 OK, well, first, let me start by thanking the organizers for the kind of invitation. It's a pleasure to be here, my first time here in Bure. And it's especially because the team of the workshop coincides precisely with my favorite topic, so this is always nice. So I would like to discuss, in a particular instance, how a quantum spacetime, along with gravity, can arise from a matrix model. And this is something that has fascinated me for a long time. I have tried to do it for a long time. But now, for the first time, I think it really seems to work, at least, more or less. But before I go into any details, you can always sort of step back a little bit and start to dream a little bit. And if you think what could be perhaps a formulation of a quantum theory of spacetime and gravity. Of course, everybody will have a different point of view and different preferences. But to explain the motivation, so let me tell you a little bit more in my personal guidelines. So first, I would say it should be simple. I mean, for the plain reason that otherwise we would never find it. OK. Now, it should probably be a gauge theory in some sense, because in Minkowski's signature, there is always the issue of time-like components and to have unitary theories. And this is more or less the only way that we know how to handle these issues. Probably, you want to have only, finally, many degrees of freedom per unit volume, because there is a blank scale and then, hopefully, gravity becomes strong and so on. And it seems reasonable. But that tells you, of course, right away that the formulation should not be given in terms of ordinary geometry. That's just, I think it's just the wrong way to start. And you have to find some kind of an alternative to that, which leads to approximately a long scale to something like geometry. And in particular, so the underlying degrees of freedom should be non-geometric. OK, let's assume that for the moment. And the other thing, which at least for me seems reasonable, is that gravity, we only know gravity in the infrared. Of course, we know nothing about gravity at very short scales. So there is no reason to start with something like gravity. We only have to require that gravity emerges at long scales so that we have consistency with observations. And so something like, you know, gravity is perhaps something like Navier-Stokes equation. And you would never start the quantize in Navier-Stokes equation. You just quantize something underlying and then you hope that they arise in a reasonable approximate way. So that's sort of rough guidelines. And so I want to sort of try to convince you that matrix models are precisely such class of models which may realize these ideas. Suddenly they're simple. There's really nothing simpler that you can possibly write on. And it is also well known that they can describe something like dynamical geometries, dynamical non-committative spaces in particular, or fuzzy spaces. And they also are known to describe gauge theories. So that's already reasonable. And the reason why they describe gauge theories is very simple. It's this kind of very basic invariance of a matrix model, which is there from the beginning. And that's sort of the seed or the starting point of gauge invariance as we know it in physics. And in principle, at least there is a good concept for quantization. You should, well, you should, what you always do with a matrix model, you integrate over the space of our matrices. And in the Euclidean case, that thing is a very reasonable object. And in the Minkowski case, probably you want to put an i here. Then it's OK. And a little bit more problematic, and we have already heard in this morning a lot about these issues. And I will not really talk about the quantization here anymore. So I will focus about semi-classical aspects and how to get that. But of course, this is very important to keep this in the back of the mind. And in fact, this is a very important selection principle. So even if you don't want to have anything to do with string theory, you can just start with something like that in any dimension and so on. And you start to work out in some details this quantization, for example, at one loop or something like that. And then you find, well, in principle, even though this is a good concept, you run into very serious problems. And these problems go under the name of ultra-valid infrared mixing. And it really means that the theory that you will come end up with is highly non-local. It's somehow unacceptable. But there is one preferred model. And really, in some sense, only one which doesn't have this problem, at least not in a pathological sense. And that's precisely the maximally super-symmetric model. And that's precisely the model that Kawai-san has introduced us this morning, which I will call IKKT model. So there's really, even if you don't have come from string theory, this is a preferred model. And it's almost forced into that, almost. And then it turns out this shares actually a lot of features of string theory. But I would say, OK, it can cut away the landscape, which is also good. But then, yes? I think it is. I will come to that. That's precisely the dot-to-v matrix model. It's just a different name. OK, so good so far. But then, in the end, of course, the really interesting thing is, what about gravity? How can you recover gravity from such a framework? And that's what I would like to discuss. OK, and so let me start with the conclusion. So this is what I want to discuss in this talk. And this is contained in a series of recent papers. But it's really this one that I will focus on. And so one result will be that, OK. So I will discuss a particular three plus one dimensional covariant cosmological spacetime. So this is Friedmann, LeMettro, Robertson, Walker geometry, but a fuzzy one. And it happens to have a big bounce that just comes out. And all right. Then on top, so if you have this solution of the model and it is a solution of the model, then if you study the fluctuations, you will find a tower of higher spin excitations. And they will describe a higher spin gauge theory. But in fact, the higher spin is truncated at some in the trend. So this is not quite what you know from Versiliev theory. It's a little bit truncated. It's finite. It's regularized. And it contains all the degrees of freedom, in particular, for gravity, I claim, at least at the linearist level. OK, now another result will be that the propagation of these degrees of freedom on this background turn out to be, it's governed by a single universal metric. This is, of course, essential for gravity. Otherwise, you cannot talk about gravity. And it's not totally evident here that it happens. But it happens. And the reason why it is not totally evident is that Lorentz invariance will be only partially manifest. So I will explain this in more detail. In particular, the boosts will not be manifest. But nevertheless, they seem to effectively be realized. And this is very important, especially because of the observation of gravitational waves recently. Then I will discuss the metric perturbation in some detail. And we'll see. And OK, so here, not everything is settled. But essentially, it looks like the metric perturbations describe massless gravitons and the next-door scalar. So it's not the same as GR, but it's reasonably close. At least that's my current understanding. And in the end, so briefly, so all of this happens without even talking about anti-hippodactrists, comes from the matrix model. Now, but then at some point, it seems that you do need something like an anti-hippodaction. And there is this thing, so I will write down the linearized analog of the anti-hippodaction here. And the point is, this is expected to be induced upon quantization. You don't have to add it by hand. It will just arise. OK, so this is sort of