Okay, well first let me thank, let me start by thanking the organizers for the kind of invitation.

It's a pleasure to be here, my first time here in Bühr. And it's especially because the team of the workshop coincides precisely with my favorite topic. So this is very, this is always nice. So I would like to discuss in a particular instance how our quantum space-time 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 it's for the first time. I think it's sort of it really seems to work at least more or less. But before I go into any details, so you know

you can always sort of step back a little bit and start to dream a little bit. So and you know if you think what could be perhaps a formulation of quantum theory of space-time and gravity. Of course everybody will have a different point of view and different preferences. But to explain the motivation, so let me let me tell you a little bit my personal guidelines.

So first as first I would say it should be simple. I mean for the plain reason that otherwise we would never find it. Okay, now it should probably be a gauge theory in some sense because in Minkowski signature there's always the issue of time-like components and they 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 finitely many degrees of freedom per unit volume because there is a blank scale and then hopefully gravity becomes strong and so on and 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 at long scales to something like geometry. In particular, so the underlying degrees of freedom should be non geometric.

Okay, let's assume that for the moment. And the other thing which at least for me seems reasonable is that you know gravity, we only know gravity in 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 of

observations. So something like, you know, gravity is perhaps something like Navier-Stokes equation and you would never start to quantize the 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 are 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-commutative 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 looking a little bit more problematic and we have already heard in this morning a

lot about these issues and I will not, I will not really talk about the quantization here anymore. So I will here, I will focus about semi-classical aspects and how to get that but of course this is very important in the back of the head 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, you know, you could 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'll 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 supersymmetric 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 if, yeah, it's almost, you're almost forced into that, almost. And then it turns out this shares actually a lot of features of string theory, but I would say, okay, it can cut away the landscape, which is also good.

But then, yes? I will come to that. That's precisely that to be matrix model. It's just a different name. Okay, so so good so far, but then in the end, of course, what you really, the really interesting thing is what about gravity? So how can you recover gravity from such a framework? And that's what I would like to discuss.

Okay, and so I will actually, let me start with the conclusion. So this is what I want to discuss in this talk. And this is kind of, this is contained in a series of recent papers, but it's really this one that I will focus on. So one result will be that, okay, so I will discuss a particular three plus one dimensional

covariant cosmological spacetime. So this is a Friedmann-Lemaitre-Robertson-Walker geometry, but a fuzzy one, and it happens to have a big bounce that just comes out. 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 Vasilyev theory. It's a little bit, it's 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 linearized level. Okay, 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 invariants will be only partially manifest. So, I will explain this in more details in particular, the boost 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 okay, so here not everything is settled, but essentially it looks like there is, the metric perturbations describe massless gravidons 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 Einselt-Hilbert action, it just comes from the matrix model. Now, but then at some point, it seems that you do need something like an Einselt-Hilbert action, and there is this thing, so I will write down the linearized analog of the Einselt-Hilbert action here.

And the point is that this is expected to be induced upon quantization, you don't have to add it by hand, it will just arise. Okay, so this is sort of roughly the summary of the, the physics summary of the talk. And okay, so chronologically first, I will briefly discuss the model,

but I can be very brief here, because we've 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-commutative. And then we'll discuss the fluctuations on this background, and these fluctuations will lead to a higher spin gauge theory. And in particular, then the most interesting part, some are of course the metric fluctuations, so spin two part, and we'll see as I discuss that, and very briefly.

Discuss the linearized Einselt-Hewitt-Hilbert action. Okay, so let me start with the model. It's exactly the same as we saw this morning. It's the type 2P or IKKT model. IKKT 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, which is very nice. But actually, okay, I do a little bit, I change it a little, a little bit by hand.

I put a mass term. And, well, why, you will see the reason why I do that, simply because I want to have certain solutions. And the essence is putting a mass term behind, it really puts in a scale,

so this somehow fixes the scale. Without that mass term, 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 okay, let me do it. It's something like a soft suzy breaking term that you put into some model. So I think it's not too bad, the modification, and it doesn't break, well, it doesn't break any of the symmetries except for supersymmetry, softly.

Okay, so what are the ingredients, once again? So you have... There is also an eta-a-b in the master. Yeah, yeah, yeah, okay, 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 though actually I will then go to the infinite dimensional case, right away. There is gaging variance, of course. There is a global SO9-1 symmetry, so that means this is just the eta symbol of SO9-1. These are the gamma matrices of SO9-1, and these are Majorana-Weier spinors, and that's pretty much it. And in principle, if there wasn't a mass,

it would have maximized supersymmetry, and this is kind of a soft suzy breaking term, if you like. Okay, and again, this can be seen to arise in many points of view. It arises from the quantized chill action for the type 2p superstring. It simply arises from dimensional reduction of 10-dimensional superangles to a point.

