 Thanks a lot. Indeed, I wanted to remind you that we met a long time ago at the time when you were still thinking about this great idea of the relationship between matrices and membranes, and we have now seen the success and the use of it, and especially in this conference, I'm very glad to be here. So let's see. The topic I'll be talking about is not precisely that. I will mention the word matrix from time to time a little bit just because of the invitation, but I will be working on this corner where it is the ADS-CFT duality, and in fact we have a very nice previous talk by Bicert, so I'm glad to follow that. What I'll be doing is a much, much simplified version of that. It would be in fact not matrix models, but vector models, which would have been considered trivial even at the time when we met because the goal was always matrices, but I was surprised that these vector models, just an n-component field, they also have nice dualities, and they also relate to gravity, and in fact to put things into a contest of what Bicert did, he mentioned a set of operators, which are the simplest, which is the folded string, and it is that sector that the vector models would capture. Obviously, string theory has many other sectors, and that's the complexity and greatness of string theory, but this sector, which is essentially a dipole, leads to a picture, or leads to a type of gravity is known as higher spin gravities. They in a way are a poor cousin of string theory. They in fact should not exist really because they are not string theory, but nevertheless because of the duality with the simple set of models, the vector models, they might actually have a chance. So that is potentially another theory of gravity, which is in a way competitive to string theory, but it might have a lesser degree of complexity. So that is the set of theories I'll be studying, and it's a question of reconstructing space-time. I will mention space-time a lot to agree with the conference, and the work I'll describe is with my former students and some other collaborators. It will not only be that we concentrate lately on these vector models, but we even lower the dimension, meaning that the simplest model, which one might, or the lowest number of dimensions is 1D, and this is the such devier-kittire model, which is being studied a lot. So I will actually start with that 1D model, and then maybe if I have time I move up in dimension. But let's see the details of the 1D model. So the 1D model really looks very simple. It consists of Majorana fermions, which just depend on time, so there is a Hamiltonian, but the slight complexity comes in the fact that the coupling constant, for example, of a quartic coupling between them is a random variable. So that, in a way, is an aspect of, that's where matrix, some aspect of matrices come in, but it's a random variable with a simple distribution, like a Gaussian, and that essentially is a model. It can be Majorana, or it can be other fermions, or what happens is in this type of models that were introduced for studying spin glasses, but there is a phase that is no replica symmetry breaking. You need replicas because the averaging has to be done with replicas, and I introduced another index here, which is A, which goes from 1 and little n, while this capital N is taking to infinity, the little n is taken to zero, really, in this replica limit. But there will be a phase of this theory which is being studied, where actually the replicas will not play a role. That I will comment next. But again, going back to the model, it started out with a random coupling with a Gaussian distribution. If you integrate out J, which is very visible, you will get this squared, but it will be at time t, then multiplied at time t prime. And that is how you get the following formula, which is now, we have the kinetic term for fermions, which is very simple. And then we have this combination of the Majorana one-dimensional fermions at time t1, and another one at time t2. And it was power four, but now it's squared, so it's really power eight. But what you have here is a contraction over that color one to n, and this forms an object which is O n invariant. So that's why this is an O n vector model in the sense that we might think of some various field theories. The vehicle of studying this or rather the observables in this theory, and that again ties in with the previous talk, would be this operator's phi i of x, any number of derivatives here k, derivative s minus k, and phi i of x, again, which is a local operator with any number of derivatives. So those are the operators, which you remember by Seth mentioned that I correspond to maybe that picture. Just that this is also vaguely equivalent to a bilocal, when once the bilocal is expanded in Taylor series, so I will not carry this up, I will, it's just maybe notational, so I'll be all the time concentrating on bilocal. And you see in this case that this action already involves the bilocal. So the SYK model brings in non-locality immediately, and that came in because of this random coupling. That plays a fairly big role for the relevance of this model to, it exhibits chaotic behavior. That is why the model is selected as maybe the simplest model of black holes, that is why the concentration on its study, but it might be related to this non-locality, which is visible very much here. But I would anyway proceed with non-locality because my observable will be these objects, which are bilocals. So