roughly the summary of the talk. And OK, so chronologically, first, I will briefly discuss the model. But I can be very brief here, because we heard lots of details in the morning. And then I will spend some time discussing a particular four-dimensional covariant quantum space. And in fact, so this is fuzzy four-dimensional hyperboloid. Of course, this is Euclidean, but nevertheless, that's the starting point here. And from that, we will come to a three-plus one-dimensional cosmological spacetime, still fuzzy, non-committative. And then we'll discuss the fluctuations on this background, and these fluctuations will lead to a higher spin gauge theory. And in particular, the most interesting part, some are, of course, the metric fluctuations of spin two part. And we'll see as I discuss that. And very briefly, discuss the linearized answer. We have a Hilbert action. OK, so let me start with the model. It's exactly the same as we saw this morning. It's the type 2B or IKKD model. IKKD stands for these four gentlemen who came up with it more than 20 years ago. And two of them are sitting here in the audience, but actually, OK, I do a little bit. I changed it a little bit by hand. I put the master. And well, you will see the reason why I do that, simply because I want to have certain solutions. And the essence is putting a master behind it. It really puts in a scale. So this somehow fixes the scale. Without that master, there wouldn't be any geometric scales here. And this just puts in a scale. I'm not completely sure if it is really essential to do it. But OK, let me do it. It's something like a soft Susie breaking term that you put into some model. So I think it's not too bad, the modification. And it doesn't break any of the symmetries, except for supersymmetry, softly. OK, so what are the ingredients, once again? So you have a master. There is also an eta AB in the master. Yeah, yeah, sorry. Yes, it's an eta. It's an eta. Yes, yes, thank you. Exactly. So what are the ingredients? First of all, you have 10 Hermitian matrices. Think of them as very large, but finite matrices, even so actually, I will then go to the infinite dimensional case right away. There is Gitching variance, of course. There is a global SO91 symmetry. So that means this is just the eta symbol of SO91. These are the gamma matrices of SO91. And these are Majorana-Weilspinors. And that's pretty much it. And in principle, if there wasn't a master, it would have maximal supersymmetry. And this is kind of a soft Susie breaking term, if you like. OK, and again, this can be seen to rise in many points of use. It arises from the quantized chill action for the type to be super string. It simply arises from dimensional reduction of 10-dimension super Yang-Mills to a point. And OK, so that was already discussed. Now let me consider this as a classical model. So as a classical model, you have an action. You derive equations of motion. The equations of motion have this simple form. This is exactly the form that Joachim discussed in great detail this morning, including this inhomogeneous term. And I will abbreviate the double matrix commutator contractive with eta by this box symbol. It is because it is the matrix analog of the Dallon-Beirchen or Laplace operator, if you like. OK, so these are the equations that we want to solve. And in principle, then you should go ahead and quantize it, but I will not do this today. OK. So then the strategy is the following. OK, so first of all, you have to find solutions. And of course, there are lots of solutions. We have already heard this. And this is the part which is in some sense completely ad hoc. So at the moment, there is nothing I can say, well, this solution is better than that. So that's really ad hoc. And that's a little bit, of course, like in string theory. But it's not quite as bad, first of all. And second, why is it not quite as bad? Because there is actually, of course, there is the underlying matrix model, which is defined at the number derivative level. And you can hope that eventually there really is a mechanism which tells you which background is preferred. And I think tomorrow there will be a talk about this now, but there will be aspects. But for the moment, OK, so now let me just guess some interesting solutions and see if I can get interesting physics from at least some particularly nice solution. So that's the idea. So look for solutions which are better of spacetime. Then on these solutions, you can study the fluctuations. There is a systematic procedure. And these fluctuations are automatically governed by a gauge theory. But in fact, the fluctuations, of course, also describe under dynamical geometry. It's obvious, because the background is geometric. You fluctuate the background. You fluctuate also the geometry. So this is obviously some kind of a gauge theory of geometry. So you wonder, OK, this really somehow smells like some sort of gravity. But it's not clear if it will be Einstein gravity, of course. OK, and then in principle, you should do that. But if you then do the matrix integral, the matrix integral will, of course, include an integration over the geometry. So in principle, you have formulated at least a reasonable concept of integration over geometries here. OK, now again, so there are lots of solutions. And I'll focus on a particular type and then on a particular solution. So one large class of solutions that you have here are quantized symplectic spaces. Not all solutions are of this type, but many are. And this is, so let me call this also fuzzy space. It's the same for me. This is the same thing. But the point is, these are not just quantized symplectic manifolds. They are really embedded in target space. So that's the crucial thing. It's a manifold embedded in target space. It's something like a D-brain, actually, in string theory. It's exactly what it is. And this is what is described by matrices. OK, in what sense? So if you have a quantized symplectic manifold, it means that the algebra of functions is quantized, is realized as endomorphism algebra on some Hilbert space. So that's the kind of fundamental playground here. A particular function on this manifold is given by the embedding function. So if the functions are sub-manifolds, you have the embedding function of little x, and you can quantize them. And this is the capital X. This is the matrix living in that space. And this is the matrix that enters in the matrix one, at least that gives you this class of solutions. And OK, so that means if you have such quantized embedded manifolds, then you can take, for example, the common data of this matrix and the common data. You should think of it as a quantized boson bracket living on that space. So in general, it will be x dependent. OK, so whenever you see theta AB, that sort of comes from these common data of matrices. And think of it as a boson tensor. OK, so this is sort of the generic picture. And here are two canonical examples. The simplest one is known as Moyal-Weil quantum plane. This works in any even dimension. So it's just a Heisenberg algebra, of course. The common data of the axis is a unit matrix. So it's just a compact quantized symplectic space. And it's a solution of the equations of motion, obviously. We have also heard this already. And the nice thing, one nice thing about this example is that it admits translation. So you can just translate x by x plus, which are unit matrix. And that's OK, that's still a solution. So you have translation invariance manifest, but rotation invariance is broken. And it's obviously broken because there is an explicit tensor on the right-hand side, which is something like an electromagnetic field. So that's a problem, that's an issue. Now, the other canonical example is the two-dimensional fuzzy sphere, as introduced by Jens Hoppe. And this is just given by any