And okay, 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 contracted with eta by this box symbol, because it is the matrix analog of the D'Alembertian or Laplace operator, if you like. Okay, 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.

Okay. So then the strategy is the following. Okay, 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 you 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 for so far.

And second, but why is it not quite as bad? Because there is actually, of course, there is the underlying matrix model, which is defined at the non-perturbative 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 non-perturbative aspect. But for the moment, okay, so now let me just guess some interesting solutions and see if I can get interesting physics

from at least some particularly nice solutions. So that's the idea. So look for solutions which are greater of space time. Then on these solutions, you can study the fluctuations as a systematic procedure. And these fluctuations are automatically governed by a gauge theory. But in fact, the fluctuations, of course, also describe a dynamical geometry.

It's obvious because the background is geometric. If you fluctuate the background, you fluctuate all of the geometry. So this is obviously some kind of a gauge theory. Of geometry, so you wonder, okay, this just really somehow smells like some sort of gravity. But it's not clear if it will be Einstein gravity, of course. Okay, 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. Okay, 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 okay, many are. And this is, so let me call this also fuzzy space. It's the same, for me, this is the same thing. So, but the point is, these are not just quantized symplectic manifolds. They are really embedded in target space. So that's a 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. Okay, 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 submanifolds, there you have the embedding function of little x and you can quantize them and this is the capital X. This is then a 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.

So that means if you have such quantized embedded manifolds, then you can take, for example, the commutator of this matrix and the commutator, you should think of it as a quantized Poisson bracket living on that space. So in general, it will be x-dependent. So whenever you see theta AB, that sort of comes from these commutators of matrices.

And think of it as a Poisson tensor. So this is sort of the generic picture. Here are two canonical examples. The simplest one is known as Moyal Weyl quantum plane. This works in any even dimension. So this is just the Heisenberg algebra, of course. The commutator of the x's 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 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 Elon's 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 Casimir relation, which is something like, it describes a sphere. And for large representation, the commutator of the matrices become,

actually, if you fix this, then this is like one over n. So it goes to zero. And the nice thing about this example, so this is of course also, it's just a quantized simplex, two-dimensional sphere is symplectic, quantize it, that's what you get. And the nice thing about this example is that it's fully covariant and SO3. Obviously, these relations are, they admit an SO3 rotations.

Okay, so that's all I want to say about these things here. And okay, 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-commutative 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, you know, these are Hamiltonian vector fields, these are just partial derivative operators, essentially.

Now, that means if you have this background, and if you add an arbitrary fluctuation, so this lives then 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. And 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 Yang-Mills connection. So the fluctuations, think of them as a Yang-Mills gauge field. And then the action, of course, is essentially a Yang-Mills type of gauge theory, and for free you get the inhomogeneous transformation law of this. 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 moral while, but essentially this goes through for all fuzzy spaces. Okay, so that's one thing. But on the other hand, I told you, 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 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 lot of people who have, a number of people who have studied that, starting with Rivales and Yang 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 a generic symplectic 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 here 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 Rivales first, quite a long time ago, is that if you do this on my Alweil,

in fact, you do get two rich and flat metric fluctuations. This is 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. Well, first of all, the matrix model action

will certainly not give you the Einstein equation. So if anything, you have to appeal to induce gravity, a la Saccaro, 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 this 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, you cannot hide this thing anymore. It somehow, 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 problem. By the way, what means the two, in the two Ricci flat metric fluctuations, what do you mean? Two independent transversal traceless, two physical modes of the gravitons. And we are in which dimension?

So these are the good degrees of freedom. Yeah, in principle, that's fine. Yeah, exactly. That's why it's fascinating. So far, it's 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 mu nu on the right hand side,

it will not, so you 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 will get a huge cosmological constant. 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 I 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. Sorry, what is the notation m for theta?

Where? Oh yeah, okay. Sorry, forget the theta. 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 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. It does not admit the full local Lorentz or 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 the sphere S4n, and that is completely covariant under SO5. So how can this be? And this space was introduced a long time ago by Grose, Klimchick and Bressner.

