Guy Otto Witten never mind about the details of the theory so they are the so called good quiver theories in three dimensions good means essentially that you can completely there are enough matter fields to totally Higgs the whole gauge symmetry this gives you some constraints on the quiver data and the dual geometries are known exactly analytically first their local form in a very nice paper on classical integrability by the UCLA group and then the global constraints and the exact mapping in a later work of so we know exactly the map and that's why we could work this thing out let me just I won't show you the formula again you can find them in papers let me just make a few comments one is that there are no continuous free parameters in these theories all the data is in the discrete numbers the quiver, the ranks of the gauge groups and the dimensions of the fundamental representations or equivalently on the gravity side they are all in quantized brain charges so no continuous parameters but they precise one to one map on the two sides so I won't discuss this in any detail however I just want to show to you what is relevant for my purposes today the metric let me tell you about the metric on the gravity side well symmetry dictates almost the generic form you know it has to have an ADS4 that's the conformal group it's an equal force so there should be an SO4 isometry and this is realized by two two spheres so this is in a sense fixed by symmetry and the whole thing of course we are in ten dimensions that's strength theory but we are going to work over some Riemann surface sigma and the way the solutions work out so the metric is of this form there is the ADS metric there is the transverse space which has this particular form two two spheres fibered over a Riemann surface the Riemann surface will be a disk topologically and there is a warp factor e to the 2a around sigma it turns out the geometry is totally determined by five brain singularities at the boundaries of this disk the boundaries of the disk are not boundaries of the geometry they are boundaries of the parameterization of the geometry so the geometry of course has no boundary but five brain singularities totally fixed this geometry in terms of two positive harmonic functions on sigma but I won't say more on this just believe me and the size of the six extra dimensions is small compared to the radius of the ADS form no so you will see this yes so what is the decoupling limit we are interested in so remember in the field theory somehow naively we want to take small rank for the mediating YH symmetry so the rank is large but much smaller than something that the ranks of the the two blobs on the two sides anyway since we know the dictionary we just worked out the fact that as you expect there is no other degeneration limit the degeneration limit is to split the singularities in two parts pinch the Riemann surface in the middle and then find the geometry like the one I show here so here is the geometry basically the left half is some ADS-4 compactification of strength theory the right half is some other ADS-4 compactification I keep with the notation of lower and uppercase letters for the two and what is in between what is this gauge mediation geometrically well it turns out to be a very well known solution called the Janus throat it's a deformation of ADS-5 crosses 5 geometry that I will tell you a few things more about so this is the degeneration limit in the gravity side and let me hear comments so there was this paper of Thibault and the Uncogan it didn't have too many results but it was a very lucid description of what can happen sorry Thibault but I think you agree No it was very helpful for me at least for understanding things because you know that was early times of localized gravity and multi-gravity Thibault and the Uncogan asked the question under what conditions can we find a spectrum like the one I want namely a massless and a very slightly massive gravity on a gap and then further the states well they point out that that's impossible if the internal manifold is a rich flat and then they make the comment that there is a trigger bound on the eigenvalues of the Laplacian it's a very nice bound it tells you that the lowest mass or eigenvalue is bigger than the infimum of the following thing you cut the manifold in all possible ways with S gamma 2 are the two pieces you take the ratio of the volume of S with the minimum gamma 1 or gamma 2 you take the infimum and this bounds from below the lowest mass of the would be gravity so they pointed out nicely that that's the only way something can work out and this is in a sense what happens now in a sense because there are twists here are the two basic twists the first is that there is warping so the real operator is not the Laplacian or the compact space but the modification thereof the fact that this is the universal spin 2 spectrum operator independently of any matter fields you see it doesn't depend on anything but the geometry even though you have fluxes you have scalars and so on this is can be shown it's shown in these two papers actually to be the case always so there is a universal operator however it's not simply the Laplacian it depends also on this warp factor so I don't know if there are trigger bounds or can be easily obtained for this kind of operators I haven't really looked into this however this is one of the differences the second crucial difference actually is that it turns out to be a second relevant scale in this problem here are the relevant scales so there is the size of the two blobs of the compact space now Tibo asked whether this is different from the ADS radius the answer is no actually that's the well known scale separation problem we don't have solutions in which you can really decouple the Kalucha Klein scale or more there as the ADS radius same on the other side in between there is a throat and the radius of the throat can be indeed parametrically smaller because it's controlled by the rank of the mediating gauge group so this is your S here but there is this new parameter which I would call the length of the throat and which has to do simply in strength theory with the way the dilaton varies from one to the other side of the correspondence now you may say what is the dilaton the dilaton in CFT is a bizarre thing it's fixed it's not free but it's the ratio in a sense of electric to magnetic global groups or the numbers of FENES-5 to D5 brains so you can define it but it's not at all clear why it should play the role it does but where the two ADS is linking say if I think about ADS the ADS is throughout it's fibered so here it's basically let's say a constant radius ADS here too in between it's fibered so as to make something that's like ADS-5 or more precisely the Janus deformation of ADS-5 where