irreducible representation of SO3 or SU2, probably a large one. And then the three generators, they satisfy the Kazami relation, which is something like, it describes a sphere. And for large representation, the common data of the matrices actually, if you fix this, then this is like 1 over n, so it goes to 0. And the nice thing about this example, so this, of course, also, it's just a quantized simplec, two-dimensional sphere, simplec, they quantize it, that's what you get. And the nice thing about this example is that it's fully covariant under SO3. Obviously, these relations are, they admit an SO3 rotations. OK, so that's all I want to say about these things here. And OK, and then on any of these type of fuzzy background solution, you can, you should consider fluctuations, of course, and ask what is the physics of these fluctuations. And very generically, what you get here is a non-committative gauge theory, and also this we have already in the morning. Let me very briefly go again through it because it will be important. And the starting point is a simple observation that if you have these matrices, they define naturally derivations on the algebra. And these derivations, you should think of it as part, you know, these are, these are Hamiltonian vector fields. These are just partial derivative operators, essentially. Now, that means if you have this background x, and if you add an arbitrary fluctuation, so this leaves them in the full endomorphism algebra, then you can consider the inner derivation generated by this fluctuating background. And if it acts on a scalar field, for example, then essentially this is a covariant derivative operator. If you work it out, you see very quickly, it's just this thing. So the objects are naturally in their joint representation here always. And then, if you compute the commutator of two fluctuating background matrices, essentially this is the field strength of a Young-Mills connection. So the fluctuations, think of them as a Young-Mills gauge field. And then the action, of course, is essentially Young-Mills type of gauge theory. And for free, you get the inhomogeneous transformation law of these. So this transformation arises from the trivial, sorry, from the trivial transformation in the adjoint of the full matrices. Okay, so this is very nice and very standard. It's written for a more alive, but essentially this goes through for all fuzzy spaces. Okay, so that's one thing. But on the other hand, so these are geometric matrices. So fluctuations of these background, they should somehow also describe dynamical geometry because they describe the geometry itself. So this is a kind of a very interesting duality between gauge theory and fluctuating geometry here, which has fascinated me for a long time. And there is a number of people who have studied that, starting with Rivelles and Young a couple of years ago. And let me very briefly discuss here what's going on. So the question is, if you have this background, so it's a generic symbolic manifold in this matrix model, and if you let it fluctuate, what kind of gravity-like theory does it describe? Or does it have anything to do with gravity? Now the first observation is the following. For very generically, you have this matrix Laplace operator and you can always rewrite it as a Laplace-D'Alembert operator with respect to an effective metric. And the effective metric is exactly the kind of thing which the string U is considered as an open string metric. And so that means any background describes a geometry. And obviously if you fluctuate the background, you will also get the fluctuation of the effective metric. And in fact, there is a nice, very fascinating observation by Rivelles first, quite a long time ago, is that if you do this on my alveol, in fact, you do get two richer flat metric fluctuations. It's quite kind of remarkable. But the thing is you don't get the full Einstein equations. So this doesn't really, at this point probably, yeah, I mean, I think I'm still not totally sure, but I don't think this will give you real physical gravity, or at least there are some issues. But first of all, the matrix model action will certainly not give you the Einstein equations. So if anything, you have to appeal to induced gravity, ala-sacarov, which means you quantize it, and then of course you get an induced Einstein-Hilbert action, and then it might work, but there are issues, there are problems, and two of the problems that you find here are the following. The first problem is that there is always this symplectic form, and the symplectic form of course breaks Lorentz invariance. Now, this is actually not as bad as it seems, because there is no charged object under this thing, and in some sense it's just absorbed in a metric. But nevertheless, if you go to one loop, the loops, loop computations, they will contain all very high energy modes that you have. And at high energies, then you really cannot hide this thing anymore. It will show up in the loop contributions, unless you have maximal supersymmetry. And then you kind of, well, you will induce also such terms in the effective action. So if you do the heat kernel computation, which is in one explicit example, you really get these kind of terms also, not just the induced Einstein-Hilbert action, and then this doesn't look very nice, I mean, you don't really want to have this probability. By the way, what means the two in the two richly-flat metric fluctuation? What do you mean? Two independent transversal traceless, two physical modes of the gravitons. And we are in rich dimension before. So these are the good degrees of freedom. Yeah, in principle, that's fine. Yeah, exactly. That's why it's fascinating. So far so good. Not the full Einstein equation, if you have two. Okay, so no, no, okay. So what I mean is, with full Einstein equations, I mean the inhomogeneous part. So if you put T-mini on the right hand side, it will not get the Einstein equations. That's what I mean. But that part is fine, at least seems fine. Yeah, so one of the reasons, one of the issues is that you will generically get this. And of course, you'll also get a huge more cosmological constant of, okay. Anyway, so for me, this is the main issue here. And I would like to, there are many reasons why I don't really want to have explicit symplectic structures. So you want to get rid of that. And the point is, there is a resolution to these problems. And I think both of these issues seem to be resolved on covariant quantum spaces. And that's what I want to discuss here. So what is the notation m for theta? Is this correct? Where? Oh yeah, okay, sorry. Forget the data. It means that's a quantized symplectic space. Theta is the Poisson structure. And that's the four dimensional quantized symplectic space arising from a Poisson structure, Poisson tensor theta. So it should be symplectic omega, whatever. Okay, so now let me explain the covariant quantum spaces. And I really want to be in four dimensions. So again, the issue is, in four dimensions, any symplectic form of course breaks, does not admit the full local Lorenzo Euclidean invariance. And it seems impossible to get around this. But in fact, there is a standard example well known in the literature, which is the four dimensional for this sphere, S4n. And that is completely covariant under SO5. So how can this be? And this space was introduced long time ago by Gorsuch, Klimtrick, and Bresnider. And it's very interesting object that has generated a lot of literature. But the thing is that the space of functions on this thing is much more than what you like. And it turns out, this thing really describes a high-spin gauge theory. And it's kind of a complicated object. But that's what it is. So this thing will lead to a high-spin gauge theory. But I want to skip, for lack of time, I want to skip this