And it's a very interesting object. It has already 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 highest spin gauge theory. It's kind of a complicated object. But that's what it is. So this thing will lead to a highest 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 fuzzy hyperboloid. That is completely analogous to this one, but it's non-compact. It will also lead to a highest 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 upon. And this is a cosmological type of space-time. And that's an interesting object, I believe. And we will 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, when you have all these references, probably they did some kind of work on the fuzzy form. Yeah, yes. How about this one point vector reference?

Which one? Zhang Hu, 2000. Yeah, this one. That's a paper actually on a four-dimensional quantum Hall effect. It's a very interesting paper. So how is the connection? Oh boy. Not very quickly. But I mean, it's a non-abelian version of the quantum Hall effect in four dimensions.

And it exploits exactly the extra structure that you have here. And it's kind of, you study 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 fuzzy hyperbolic.

OK, so what is this object? So you start with SO4,2 conformal group. And so let me call the generators MAB of this conformal group. This is the Lie algebra of SO4,2.

And now for fuzzy 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 SO4,2, which they are. Those are the so-called discrete, short discrete unitary series of representations.

So in physics literature, they are known as mini-reps or double tones. And they come parametrized with an integer n, with a positive integer n. So these are very special representations. Some of the special properties are the following. So those are not just irreducible under SO4,2. In fact, they remain irreducible under SO4,1.

And that will be very important. All of their multiplicities are one. And they can be 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 have their highest weight representations.

And there is a particular generator, which is kind of an energy generator, if you like. And this is bounded below and then goes in integer steps. And in fact, at the lowest weight is an n plus one dimension irreducible over SO2L. And in some sense, it describes a fussy two sphere, actually.

OK, so that's the kind of structure that we need for this. Representations. And then, how do we get the hyperbolic weight? 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. At the AB is now the SO4,1 tensor. And this constraint is now actually unit matrix. And that expresses the fact that this is an irreducible representation of SO4,1 and not just of SO4,2. So this is why you have to choose these representations.

And obviously, it means, well, this describes a hyperbolic weight, a one-sided hyperbolic weight. This is the time-like direction. OK, so that's that. And then if you compute the commutators of these generators, you get the remaining generators of SO4,1. And this is what they usually call theta. It's the same thing.

It's just the algebra generator. So by the way, this means from a structural point of view, this is something like a Snyder space, for those of you who were related to that. Is that the quadratic carcinoid of this subalgebra? That is, no, which one? That, no, of course not, no. No, this is just SO4,1 invariant constraint.

And because it's irreducible, it is really proportional to the unit matrix on this representation. So this is very special. Anyway, so this means this is kind of one-sided hyperbolic. Of course, this is a Euclidean space. It's obvious down here, so this is not Minkowski. And it's covariant on the SO4,1 by construction.

This, by the way, is known as Euclidean AdS4. 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 they are, those are spinorially representations of SO4,2, which is SU2,2. And take a Fock space representation of these guys and fix the particle number. So this is the particle number n of the Fock space representation of these guys. And that gives you precisely these short discrete 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 Fock space are just given by these sandwich operators, as usual. And the x's are given by this object. This is the gamma matrix of SO4,1 now. And that tells you what's really going on, because that tells you,

well, that space is really quantized CP1,2, 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, so here you have four complex operators, so like four complex numbers, and that there is a, the number is fixed, so that means the radius is fixed,

and somehow this way you see that this is CP1,2. And it's also a quadrant orbit. So if, however you look at it, that's what it is. And it means, so it's not just, there is not just this hyperbolic, but at each point of the hyperbolic, there is sort of internal two-dimensional sphere. That's, that's really the geometry that's going on.

And the full space of functions that you have, that is realized in the anamorphic 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. Okay, but actually this is interesting and nice, because what does it mean? It means that, okay, if you now study fluctuation, 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 Kaluza-Klein theory, but it's different in a very crucial way,

because this is an equivariant bundle. So the whole bundle transforms under SL4,1, and the SL4,1 acts non-trivially on the internal space. And that means, if you do a harmonic decomposition of this internal space, these functions will transform non-trivially under the stabilizer group of wherever you sit on this

thing. So the local Euclidean rotations here act non-trivially on this internal thing. This means that the higher harmonics here are actually higher spin modes from the four-dimensional point 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-trivially 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 vibrations, or is this downstairs also symplectic? No, yeah, very good, very good. So no, only the bundle space is symplectic, this H4 is not

symplectic, not at all. And in fact, if you, what happens, so if you work out the four components of the symplectic or of the Poisson tensor field, and it rotates around this field, if you average it, it cancels completely. So the four-dimensional Poisson tensor is completely cancelled, it's averaged out. And this is why I can have full covariance,

that's the whole point here. Okay, 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. It 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, okay, 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 a whole, you get sort

of a tower of higher spin modules, if you like, in a semi-classical sense. But it turns out, 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, and it's like for the fuzzy sphere. Same thing happens here. You have a question?