the dilaton varies also across ADS-5 times S5 you want to shrink the throat I am shrinking the throat but I can no remember there is an S5 here right so it's not it's not a circle no no there is so there is you know this thing is an S5 it's six dimensions so there are these two parameters and now well the geometry of this throat is exactly known so the radius is quantized as I said because it's related to the gauge the rank of the mediating gauge group in the pinching limit the throat radius must be much less than the ADS radii but both can be very large so there is some classical gravity all I mean is the ratio not sure because the curvature is going to be much larger much sorry the curvature is going to be much larger yes but they can both be much larger than string or plank scale but the ratio has to be very small that's all that I need and now you can basically solve the spectral problem so the way we solve it is by just doing the obvious thing namely we look for a solution inside the throat that goes to constant values on the two sides now this is not a normalizable or the ADS-5 geometry it goes to constants but it is normalizable because the geometry is cut off at the two sides by the two radii little L and capital L so this is the mode you can compute it and here is the expression for epsilon you see that geometrically it looks a little like the trigger bound but it's a different power so it actually is much bigger than the trigger bound this is mass squared so it's the 8th power of the throat so the mass is the 4th power of the throat over the 5th power of the ADS big space but there is this correction factor which depends on delta-phi and which is something that goes from when delta-phi is 0 from 3 to 0 when delta-phi goes to infinity now this seems to be something that allows me to tune continuously to 0 epsilon actually the approximation in which we computed this does break down however somewhere before when the volume of the throat becomes of the same order as the volume of these blobs so I cannot quite take it to 0 but I can take it very far along 0 but there is a different way to write the same thing which touches basis with the Aharoni et al calculation so epsilon is precisely 1 over c plus 1 over c I re translated now the geometric stuff in terms of CFT data but now there is this effective lambda remember there is no marginal double trace coupling but I can rewrite it as some effective lambda and the effective lambda is the product of two things you see there is n which is the mediating rank n squared over c so this can be very small and this simply tells you that the degrees of freedom mediating the interaction are much less than the degrees of freedom on the two universes and this parametrically suppresses epsilon and then there is always this second thing that has to do with the dilaton variation so one is a discrete the other is a continuous parametric way to take lambda to 0 and has to decouple these two CFTs and where is the curvature in black units the curvature I repeat all curvatures are much bigger than Planck what is the curvature in terms of what no but the L you know curvature is 1 over L wherever you see L the inverses curvature all the cells are sufficiently big the units of Planck unit we need to implant units right yeah yeah but I suspect it will also be as forward over c no let's discuss it afterwards no no there is no there is no problem with this I mean the whole geometry is large and classical because the only thing that enters I repeat these relative scales you don't care you can blow up the thing you know just multiply all the brains in this configuration by an arbitrary large factor it's the same geometry except that all the scales are blown up let's talk let's talk about it in the break if you don't mind so so this is the embedding now I have just one last comment so which in a sense is comforting because it fits with expectations in a rather interesting way but it also shows the limitations of what we have done let me go back to effective field theory now the mass I'm talking about is always much less than the ADS radius I mean the inverse sorry there's a minus 1 because m times L is square root epsilon that's very small so the meaning of effective field theory is not very clear but there is a restricted sense in which one can talk about this thing namely just look at the non-linear Stuckelberg action of this coupled matrix and see when does strong coupling set in that's how the limitation of effective field theory was derived in way in the other case if you do you know you know but in the other case if you do this you get the same cut of intermediate scale except one power of the gravitational mass is replaced by one over the radius of ADS this is in this paper of the armetal so there is clearly a cut of scale which is in between the plank scale and the gravitational mass in units of ADS radius if I translate this to scaling dimensions basically what this tells me is that if I have a spin 2 operator with scaling dimension epsilon, anomalous scaling dimension epsilon then I should have a breakdown of my theory something should happen at anomalous scaling dimensions epsilon times c to the one third that's just a simple translation of this bound now there are good reasons to believe something that should happen there by the way this scale you see is of course much bigger than epsilon both because there is c which is large and because it's epsilon to the one third but I'm assuming for now that it's much smaller than one which is the kalucha client scale or the multi particle bound state so it's somewhere in between now there are arguments no real proof actually but arguments that this breakdown is a severe one in the sense that it cannot be corrected by putting extra scalars vectors or fermions you have to go to higher spins and if this is true it means that you should find the condensing tower of higher spin 2 operators when you try to take epsilon to 0 that arrives before or the latest scaling dimensions so you can look at the problem hold c fixed but since you are right I just want to be precise what do you mean exactly by improving the whole high spin towers for UV completeness what I'm saying here of course we want some UV complete theories what I'm saying if one was powerful enough one could ask this as a conformal bootstrap question what I'm saying is there is a spin 2 operator a geometrically small scaling dimension that I can take to 0 and I'm asking what happens in the limit so the limit there is good reason to believe this limit is singular In the bulk the length scale linked to this easter is what it's compared to LED acids here it is so it's the mass of the graviton the mass of Planck but leds to the one third I know but if I take her ball I mean if I take you know keep everything