case. And I will jump right away to the non-compact case of the four-dimensional Fossi hyperboloid. That is completely analogous to this one. But it's non-compact. It will also lead to a high-spin gauge theory. But this is more interesting because if you then start with this thing and apply a suitable projection, then you get a space with Lorentzian signature. And that's the interesting space I want to focus up on. And this is a cosmological type of spacetime. And that's an interesting object, I believe. And we'll study fluctuations on this space. And this will indeed contain spin-2 modes. And that seems to lead to pretty close to real gravity. I have a question on the other case. Can I ask it if not? Well, you have all these references, probably. They did some kind of work on the Fossi formula. How about this one point-factor reference? Which one? Zang Hu 2000. Yeah, this Zang, that's a paper actually on the four-dimensional quantum holography. It's a very interesting paper. So how is the connection? No, boy. Not very quickly. But I mean, it's a non-Abelian version of the quantum hard effect in four-dimensions. And it exploits exactly the extra structure that you have here. And it's kind of near-Saudi-Landau levels and things like that. It's a very nice paper, I can. But let me not try to explain it in more detail here. OK, so I will try to focus on this part. But I have to start with the four-dimensional Fossi hyperbolic weights. OK, so what is this object? So you start with SO 4,2, conformal group. And so let me call the generators MAB of this conformal group. This is the Lie algebra of SO 4,2. And now for Fossi spaces, it's not just about algebra. You have to choose a representation. And it's very important to choose the correct representation. The correct choice is given by the simplest unitary representations of SO 4,2, which they are. And those are the so-called discrete, short-discreet unitary series of representations. So in physical literature, they are known as mini-reps or double-tones. And they come parameterized with a positive integer N. And these are very special representations. Some of the special properties are the following. So those are not just irreducible under SO 4,2. In fact, they remain irreducible under SO 4,1. And that will be very important. All of their multiplicities are one. And they came obtained by a minimal oscillator representation. So by all accounts, these are the simplest representations that you have. They are also known as holomorphic series, I believe. And they are the highest weight representations. And there is a particular generator, which is kind of an energy generator, if you like. And this has bounded below and then goes in integer steps. And in fact, so at the lowest weight is an N plus one dimension irreducible over SO 2L. And in some sense, it describes a fuzzy two-sphere, actually. OK, so that's the kind of structure that we need for these representations. And then, how do we get the hyperboloid? Well, you take from these generators, you fix one index, put it one time-like index. So five, remember, was the time-like direction here. And then call this thing xA. Now, you can compute this thing, the constraint. And the AB is now the SO 4,1 tensor. And this constraint is now actually unit matrix. And that expresses the fact that this is an irreducible representation of SO 4,1, and not just of SO 4,2. So this is why you have to choose these representations. And obviously, it means, well, this describes a hyperboloid, one-sided hyperboloid. This is the time-like direction. OK, so that's that. And then, if you compute the commit data of these generators, you get the remaining generators of SO 4,1. And this is what I usually call theta. It's the same thing. It's just the algebra generator. So by the way, this means from a structural point, if you, this is something like a Snyder space, for those of you who were related to that. Is that the quadratic Casimir of the subalgebra? That is no. Which one? No, of course not. No, this is just, this is a SO 4,1 invariant constraint. And because it's irreducible, it is really proportional to the unit matrix on this representation. So these are very special. Anyway, so this means, this is kind of one-sided hyperboloid. Of course, this is a Euclidean space. It's obvious down here. So this is not Minkowski. And it's covariant under SO 4,1 by construction. This, by the way, is known as Euclidean ADS-4. OK, that's that. And then, well, actually, it's interesting to notice that there is a nice oscillator construction of this object, of this representation. Namely, you start from four bosonic oscillators. So those are spinorial representations of SO 4,2, which is SU 2,2. And take a fox-based representation of these guys and fix the particle number. So this is the particle number n of the fox-based representation of these guys. And that gives you precisely these shortest create unitary representations that I was talking about. Because now, if you know that, then you can, that means that the generators of the Lie algebra in this fox-based are just given by these sandwich operators, as usual. And the x's are given by this object. This is the gamma matrix of SO 4,1 now. And that tells you what's really going on. Because that tells you, well, that space is really quantized, CP12, which is just a non-compact version of CP3. So it's a six-dimensional complex symplectic space, which happens to be a S2 bundle over H4. Why is that the case? Because here you have four complex operators, so like four complex numbers. And the number is fixed, so it means the radius is fixed. And somehow, this way, you see that this is CP12. And it's also co-adjoint orbit. So however you look at it, that's what it is. And it means, so it's not just this hyperboloid, but at each point of the hyperboloid, there is sort of internal two-dimensional sphere. That's really the geometry that's going on. And the full space of functions that you have, that is realized in the n-emotion algebra, is really the space of functions on the entire bundle and not just on this thing. This is why it's a more complicated object than usual symplectic spaces. OK, but actually, this is interesting and nice, because what does it mean? OK, if you start now study fluctuations, so functions on this full object, which is the bundle, so locally it's just a product of these two spaces, it means functions here will be functions on H4, which are also harmonics on this internal two-space. So it's something like a Calusa client theory, but it's different in a very crucial way, because this is an equilibrium bundle. So the whole bundle transforms under SO4 comma 1, and the SO4 comma 1 acts non-trivial on the internal space. And that means if you do a harmonic decomposition of this internal space, these functions will transform non-trivialy under the stabilizer group of wherever you sit on this thing. So the local Euclidean rotations here act non-trivialy on this internal thing. This means that the harmonics here are actually higher spin modes from the four-dimensional points of view. This is the reason why you always get the higher spin theory here. So if you sort of think of these internal excitations here, then they will sort of transform non-trivialy under local rotations. So that's the reason why I get higher spin theories here. Let me make it a little. Is there some symplectic vibration? So is this board downstairs also symplectic? No, yeah, very good, very good. So not only the bundle space is symplectic, this H4 is not symplectic, not at all. And in fact, what happens? So if you work out the four components of the symplectic or of the Poisson tensor field, and it rotates around the sphere, and if you average it, it cancels completely. So the four-dimensions of Poisson tensor is completely canceled, it's averaged out. And this is why