So I'm 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 usual 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 looking at papers written by Samson and company. And they looked terribly uninteresting to me

at the time. So, you know, quantizing quad joint 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-Witten 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-Witten model, but probably also English grammar was regressing. Because for reasons that I never understood, the first title had Vesumino-Witten or capital.

Then in the second paper, just a week later, Zumino and Witten were relegated to lowercase letters, but everything else was a capital. I can tell you that I also complained. 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. And 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. There is a landscape of Samsons. And of course, there is our universe, where he's a great physicist. But apparently, he could equally well have been a champion of tennis, a champion of chess,

I'm told, a top violin player, even some less recommended options, like this one. But I was wondering whether there's also Samson's swampland. 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 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. In quantum gravity, 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, 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 will be a talk by Gregory Gabardazzi 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 diffeomorphic, 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, which include g mu nu at little x and the pullback of the capital

g mu nu, gmn, 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 these kind of couplings is very small in a way to be made precise.

So the question is, does this make sense? And 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 different variances, and it was called FG gravity, and the name has stuck even today, was Isham-Salam-Stradii 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,

mu squared. I'm assuming the coupling here is non-derivative in the metrics, by the way, for reasons that I'm not explaining here. So mu 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, it 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 Derang, Gavada, Zetoli, and Hassan, and 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, decouple, and then you are just left with a massive graviton. So massive gravity is just a special limit of this bimetric 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, which is basically some geometric mean of the mass of the graviton on the Planck scale. And this was first pointed out in this very nice paper

of Arkani-Hamed et al., and there is much more work later. OK, 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 Liguon-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 multi-gravity 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,

were the Randall Sultrum and the DGP gravity. There is this paper that I will mention a little later by Thibaut and Cogan. 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 postdoc 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. That's the dictionary. And we also know that if the energy momentum tensor is conserved, this means that its scaling 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. 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. OK, 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 weakly in the sense that the graviton will just obtain a very tiny mass. That's the question.

So as I said, if delta L is 0, 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 1. And there is the orthogonal combination, which can acquire some anomalous scaling dimension and which is not conserved. This means that there is a change 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 1. Actually, parametrically smaller than 1. It has to be able to go to 0. And at the same time, all other spin-2 operators must have dimensions that are separated from this by a gap.

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 Kaluza-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 0,

whereas this operator goes formally simply to the little tij, which now has an anomalous dimension. 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 Haroni, Clark, and Carr, 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 buts in this. The first one is that this, being a double trace operator, is down 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 string theory? We know that from Witten's prescription that in gravity, they correspond

to modified boundary conditions in AdS, or an alternative quantization. But there is no string 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 string theory embedding.

You 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, Aharoni 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 1 plus 1 over the central charge of theory 2. And you will see that this same type of formula will arise, but differently 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. So that's what we'll do. They live in the same space, or they touch at a boundary? They live in the same space here, a flat space.

Here, the sifties are just in flat space. There's no gravity in the sifties. That's just sifties. 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.

I was just wondering if you've had two times, as well as two operators, two sets of operators. Well, you have two diffomorphic 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. It's somewhere else. OK, now as you will see, what this allows us to do is, first of all, to treat things classical in gravity, no quantum effects, and to embed things in string theory in a very controllable manner.

OK, so that's what I will show to you. But before even starting, one can do some very simple, almost freelance 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 string theory, things controllable?

Well, let me tell you a few things about some kind of algebraic constraints or classification, in the same way that Nam classified conformal field theories. First of all, particle states in anti-deceit,

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, 0 and a half are special, have scaling dimension S plus 1. So for S equals 2, the graviton, this is delta equals 2, 3, remember? And this is then a short representation,

basically the conservation law of the corresponding current truncates, gives null states, so truncates the representation. Now, Porat pointed out in 2001 and 2003 that the Higgsing 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. So in particular, the short graviton multiplet, together with the short Stukelberg multiplet, can recombine into a long graviton multiplet.