So you only get the integer spins? Yeah, you get integer spins, integer spins. Of course, if you do that in the fermionic 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 I get. Okay, now explicitly. So what is C0? So these are scalar functions on

the hyperboloid. These are just any functions that you get if you 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 theta generators.

Now, data turns out to be a self-dual two-form, and you can be realized or represented by this kind of a rectangular Young tableau. And more generally, so the most generic or general object here in the N-warsam 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 two times two 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

really just precisely the content that you find here. So bottom line is you get your higher spin modes, and these higher spin modes are really sort of would-be Kaluza-Klein modes, which arise from these internal modes on the S2. Okay, so that's reasonably makes sense. And by the way, we'll see, so why do I call this, this is really spin one, even though it comes

in this disguise, but you'll see later on why it's really spin one, and so on. Okay, now, but that's actually not really what I want to do, I want to do Lorentzian stuff. So, and we'll see, starting from what I just told you, you do some little projection, and then you get a homogeneous, isotropic, Friedmann, Lemaître, Hobbes and Walker universe.

And yeah, so we'll discuss the higher 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 hyperbola, which I just told you, is actually a resolution of the model. And, you know, it's kind of, it just needs five matrices, which means it's embedded in R1-4

target space. Now you project this R1-4 target space along one line transversally like that. So here is your starting hyperbola, you just squash 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 a Minkowski 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 one and keep these, 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. Okay, so that's the solution I want to discuss. It looks extremely simple, it is extremely simple. So okay, what are the basic properties? Okay, now I 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 okay, 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 isometry group of the space-like hyperbola, three-dimensional hyperbola weights that you have here. So this separates, this kind of gives you a separation into three-dimensional hyperbola weights, and that's

exactly k equal minus one, Friedmann-Robertson-Walker spacetime, 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, 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, that would be the middle of the 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. Okay, so it would be some kind of a k equal minus one cosmology. You may not like this, but anyway, let's just go ahead. Now, okay, let me,

from now on, I will look at the Poisson level. And now, okay, let me be more explicit. So if I work on this space, there is no more SO4 comma one symmetry, so I better write down every single language which is compatible with the symmetry which I have, which is SO3 comma one.

So let me separate the generators as follows. So the x, x mu are the same object as before, but mu is now from zero to three. Now, there is another vector operator in the algebra, which is this one where I fix one index to be four instead of five. And let me call this t mu. And then the algebra, so 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, there is a sinh factor and eta is, this is the cosmological time parameter, and you cannot get rid of that. So that's, that's, that's here. The x comma x is theta and the t comma t is

also theta. So this is the Poisson structure. And then there's a bunch of constraints. So you can work them out. This is really, this follows from the representation. So that's only if you have chosen the correct representation. And okay, well, let me focus on these two guys. So the t's. So you have t times t is, okay, 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 you have this fiber bundle explicitly over each, so x describes the hyperbolic, or sorry, the squash hyperbolic, 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, any other more independent generator theta because, you know, that's the whole bundle. So in fact, you can express theta in terms of x and t like that. That, that follows from the self-duality. So in fact, you can forget about data from now on because really you have x and t and that describes the whole thing.

Okay. Now, okay, 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, if you can still view everything as functions on the hyperbolic H4, because that's where 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 hyperbolic and vice versa. So there is a nice mapping going back and forth from the usual tensor calculus to this kind of funny, funny encoded thing. And that just uses Poisson bracket. It goes as follows. Okay. If you start with the totally symmetric,

blah, blah, blah, tensor field theta, you take Poisson bracket with the x generators and contract them, then you get an object in this algebra. Fine. And 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 there is, it's just a different

way of encoding higher spin fields. So that's all very nice. But on the other hand, you know, this is a bit, you really want to work on the, on the Minkowski signature and here you have only less symmetry. You have only SL3-1 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 obviously, again, it's a symmetric tensor field. These are a little bit different. It's quite non-trivial to go from here to there anyway. But from 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 that these tensors are, there is an internal gauge invariance. And you can in some sense, you know, so the, you can fix, you can choose these guys always to be in space-like gauge, which means that they are not, 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 goals and you have, you'll have big problems that you should, that's one of the issues that is asked here. And that's very nice because, so that means from these internal degrees of freedom, you won't, you

won't, this is good. Okay. That's one thing. All right. So now let's go ahead and study the fluctuation. 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 discuss,