I can have full covariance. That's the whole point here. OK, so we expect to get some kind of a higher spin theory. So let me make it a little bit more explicit in the fuzzy case. How do you work with this in the fuzzy case? Because you have matrices. Turns out a very good way to work with it is using coherent states. Now, this is a quantized quadrant orbit, so there is a natural coherent states. And let me denote them like this. And that means you have a quantization map, which means that, OK, you want to understand the endomorphism algebra. Endomorphism algebra here for the non-compact case, probably you should go to Hilbert-Schmidt operators, but anyway, that's in detail. Now, the point is you can realize all of these operators in this kind of a symbols with these sort of classical functions on the bundle, and here you just have these coherent states. And this you can now decompose into higher spin sectors. And in this sense, from the full space of fuzzy functions, you get sort of a tower of higher spin modules, if you like in a semi-classical sense, but it turns out. Now, yeah, so one important thing is this tower goes only up to n. So it doesn't go to infinite spin. It goes only up to n. This is a typical phenomenon that you have for these fuzzy spaces. It's like for the fuzzy spheres. Same thing happens here. You have a question? So are you only get the integer spins? Yeah, you get integer spins here. You also generate half integer spins. Of course, if you do that from your x-sector, you will get half integer spins. I'm only focusing on the bosonic sector here. But from the bosonic sector, no, that's what you get. OK, now explicitly, so what is C0? So these are scalar functions on the hyperboloid. These are just any functions that you get if you've write down polynomials or whatever other functions of the x-generators. Now, the first non-trivial one is the C1. Now, C1 is if you have not only functions of x, but you add one of the extra data generators. Now, data turns out to be a self-dual do-form, and you can be realized or represented by this kind of a rectangular young tableau. And more generally, the most generic or general object here in the end of awesome algebra is a function on the hyperboloid taking values in n of these generators. And this n of these generators you can represent by this kind of a rectangular 2 times 2 line young diagrams. And that actually happens to be precisely the kind of thing that you see in higher spin type of theories. And the reason why I get only these diagrams is there are kinds of constraints here which I haven't written down yet. But that's precisely the content that you find here. So bottom line is you get here higher spin modes. These higher spin modes are really sort of would-be kaluzak line modes which arise from these internal modes on the S2. OK, so that reason normally makes sense. And by the way, we will see, so why do you call this? This is really spin 1, even though it comes in this disguise. But you'll see later on why it's really spin 1 and so on. OK, now, but that's actually not really what I want to do. I want to do Laurentian stuff. And we'll see. Starting from what I just told you, you do some little projection. And then you get a homogeneous esotropic treatment for the mental robots and walker universe. And so we'll discuss the high spin modes. So how do you get the Minkowski signature? It's extremely simple. You just do a projection. You do a projection in target space. So this hyperboloid, which I just told you, is actually a solution of the model. And it just needs five matrices, which means it's embedded in our 1, 4 target space. Now you project this 1, 4 target space along one line transversely like that. So here is your starting hyperboloid. It just quash it onto the plane here. And because the time-like direction goes like here, it's kind of obvious, optically, from looking at it. This is going to have Minkowski's signature. And in fact, it will be sort of a two-sheeted squashed thing. And algebraically, this projection is realized very simply. Just drop one generator. So I had five generators before. Just drop one Euclidean 1 and keep four of them. And this describes precisely this project of a projection. And it's very simple to see that these are solutions of the model. OK, so that's the solution I'm going to discuss. It looks extremely simple. It is extremely simple. So OK, what are the basic properties? OK, now I've flipped the picture. Sorry. So this is now the time-like direction. So what you find is this kind of a two-sheeted thing. And OK, first of all, there is a manifest SO3,1 symmetry. Because I've dropped one generator, so it's no longer SO4,1. It's only SO3,1. And this SO3,1, that is the isometric group of this space-like three-dimensional hyperboloids that you have here. So this separates, this kind of gives you a separation into three-dimensional hyperboloids. And that's exactly K equal minus 1 Friedmann-Robertson-Walker space-time, which happens to have a double cover. And it means that, yeah, so the metric just for the symmetry, any metric with the symmetry, you can always write in this kind of form with this is kind of a cosmic evolution parameter now. Now, if the effective metric was just the induced metric that you see here, it would be trivial. It would be the Milner universe. And then this is kind of boring. But it's not. The effective metric is different. But it will always have this kind of form because this symmetry is really manifest. OK, so it will be some kind of a K equal minus 1 cosmology. You may not like this, but anyway, let's just go ahead. Now, OK, let me, from now on, I will work at the Poisson level. And now, OK, let me be more explicit. So if I work on this space, there is no more SO4,1 symmetry. So I better write down every single language which is compatible with the symmetry, which I have, which is SO3,1. So let me separate the generators as follows. So the x, x mu are the same object as before, but mu is now from 0 to 3. Now, there is another vector operator in the d algebra, which is this one, where I fixed one index to before instead of five. And let me call this T mu. And then the Poisson algebra is this one. So T comma x, this is something like canonical commutation relation. So the T's are something like momentum generators for the x, because there is an eta, but not quite. There is a sinh factor, and eta is, this is the cosmological time parameter. And you cannot get rid of that. So that's here. The x comma x is theta, and the d comma t is also theta. So this is the Poisson structure. And then there's a bunch of constraints. We can work them out. This follows from the representation. So that's only if you have chosen the correct representation. OK, whatever. Let me focus on these two guys. So the T's. So you have T times T is, OK, it's a function of time. And T is orthogonal to x. So that means that the T describes a space like two-dimensional sphere. That's the meaning of the T. And that's exactly the fiber structure that I told you before. So now I have this fiber bundle explicitly over each. So x describes the hyperboloid. Oh, sorry, the squashed hyperboloid. And you still have, at each point, you have this two-dimensional sphere. And that's precisely described by the T. And then there is a bunch of other relations. But there shouldn't be any more independent generator theta because that's the whole bundle. So in fact, you can express theta in terms of x and t like that. That follows from the self-quality. So in fact, you can forget about data from now on. Really, you have x and t. And that describes the whole thing. OK. Now, OK, so let's again look at the higher spin modes. So the higher spin modes are these guys which have s explicit theta or t generators. Now, you can look at these. Somehow, you have to learn how to work