So for spin 2, this means that you take the massless spin 2 graviton, a massive vector, which plays the role of a Stukelberg, 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, SO23 becomes OSP4N, where N is now the number of supersymmetries. And now, we are lucky that all unitary highest weight representations of all supergroups have been very nicely compiled in this work of Cordova-Dumitrescu and the inter-elegator,

based on a lot of previous work. 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 2 super multiplet.

The Stukelberg field is Stukelberg super multiplet, which always includes spin 3 half states, actually. And it should be possible for the two to combine into a long spin 2 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 4 and when the number of supersymmetries is more than half maximal, actually.

So in this case, this massive gravity is just excluded. That's just kinematics. So this rules out certain cases. Not always obvious. Here is a table. But for instance, there is no massive gravity with n equal 5 or higher supersymmetries in AdS4,

just for kinematic reasons. Now, what's allowed in principle is therefore four-dimensional gravity with less than four supersymmetries or equal. 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 supersymmetry. So they are more restricted than the weak gauging.

And this, I won't describe this. You can see it in this recent pre-print of last year. Basically, you can just show that if you take 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 AdS5 gravity with half-maximal supersymmetry and massive AdS4 gravity with again half-maximal supersymmetries are the two maximal situations

by analogy with, again, non-six-dimensional superconformal symmetry. And those are the two I want to concentrate on. Are there relations between them? I mean, you're doing it like a multiplication. It's a good point.

So I'll make a comment in the end on this. Let me simply show you very explicitly the Higgsing of the n equal 4, four-dimensional supergraviton. As I said, you have the graviton multiplet. It has 16 bosonic states. It has six vectors, two scalars, and the graviton.

The Stuckelberg guy has 112 bosonic degrees of freedom. It includes four gravitini, actually. And if you combine the two, you make a long 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 Cramer and Zulia. 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 several lists

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 string theory. But are there any questions on this before? Yes. This recombination is a continuous process, right?

Yes. But here you don't have a continuous parameter. So what? 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. Yeah, you can make a rational number, but you can never make it continuous. Yeah, 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 1 and there is no way with a parametrically small deformation that you can make. So let me show you now the embedding in string theory. So as I say, I will consider the two maximally

supersymmetric allowed cases. There is some quiver 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 on 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. So now you have this one big quiver. 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-deceitering.

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 0, the two things decouple.

But the problem is, indeed, that I don't have now an ideas 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 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, I mean, you said it's super conformal total theory, right? Yes. So a bunch of gauge couplings there.

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 at where? 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 a convex 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. The rank is small, then the gauge coupling is again weak.

It's the same as in 4D. No, it's three-dimensional gauge groups. It means still the gauge coupling is going to be weak. It's like that with limit of large enough, and three-dimensional gauge theory was large enough. They don't have a gravity tool. Okay, so you will see that there is, okay, so again, hold on. I mean, it's the ratio of ranks that enters.

So of course, I want to stay with gravity dual, so. Okay, 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 four holographic dual pairs that are pretty much well understood now.

On the CFT side, they were conjectured to exist by Gaioto 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 fix 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, Dokier Estes-Gutpel, and then the global constraints

and the exact mapping in a later work of Assel Estes-Gumis and myself. So we know exactly the map, and that's why we could work this thing out. I won't show you the formulae. 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 of 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 a 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 n equal four, so there should be an SO4. Isometry, and this is realized by two spheres. So this is, in a sense, fixed by symmetry. And the whole thing, of course, we are in 10 dimensions, that's strength theory.

So it has to be fibered 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 spheres fibered over a Riemann surface.

The Riemann surface will be a disc topologically. And there is a warp factor, e to the two a, which is simply the scale, the radius of AdS, that can vary as you go around sigma. It turns out, I won't show it to you, the geometry is totally determined by five brane singularities at the boundaries of this disc.