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 mass term, so you need a mass term 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 to always do separation

into higher spin sectors. And then for this background, this is precisely exactly preserved even at the non-commutative level. So there is, there's a spin Casimir you can write down, which exactly commutes with that. And it means that all this nice separation into higher spin sectors is, is 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 x solution, but this thing is nicely under control. So let's, let's go ahead with that. Okay. All right. So then we can expand this into these higher spin modes. And I told you before that any of these boxkit generators encodes Laplace, in fact,

a d'Alembert operator here, and you can work it out. So you can work out the effective Friedmann-Robinson-Lemaître metric here. And it turns out, okay, you can sort of do it almost explicitly. At late time, this, what you get is sort of an asymptotically hosting universe.

So you may or may not, this is not of course, the LambdaCDM 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 a 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 or 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. No. Okay. A flat that would be Milne, but it's not.

Okay. Okay. 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, okay, there is a gauge transformation. I will discuss this

a little bit, and then 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 fluctuation spectrum. 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 you can diagonalize. And that's what we did. And it's actually quite, it's not easy to diagonalize. It took us a long time. And the only reason why you managed to solve it is because there is this

big underlying SO4,2 symmetry. So that's extremely useful even for it's not a symmetry. Okay, so what are the, the point is the following. So we take this background, and so we make the following answers for solutions. There is two types of modes, a plus and a minus, which are sort of parameter. You take a sort

of a generic spin as a field in an anamorphism algebra and out of that you need to 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 now, not the T's. So the point is commutated with Poisson brackets with X. It raises and lowers the spin by plus minus one. 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, these guys 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 I find, I just need to diagonalize the box operator. So this is a scalar box operator basically on my spatial 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. Okay, so this is the use of D square equals zero. You have to gauge fix A of course, by the way, yeah, so I didn't tell you the gauge fixing, but there is a straightforward way to gauge fix and then you can do all that. And you have to factor out the pure gauge mode.

This is very much like Yang-Mills because the whole theory is formulated like Yang-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 notice it goes on 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 vectors of fluctuation. And the fourth mode is a time-like mode, off shell. And here we were not able to diagonalize this thing. 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 mixing here,

which I don't completely understand. Okay, so there is a conjecture that there are no goals here. And I think the conjecture is reasonable, but we haven't established it. Are you saying for each spin? For each spin, yes. Four modes finally? No. Okay. There are two physical modes. The plus and the minus. Yeah. So there are two independent physical modes for each spin, yes. In this, somehow, in our fluctuation spectrum.

And off shell there is two more, but this is just off shell. Up to any qualities cut off. Can you put back the previous transparency, you wrote it down. Yes. So d square is the Laplacian square. Yes, that's the vector Laplacian, yes.

And why the negative mode? Plus and minus, what that is? Yeah. No, what do you mean? Yeah, there are two modes. No, 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 n, 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 goes 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, there is the cosmic scale parameters 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 suddenly goes to zero. But at early times, there may be an issue. Yes. Yes. Okay. Anyway, so that's, that's what we find. So there is,

this would be tachyons. Yeah. Yeah. Yeah. Yeah. That's, that's the issue which may, may play a role here. Yes. I'm not, I'm not sure. It depends on these cutoffs and so on. Okay. So that's, that's what we have here. Okay. 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 sort of, you cannot really support the propagating time-like mode, but the argument is not completely at tight. And I think one should, 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, it's a fully Lorentz, it's 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 as this thing enters into propagation. That would be a disaster, but it doesn't happen. Okay. All right. And then briefly, something about gravity. Sorry. Yes. Okay. So let me sketch it. So first of all,

so to extract the effective metric that's always done from the kinetic operator and there's sort of a straightforward procedure to do that. If you have the effective metric and of course, then if the background is fluctuating, then also the effective field band and the