with this. That's the thing. And there are two useful points of use here, it seems. First of all, you can still view everything as functions on the hyperboloid h4 because that's what everything comes from. That's useful because you have more symmetry. And this is more powerful. And then you can actually represent each such object explicitly as a totally symmetric traceless divergence free tensor field of rank s on the hyperboloid and vice versa. So there is a nice mapping going back from the usual tensor calculus to this kind of funny encoded thing. And that's just using the Poisson bracket. It goes as follows. OK, if you start with the totally symmetric, blah, blah, blah, tensor field theta, you take Poisson brackets with the x generators and contract them. Then you get an object in this algebra. Fine, so now this lives in this spin s sector. And conversely, if you start with this higher object and take some explicit Poisson brackets, you recover your totally symmetric tensor field. So it's just a different way of encoding higher spin fields. So that's all very nice. But on the other hand, this is a bit, you really want to work on the Minkowski signature. And here you have only less symmetry. You have only s of 3, 1 covariance. And there is another natural way to represent your objects, namely, in terms of the t generators. And kind of obviously, now, a spin s object is something like that, which has s t generators of type t. And that's obviously, again, it's a symmetric tensor field. These are a little bit different. It's quite nontrivial to go from here to there anyway. But for many points of view, this is more useful. The reason is because you have this constraint. So t is orthogonal to x. And that means the t's tensors, there is internal gaging variance. And you can, in some sense, you can choose these guys always to be in space-like gage, which means that they have no components in time-like direction. So these are really space-like tensor fields. And that's very important. Because if this wasn't the case, you would have ghosts. And you'd have big problems. That's one of the issues that it has here. And that's very nice. Because so that means from these internal degrees of freedom, you won't get ghosts. So this is good. OK, that's one thing. All right, so now let's go ahead and study the fluctuations. So first of all, we have the background solution. And now actually, I will sort of change my mind. And I will discuss. So the x generators are solutions. But the t generators, they are also solutions of the model. In fact, I want to focus on the t solutions today. So this is actually kind of the momentum picture that you like to advocate. So you may like this. The reason why, I will explain the reason why I do that. So first of all, it's a solution because you have a master. So you need a master here. But the nice thing is that if I take this as a background, then the effective Laplace operator respects the spin. And so I have always this separation into higher spin sectors. And for this background, this is precisely preserved even at a noncommutative level. So there is a spin casimir you can write down at which exactly commits with that. And it means that all this nice separation into higher spin sectors, you can rely on that. Now that doesn't seem to work on the x solution. That's the reason why I sort of switched to this picture. So I'm not sure what happens in the x solution. But this thing is nicely under control. So let's go ahead and start. All right, so then we can expand this into these higher spin modes. And I told you before that any of these box generators encodes a Laplace. In fact, a Dallon-Bear operator here. And you can work it out. So you can work out the effective Friedman-Robertsen Lemaître-Walker metric here. And it turns out, OK, you can sort of do it almost explicitly. At late time, what you get is sort of an asymptotically coasting universe. So you may or may not. This is not, of course, the lambda CDM that you get. But somehow, I believe this is not too bad in some sense. It's kind of a reasonable approximation to what we see. And there is also singularity in the beginning. And it's a very strange algebraic singularity that you find here. But it really is a singular thing. So it's a bit strange. I don't want to discuss if this is realistic, probably not. But anyway, it's not totally crazy. And it's something interesting to play with at the very least. It's flat space time in this guy? No, it's not. It's not flat. No, it's not flat in any sense. OK, a flat that would be Milne, but it's not. OK, so now what about the fluctuations? So as I told you before, this is the model. You plug in your background, which is the one that I just told you. You could do this for any of these, but now we will focus on this Lorentzian cosmological background and study the fluctuations. And the fluctuations, OK, there is a gauge transformation. I will discuss this a little bit. You can expand the action to second order in these vector fluctuations, and then you get some kind of a vector Laplacian. And that's what you have to diagonalize in order to get your physical fluctuations back. So I do this now again. It's sort of what I did before in Young Mills, but it's better to start from scratch again. Because the reason is because these are now higher spin-valued fluctuations. So that's not something very familiar. But it's a well-defined object, and it's an operator which is good, which you can diagonalize. And that's what we did. And it's actually quite, it's not easy to diagonalize. This took us a long time. And the only reason why you managed to solve it is because there is this big underlying as a 4,2 symmetry. So that's extremely useful, even though it's not a symmetry. OK, so the point is the following. So we take this background. And so we make the following ansatz for solutions. There is two types of modes, a plus and a minus, which are sort of parametric. You take sort of a generic spin-as field in an anamorphism algebra. And out of that, you make a vector fluctuation. And this is done in this way. It's using Poisson brackets with the background solution. But these are really the x's and not the t's. So the point is, combinator with Poisson brackets with x, it raises and lowers the spin by plus minus 1. So you can project either on the upper part or in the lower part. And in this way, you get two independent modes. And it turns out these are eigenmodes. So the vector operator, I can disguise, kind of these are intertwiners in some sense. It's not totally obvious, but they are. And then you can sort of put the differential operator underneath here. And then it means that you have, I have diagonalized the d-square operator. If you find eigen, it just need to diagonalize the box operator. So this is a scalar box operator, basically, on my space of function. And that's something which is well under control. That's just representation theory in some sense. So that means we have found two independent physical modes, propagating modes here. And that, now you can sort of define the physical Hilbert space of this theory. OK, this is the usual d-square equals 0. You have to gauge fix A of, by the way, yeah. So I didn't tell you the gauge fixing. But there is a straightforward way to gauge fix. And you can do all that. And you have to factor out the pure gauge mode. So this is very much like Young Mills because the whole theory is formulated like Young Mills. And I believe that we have the two physical solutions of this thing. Now, on top of these two physical solutions, there is, of course, a pure gauge mode. And the pure gauge mode always has this form. So now this is for some bracket with t, not with x. And that's obviously also a solution. But this is kind of trivial, of course, that's factored out. But then there is, now, if you go off-shell, so