The boundaries of the disc are not boundaries of the geometry, right? They are boundaries of the parameterization of the geometry. So the geometry, of course, has no boundary. But the five brane singularities totally fix this geometry in terms of two positive harmonic functions on sigma, so I won't say more on this.

Just believe me. And the size of the six extra dimension is small compared to the radius of AdS for? No, so you will see this, yes. Yes. So what is the decoupling limit we are interested in? So remember, in the field theory,

somehow naively we want to take smaller rank for the mediating gauge symmetry, but still stay semi-classical. So the rank is large, but much smaller than something that the ranks of the two blobs on the two sides. Anyway, since we know the dictionary, with Yannis, 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

for compactification of string theory. The right half is some other AdS for 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.

It's called the Janus throat. It's a deformation of AdS five cross S five 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 Thiebaud and Ian Cogan. It didn't have too many results, but it was a very lucid description of what can happen. Sorry, Thiebaud, but I think you agree. No, it was very helpful for me, at least, for understanding things,

because that was early times of localized gravity and multi-gravity. Thiebaud and Ian Cogan asked the question, under what conditions can we find the spectrum like the one I want, namely a massless and a very slightly massive gravity on a gap and then further states?

Well, they point out that that's impossible if the internal manifold is rich flat. And then they make the comment that there is a Chinger 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 one, gamma two are the two pieces. You take the ratio of the volume of S with the minimum gamma one or gamma two. You take the infimum, and this bounds from below the lowest mass of the would-be graviton.

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-two 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 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 there 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, Thibaut 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 Kaluza client scale. So this is of the same order 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 NS5 to D5 branes. So you can define it, but it's not at all clear why it should play their role. It does. Mr. Picasso, where the AdS is linking? So if I think about AdS, I mean, is this? 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 AdS5, or more precisely, the Janus deformation of AdS5, where the dilaton varies also across AdS5 times S5.

I am. I am shrinking the throat, but I can't. No, remember, there is an S5 here, right? So it's not a circle. No, no, there is. So 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 rank of the mediating gauge

group. In the pinching limit, the throat radius must be much less than the AdS or AdI, but both can be very large. So there is semi-classical gravity. All I need is the ratio. I'm not sure, because the curvature is going to be much larger still. Sorry, the curvature is going to be much larger. Yes, but they can both be much larger

than string or Planck 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 mode of the Janus or the AdS5 geometry. It goes to constants. But it is normalizable because the geometry is cut off on the two sides by the two AdI, 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 Chigurh bound, but it's a different power. So it actually is much bigger than the Chigurh bound. This is mass squared, remember?

So it's the eighth power of the throat. So the mass is the fourth power of the throat over the fifth 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. Here is a different way to write the same thing, which touches bases 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 safety 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 hence 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 inverse is curvature. All the cells are sufficiently big.

In units of Planck units. We need to implant units, right? Yeah, yeah. I suspect it will also be n squared over c. No, let's discuss it afterwards. No, no, there is no problem with this. I mean, the whole geometry is large and classical. Because the only thing that enters are, I repeat, these relative scales.

You don't care, you can blow up the thing. Just multiply all the brains in this configuration by an arbitrarily large factor. It's the same geometry, except that all the scales are blown up. Let's talk, Zohar, let's talk about it in the break, if you don't mind.

So this is the embedding. Now, I have just one last comment, 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, right? 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. I mean, the quantum wavelength of the graviton

is super horizon. It's not totally clear what one means by effective field theory. But there is a restricted sense in which one can talk about this thing. Namely, just look at the non-linear Stuckelberg action of these coupled metrics and see

when does strong coupling set in. That's how the limitation of effective field theory was derived in the other case. If you do this, you get the same cut of intermediate scale, except one power of the graviton mass is replaced by 1 over the radius of AdS.