effective metric will fluctuate. So this you can get the linearized gravity in a straightforward way. And this you expect that this metric fluctuation that you get in this way couples in the standard way to energy momentum tensor. Now I have to, 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 it 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 tensors 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 tensor of these guys, you find, okay, you can do that. At least now, these computations here are done up to cosmological scale, so I don't have exact results. But the result is the following. So from these degrees of freedom, there is a special subsector, which corresponds to the physical degrees of freedom of the graviton modes. And they really seem to, 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. There is some, 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 look at the, if you compute the rigid tensor, the rigid tensor 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. And, but there is an extra scalar mode and extra scalar mode. In fact, in a priori there are two, but again, they lead to identical rigid tensors. So anyway, so this is all, 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 okay, it's not too far either. And finally, let me very two words about the gauge transformation, because I think that's interesting. Now gauge, I would like to have diffeomorphism invariance of course. Now, diffeomorphism invariance

should arise from the spin one 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 constraints. So the t's here, they don't have a time-like direction. So this is why you 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 diffeomorphisms, you can only have three. That's reasonable. And in fact, if you work out how these three would be diffeomorphisms 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 one, these three diffeomorphisms 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, so the vector

fields here that you have here, they're not quite volume preserving. They have, they satisfy this constraint, which at the late times becomes almost volume preserving, but not quite. So this kind of depicts the time-like component of the generator. So anyway, you have three instead of four diffeomorphism invariants. And finally,

given this gauge invariant, you can ask what is a linearized ion site reproduction. And again, it's somewhat complicated and there are more, there's kind of additional terms you have to write down. And 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, they all seem to be suppressed at cosmological scales. So I think the leading term is Einstein-Hibert. With the one-third, I mean. Yeah, no, the one-third is completely misleading. H is essentially zero up the corrections. Yeah, this is confusing. It confuses, but now I understand it. So H is something like one

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 geometric 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 this is qualitatively summarizing. So the pure matrix model, even without quantum effects will give you a richly flat vacuum solutions. Now, if you add this Einstein-Hibert term, you know, you should still have the same richly flat vacuum solution. So that seems to make sense to me, but you need the

induced Einstein-Hibert 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 that 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 yeah, so there is a chance at least that is the solar system that's maybe okay. At the non-linear level, I have nothing to say. The model is non-linear, but it remains to be understood. Another important thing is that there is no invariant cosmological constant.

So while that's, I cannot prove it, but we looked for gauge invariant solutions and I think the cosmological constant term is not gauge 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 Yang-Mills action. So that's, 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. I think it could be very interesting. And, you know, I think there is probably less need for fine tuning. Let me skip this. And okay, so, okay, 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 space time and matter. There is a nice class of four-dimensional covariance space times, 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 Friedmann-Robertson-Walker space-time, which features a regular big bounce 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 emerging gravity than quantized GR. Electrovalet behavior I would expect to be good because it's this maximally supersymmetric Yam-Mills type of action that you started with. So it's at least well-suited to study quantization, but it needs a lot more work just at the beginning. Okay, so let me start here. Yeah, I was thinking about the signature of this effective 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?

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 signature, you mean with this one? 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 instanton, you could have that, yes. So you would make sure that your sort of, your four-dimensional world that lives inside it has the right. Yeah, yeah, so for the moment I've just chosen interesting solutions rather than others, yes. Yes, that's, yeah. Anyway, you are optimistic by saying that you have apparently mass terms for your graviton

and 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 March line theory. They are usually simple, they have four goals, the March line effect does not work.

So one is very far from good. Yeah, yeah, so I hope that it's okay here, but yeah. Yes. Just the last question, so you were talking about the six-dimensional quadrant organs which you call CPU1, 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. Sorry, only semi-scientific. You quoted when you started talking about the space you discussed the work of Hasseber.

And the question is, is it more or less, one question is in what context he thought of it, and secondly, is it based, it's very similar, almost the same you found, or what are the subtle differences?

Okay, no, Hasseber has a paper which very generally discusses all kinds of this hyperbola 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. So in the emergent gravity, you mentioned whether you had some matter in your

Eisler equations, but is this related to the fact that you started with the, 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 if I understood the question. Matter?

You start now with a sort of vacuum background? Yes, yes, yes. Ah, yeah, yeah, okay. Do you want the tab also? Yeah, yeah, yeah. There is, you know, there is all this fermion, so in principle there is, the model contains all kinds of matter degrees of freedom, which I have switched off for the moment, and principle is there. And then of course you would like to understand

how does this matter influence the gravity. But in principle your Friedmann background has matter, because otherwise, that's why I asked whether it was flat. No, my, this solution has no matter whatsoever, so that is a solution without any matter, so that's... Still, the Friedmann 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 G menu has nothing to do with Einstein. The background has nothing to do with Einstein, only the, yeah. Yeah.