now I'm discussing this off-shell, there is, of course, a fourth mode because I have four vector of luxuration. And the fourth mode is a time-like mode of shell. And here, we were not able to diagonalize these things. But it looks very much like there is no solution of this thing. So this is really just an off-shell thing. And there is no on-shell time-like solution. That's a conjecture. So I cannot prove this at the moment, but I think it's correct. And if this conjecture is correct, and there is some other things which are, there is a little bit of a mix in here which I don't completely understand. OK, so there is a conjecture that there are no ghosts here. And I think the conjecture is reasonable, but we haven't established it. Are you saying for each spin, eh? For each spin, yes. Yeah, for each spin. Four modes, finally? No, OK. There are two physical modes. And the plus and the minus. Yeah, so there are two independent physical modes for each spin, yes. In this, somehow, in all the fluctuation spectrum. And off-shell, there is two more, but this is just off-shell. Up to any quality is cut off. And the. Can you put back the previous transparency? You wrote it down, but. Yes. So D squared is the Laplacian. Yes, that's the vector Laplacian, yes. And why the negative mode? What's the negative mode? Plus and minus, what are these? Yeah. No, what do you mean? Yeah, there are two modes. No, but what's the meaning of that negative mass? Well, so first of all, this is a cosmological scale parameter. So if, I mean, you're right. So it could be that there is a slightly negative thing, but if it's then it's only a cosmological negativity, which is maybe not so bad. So it would be mildly. And the value of the cut off is what? Anything, any fixed value. So it depends on representation. No, no, it's independent. No, no, sorry. Yeah. No, well, of course, I mean, the cosmological curvature, of course, it can't go to zero at late times, whatever. So n is fixed once and for all. So in early times, you're right. In early times, there may be a substantial instability here. Yeah, so r is fixed, but there is another. The cosmic scale parameter is hidden here. So if you work this out, it really there is a one over, yeah, something like that. So at late times, whatever is here certainly goes to zero. But at early times, there may be an issue, yes. Yes, OK. Anyway, so that's what we find. So there is this would be dachions, yeah, yeah, yeah. That's the issue which may play a role here, yes. I'm not sure. It depends on this cut offs and so on. OK, so that's what we have here. OK, there is good hope that there are no ghosts, but there is in fact, there is a general argument by Hikaru. The point is that you have only one time like matrix and you can always diagonalize. And then it's sort of, you cannot really just support the propagating time like mode, but the argument is not completely at tights. And I think one should make sure that this is true. Anyway, so there are two propagating degrees of freedom for each spin. Now the point is that the propagation is really for each of these spin modes, the propagation is governed by the same operator. And this one really has kind of is a fully Lorentz, is an ordinary wave operator. So this thing is really the propagation respects sort of Lorentz invariance, even though there is no manifest boost invariance in the model. So that's, thankfully, this thing works out so far. And so in principle, because there is a cosmic background, the cosmic background in some sense defines your time like vector field. And it could have happened that this thing enters in the propagation. That would be a disaster, but it doesn't happen. OK. All right. Then very briefly, something about gravity. Sorry? Yes. OK. So let me sketch it. So first of all, how do you extract the effective metric that's always done from the kinetic operator and is sort of a straightforward procedure to do that? If you have the effective metric, and of course, and then if the background is fluctuating, then also the effective field band and the effective metric will fluctuate. So that is, you can get the linearized gravity in a straightforward way. And you expected this metric fluctuation that you get in this way couples in a standard way to energy momentum tensor. Now, you have to be careful about the conformal factor to really get rid of the effect. The conformal factor, I didn't discuss properly yet. And one should do it, and one can do it. And there is a sort of, you can work out precisely the conformal factor, how this is determined by the background objects. And once you do that, then you get this kind of coasting and initial geometry. But let me skip that. Now, what about rigid densors and curvature fluctuations? So let me consider this linearized fluctuation to the effective metric. So this is the kind of the graviton or the gravity fluctuation. And then from the mode analysis, which I just told you, it follows you have two physical modes, which enter into the fluctuating metric. One of them looks very much like a massive graviton a priori. But it's different because the symmetry is broken to SO3, 1. And there is another one. And there are three pure gauge modes rather than four. And I will explain that. Now, if you compute the linearized rigid densor of these guys, you find, OK, you can do that. At least now, these computations here are done up to cosmological scales. I don't have exact results. But the result is the following. So from these degrees of freedom, there is a special sub-sector which corresponds to the physical degrees of freedom of the graviton modes. And they really seem to, well, so they are richly flat up to possibly cosmological scales. And so this is the usual graviton sector or gravity fluctuation sector as in GR. The other degrees of freedom, which you at first look like they come from a massive, they would describe a massive graviton, they don't, because if you compute the rigid densor, the rigid densor is still zero. And I think it means that these are actually, at least their contributions to the gravity, to the metric fluctuation is trivial. So the effective metric fluctuations here are only those of massless gravitons, not of a massive one. It was not obvious from the beginning. But I think I believe that this is the correct conclusion. But there is an extra scalar mode. In fact, there are two, but again, they lead to identical rigid densors. Anyway, so this needs to be cleaned up a little bit, as you can notice. But I think the conclusion is you have two propagating gravitons as you like to. And there is an additional scalar mode, which you don't have in GR. So it's not the same as GR, but OK. It's not too far either. And finally, let me very talk about the gauge transformation, because I think that's interesting. Now, I would like to have different morphs and invariants, of course. Now, different morphs and invariants should arise from the spin 1 gauge transformations. And so these are gauge transformations which have this canonical form, but where the generator is a vector field. And vector fields are naturally realized in this sense here. But the point is that the t's satisfy a constraint. So the t's here, they don't have a time-like direction. So this is why I have only three independent vector fields, rather than four. And the underlying reason is really because there is an underlying symplectic volume. And the underlying symplectic volume is preserved. So you cannot have four different morphs, and you can only have three. That's reasonable. And in fact, if you work out how these three would be different morphisms act on the metric fluctuations, which I just told you. And now I have to take into account the correct conformal factor that I didn't tell you before. But if you take into account the correct conformal factor, you see