This is in this paper of De Rame et al. So there is clearly a cut of scale, which is in between the Planck scale and the graviton 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, a normal scaling dimension epsilon, then I should have a breakdown of my theory. Something should happen at a normal scaling dimension epsilon times c to the 1 third. That's just a simple translation of this bound.

Now, there are good reasons to believe that this 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 1 third. But I'm assuming for now that it's much smaller than one,

which is the Kaluza-Klein 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 power of higher spin-2 operators when you try to take epsilon to 0 that arrives before or the latest at these scaling dimensions.

So you can look at the problem, hold c fixed. But since you erased, I just wanted to precise. What do you mean exactly by including the whole higher spin towers for UV completeness? What I'm saying, yeah, yeah, here, of course, we want some UV complete theory. So 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 with a parametrically 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 E star is what it's compared to LADS is. Here it is. So it's the mass of the graviton, the mass of Planck over LADS to the 1 third.

I know. But if I take Hubble, I mean, if I take that. You know, keep everything fixed and take little m 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. As you see at physical length, I mean, if we do phenomenology just to understand,

oh, is it beyond the horizon? No, no, it's at arbitrarily 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? Is it? Times this other parameter.

Here it was, right? So it's m squared over c. But it's multiplied by this thing. How is it possible to make it much smaller than 1? Well, so indeed. Just mean in the next slide, go to the next slide. Yeah, so m 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 1 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. I'm just worried about the next slide, not this one. Next slide. Here if we plug epsilon equals n squared over c, Delta star minus 3 is always much bigger than 1. No, there is 1 over delta phi. Exactly, so delta phi is much, much bigger than n squared. Yes, exactly.

So OK, so you look now at this limit. Fortunately, the full spectral problem in the Jans geometry can be solved. It was reduced in this old paper of mine with John Estes to Hens equation. That's the next in the series of canonical Fuxian equations after hypergeometric.

They 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 over delta phi to the square.

These are basically linear kaluza train 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 weakly 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 n equal 4, d 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 Hawking quanta to some other reservoir. And we can maybe do it in a controllable way, something we are working on currently with my student, Vasilis. 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 Russian to Georgian, this middle name became one symbol.

So it's a Russian tradition to put your father's name in the middle one. So this is not, so this is simply part of Sampson in Georgian. Georgians like to end everything by e. So that letter is e, so it's Sampson e. Happy birthday.

Any questions? Yeah, of course I didn't really understand anything. But why would one want to have a second traffic

come with a small mass? Is there any indication that I should go to all these complicated moves to end up with? OK, a, 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. That's a deformation of GR, which seems to be part of the landscape.

And whether it's part of the real world, I don't have any wisdom or claim indeed. So if we go back to the beginning of your talk, the one that's a totally good non-linear couplings.

So would you say there will be these special couplings appearing in the AFT of this massive gravity term with the mass of e? I don't 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 n equal 4 supergravity

makes them, if they are, they're 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 and an n equal 8 supergravity in some way that reproduces

maybe these GR GT couplings. But we don't have any computation. Did I understand correctly that in the large delta phi limit, something breaks down?

Yeah, you see, what breaks down is simply our way of computing the result. Because how do you compute really 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 1 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 phi becomes sufficiently large, it should break down. And it better break down, because then we will be doing better than the Chigurh bound. Now, you may say, well, yeah, but it's not exactly the same operator.

Still, I'm suspecting there is something equivalent to the Chigurh bound for this operator. But later, when you compute the BPS states, you don't use this approximation? This calculation is just the full Janus throat. So things go to 0 at the two ends.

They are normalizable states inside the throat. So I don't, whereas the other is not normalizable, yeah. You mentioned this application to be separating black holes. So then you would put a black hole along the ADSes.

Is there some entanglement entropy? Intimidation is just the throat. That's a very interesting question, yeah. 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 would do is just cut in the middle and apply Ryuktaka Yanagi. It's divergent, but it is as it should be proportional to n squared. So 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, yeah. OK, so let's thank Costas again.