fixed and take little length to 0 so you can make it as low as you want I mean that's the whole point so there is a singularity in the limit of m going to 0 that you see at physical length if we do phenomenology just to understand or is it beyond the horizon no no it's at arbitrary short scales if I take continuous with the mass In your theory epsilon is m squared over c Say again In your theory epsilon is m squared over c times this other parameter here it was yeah it's m squared over c but it's multiplied by this How is it possible to make it much smaller than one Well so indeed Yeah so n is quantized I cannot take it continuously to 0 if I keep c fixed but I can take delta phi to infinity and then this goes like one over delta phi So you need to take delta phi to be much bigger than n Yeah yeah so I need delta phi large I mean of course I can make it small by taking n over c small but this is not a continuous parameter but I have delta phi No I'm just worried about the next slide not this one Here if you plug epsilon equals n squared over c delta star minus 3 is always much bigger than 1 No there is one over delta phi Exactly delta phi can be much much bigger than n Yes Exactly So you look now at this limit fortunately the full spectral problem in the Janus geometry can be solved but it was reduced in this old paper of mine with John Estes to Hen's equation that's the next in the series of canonical Fuxian equations after hypergeometric there are very powerful numerical algorithms to do it you look at the spectrum and you discover not surprisingly that indeed there is a tower of non-BPS spin-2 modes that condense indeed with the power of 1 to the square these are basically linear Cauchatrine modes that's below the breakdown scale so in a sense the breakdown is well taken into account in this calculation notice that these are non-BPS all these are long multiplets they are non-BPS they are not protected by supersymmetry and therefore not necessarily visible in any weekly coupled CFT dual but I believe that the generic effective field theory arguments are such that a similar singularity should arise in the double trace couplings and it would be very interesting to try to understand it because it may help us also understand these double trace couplings so that's the end of my talk just summarize quickly so there exists an equal 4D equal 4 solutions in which the graviton has a parametrically small mass that's part of the landscape the dual CFTs interact via gauge messengers with controllably small leak out of energy but the decoupling limit is singular and one may ask if this is also this case for the double trace couplings and the last thing I want to mention is that this very controllable communication between two universes may actually be useful in recent models of black hole evaporation trying to understand the information paradox because we can leak out the hot end quanta to some other reservoir and we can maybe do it in a controllable way something we are working on currently with my student Vasiles so that's all, thank you very much and now don't be impressed that's Google but I was still perplexed by how in going from Russia to Georgia this middle name became one symbol but it's a Russian tradition to put your father's name in the middle so this is not so this is simply part of Samsung in Georgia in Georgia it's like to end everything by E so that letter is E so it's Samsung happy birthday happy birthday many questions yeah of course I didn't understand anything but why do you want once to have a second traffic come is a small mass is there any indication that I should go through all these complicated moves to induct these ok you should probably ask this in Gregory's talk I think Thursday right because he will probably try to make contact with the real world I was here you know it's so hard to deform GR that I think we should be interested in any possible way that this may be done so that's simply the spirit you know that's a deformation of GR you know it seems to be part of the landscape and it's part of the real world I don't have any wisdom or claim so if we go back to the beginning of your the run about that say totally good non-linear couplings so would you say there will be these special couplings appearing in the AFT of this massive gravity term of the master system you know of course I don't know it's a very good no no we cannot compute the effective couplings but the fact that there is an equal for supergravity makes them if they are there extremely constrained you see somehow this I mean the way I sort of try to think about it is that because I need the Stuckelbergs I need an n equal 8 multiplet so there should be a way to put together an n equal 4 with supergravity in some way that reproduces maybe these DR GT couplings but we don't have any computation Did they understand correctly that in the large delta file something breaks down? Yeah you see what breaks down is simply our way of computing the result with because how do you compute the result here remember I'm solving the spectral problem in the throat assuming that it goes to constants on the two sides the reason why this is the good leading approximation answer is because the norm of the state I didn't show it is governed by volume of course so it's in a sense the constant values are one over volume to the appropriate power on the two sides if the volume of the throat becomes comparable then this is not a good approximation anymore so when delta 5 becomes sufficiently large it should break down and it better break down because then we would be doing better than the trigger bound now you may say well yeah but it's not exactly the same operator still I am suspecting there is something equivalent to the trigger bound for this operator but later when you compute the on BPS states you don't use this approximation this calculation is just the full Janus throat so things go to zero at the two ends they are normalizable states inside the throat so I don't whereas the other is not a normalizable you mentioned this application to evaporating black holes so then you would put a black hole but you know the other ADS is there some entanglement entropy interpretation of this throat that's a very interesting question so indeed I mean one way to think of this is you have these two universes the entry and the exit of the throat look like the three brains basically so you enter on one side but then you exit on the other side so it's some kind of warm brain if you want and now the naive thing you is just cutting the middle and apply a real takayanagi it's divergent but it is as it should be proportional to n squared so you know it has the correct dependence on n but I'm not totally sure if I should interpret it as entanglement but it's a good question ok so let's think cost is again