that these spin 1, these three different morphs will have exactly the form that they should have in GR. So in fact, this I found only last week. So I'm very glad that this worked out. So that's just a nice, consistent check of the framework. So it is a little bit like unimodular gravity, but it is not. But because the vector fields here that you have here, they're not quite volume preserving. They satisfy this constraint, which at the late times becomes almost volume preserving, but not quite. So this speaks to the time-like component of the generator. So anyway, you have three instead of four. Different morphs and invariants. And finally, given this gauge invariant, you can ask what is the linearized Einstein-Hebert action. And again, that's somewhat complicated. And there are more those kind of additional terms you have to write down. Well, so I think I found it. The leading term is that of the ordinary linearized Einstein action, but there is a lot of extra term, correction terms, whose meaning I don't understand. But they all seem to be suppressed at cosmological scales. So I think the leading term is Einstein-Hebert. With the one third, I mean the. Yeah, no, the one third is completely misleading. H is essentially 0 up the corrections. Yeah, this is confusing. It confuses us, but now I understand it. So H is something like 1 over cosmological curvature. So it actually, it makes sense, even so you don't see. And so this is the action that you would expect is automatically induced by quantum corrections, because as always. And then, so you have the metric degrees of freedom that you need in gravity. And as soon as you also have this term, then you expect to get at least linearized gravity out of this model. So it's qualitatively summarizing. So the pure matrix model, even without quantum effects, will give you a Ricci-Flat vacuum solutions. Now if you add this Einstein-Hebert term, you know, you should still have the same Ricci-Flat vacuum solutions. So that seems to make sense to me. But you need the induced Einstein-Hebert term to get the inhomogeneous Einstein equation. So the inhomogeneous Einstein equation, you do not get from the bare matrix model. There you need the induced gravity term. But I think it's reasonable. So I would expect, it's reasonable to expect to get at least linearized gravity at intermediate length scales. For sure, you will not get it at cosmological scales, because the background is not a solution of GR, but it's a solution here. And so there is a chance, at least, that is the solar system that's maybe OK. At the nonlinear level, I have nothing to say. The model is nonlinear, but it remains to be understood. Another important thing is that there is no invariant cosmological constant. So I cannot prove it, but we looked for gate invariant solutions. And I think the cosmological constant term is not gate invariant here. There is no such term. It doesn't exist. And that's, of course, very interesting. And in some sense, this term is replaced by the matrix model action, which is really a Young-Mills action. And that has a different meaning. And that matrix model action actually supports your cosmological background without any fine-tuning. So for sure, at cosmic scales, there will be significant differences. And I think it could be very interesting. And I think there is probably less need for fine-tuning here. Let me skip this. And OK, so just very briefly. So the point was that from matrix models, this is a natural framework to start with. And it will give you a quantized theory of spacetime and matter. There is a nice class of four-dimensional covariance spacetimes, which seem to be reasonably healthy and will not have certain pathologies and that typically lead to higher-spin gauge theories. I showed you some reasonably nice cosmological Friedman-Horwitz spacetime, which features a regular big bounds and a finite density of microstates, just by construction. Fluctuations, in principle, contain all the degrees of freedom that you need for gravity, even so you have to appeal at some point to induce gravity probably. So it's more like emergent gravity than quantized GR. I would expect to be good because it's this maximally supersymmetric Hamilton type of action that you started with. So it's at least well-suited to study quantization, but it needs a lot of more work just at the beginning. OK, so let me start here. Stop here. Yeah, I was thinking about the signature of this affected metric. Is it clear that, I mean, when you start with some background, of course, you get something. But is it clear that sort of dynamically you get the right signature? What do you mean dynamically? And when you, I mean, if you start, if you don't perturb around the background, I mean, you find solutions to the equations of motion. Could you have different signatures? You mean with different? Yeah, yeah, sure. It's true. Yeah, I could have different signatures here. Yes, yes. For example, you can have sort of a brain solution which is embedded, which is more like an instant on. You could have that, yeah. So you would make sure that your sort of four-dimensional world that lives inside it has the right. Yeah, so for the moment, I've just chosen interesting solutions rather than others. Yes, yes. Anyway, you are optimistic by saying that you have apparently master for your rabbit and then you say they are cosmological scale. So I don't worry. But you know that after even 30 years of work on massive gravity, massive gravity theories have very different properties for much like theory. They have usually six or four goals. The Wachler effect does not work, you know. So one is very far from good. Yeah, yeah, so I hope that that's OK here. Yes, that's the last question. So you were talking about the six-dimensional co-event orbits, which you call CPU-1, which is a sphere bundle. But the H4, you said, is not, say, a symplectic reduction. No, no, it's really not, as far as I understand. So the whole thing is symplectic. The whole bundle is symplectic, yes. Yes? Sorry, only semi-scientific. You quoted when you started talking about the space you discussed, the work of Hasebe. And the question is, is it more or less, one question is, in what context you thought of it? And secondly, is it based, it's very similar or almost the same you found? Or what are the subtle differences? OK, no, Hasebe has a paper, which is very generally discusses all kinds of this hyperbolo in different dimensions and all kinds of things. And at some point, very briefly, he mentions precisely the same, but only a few lines. So in that sense. Another question. So in the emergent growth, you mentioned whether you had some matter in your isolation equations. But is this related to the fact that you started with that your background was sort of matter-free? And should you consider some kind of background which contains matter to get there? I'm not completely sure. I understood the question. Matter? Yes, you start now with a sort of vacuum background. Yes, yes, yes. Oh, yeah, yeah, OK. What did you have also? Yeah, yeah. Yeah, there is, you know, there's all this fermion. So in principle, the model contains all kinds of matter degrees of freedom, which I have switched off for the moment. In principle, it's there. And then, of course, you would understand how does this matter influence the gravity. But in principle, your Friedmann background, as matter, it was otherwise. That's why I asked whether it was flat. No, this solution has no matter whatsoever. So that is a solution without any matter. So that's still the Friedmann. Usually, time exists only if. In GR. But this is not GR. So that's really, that tells you. So it's not part. The gravity is not the background Jimmy knew has nothing to do with Einstein. The background has nothing to do with Einstein. Only the frontier. Yeah. Yeah.