let's switch immediately to that bilocal description, which I just advertised. In this case it's only time, in higher dimension it will be space and time doubled. So this would be the invariant operators as far as O n is concerned. We will ignore the replicas because this is being studied in a phase where there is no replica symmetry breaking. Everything will be symmetric in that space, and what the vehicle, as I mentioned, will be these composite operators, 5x1, 5x2. And for those composite operators we are able to just write down an effective or collective action. So here is the collective action replacing the previous action. Obviously you recognize this was the kinetic term of Fermi, I was just written in terms of this bilocal quantity. This was the quartic interaction, which is just rewritten, both of those terms are trivial. And there is an extra term, that's all. That is where maybe some elements of knowledge comes in, that extra term trace log. Trace is in the matrix sense, everything will be in this matrix sense of multiplying things in bilocal space. So whatever formula I write, always involves such a multiplication if there is an issue. I call this a collective action. The reason for it is that it was introduced to reproduce the Schringer-Dyson equations. Those are equations Kawaii-San described in his talk. And if you ask the question, is that an action whose derivative you take, which would reproduce his equations, then that's precisely the question we asked with Sakita at that long time ago. And there is an answer, and we came up with a scheme known as a collective action. Obviously, we didn't plan to do vector models. Those at that time were considered trivial. In 30 years, they became non-trivial. We designed that for matrix models, Young-Mills theory. And that is a similar action for, which would reproduce those loop equations, which Kawaii described. But here, for vector models, this is the action. If you take a derivative with respect to psi, those are the Schringer-Dyson equations of the vector model. So I will not prove this. This is really an old statement. Now we'll just use it. And any, all the many works which were done on this SYK model, they are done in the large n limit, typically unless they are numerical, directly on a computer. They're always done in the large n limit. And they're always done by studying this action, really. So whoever writes a paper, sometimes there are versions of the action, which are in terms of two fields rather than one. And they are reached by something called Habard-Sretonowicz trick, which people are familiar with. But this is a more economical way of writing it, in terms of one field. So here it is. That's the action. The study then sounds straightforward. This is very straightforward. There are two features of the action, which follow from what I said before. First of all, you notice immediately that we moved from 1D to 2D. And this is in line with ADS-CFT correspondence, where a CFT in dimension D would become an ADS theory, where the dimension is increased by one. So that is, that is kind of very easily accomplished by this, by just concentrating on this set of close set of operators. And that set of operators, this is not an assumption here in the vector model. They are a close set of operators under the equations of motion, large n equations of motion. That is the second thing, which n appears in front. So it is already in a form, which is the ADS dual, meaning that the natural coupling constant is then 1 over n. So those two very relevant features are actually almost accomplished. So you might think that now all we have to do is just do some simple study of this model. It should be simple. It's one dimensional. And that's what I will describe. That study has been going on for three years. And first of all, there is an infrared critical point for strong coupling J. J is, in this case, that's the coupling constant. If J is large or infinite, these two terms, you can scale J and it will move out here. Therefore, at large J, you can just drop that term. I will call this term while the full action is collective. This is just C, meaning conformal. So these two terms constitute the conformal limit. So you can concentrate just on those two terms, which is even simpler. So let's study that conformal theory. And here, almost again, you get a very interesting feature of this action which we wrote down. Obviously, the action existed for 30 years, but this was noticed only recently. It was actually noticed in the framework of spring glasses maybe 10 years ago, but it was obviously featured now. That it has a reparameterization symmetry. Namely, if you make a reparameterization of time, t1 to f of t1, arbitrary function, and obviously t2, f of t2, and you transform this field, the bilocal field, in an obvious way so that this term will be invariant under this symmetry. You can then show that the trace log, because of the matrix structure, and things only happen at the end point, and there is a trace, it is also invariant up to an additive term of f. And that's precisely the feature of Liouville theory. So maybe those who love Liouville theory again, it came as a surprise to me that there is a one parameter family of Liouville theories, actually, with similar features. And this is that one parameter family if you replace that four by q, so that that interaction is not quartic, but it is higher order and the order q. So if you replace this by q, four is replaced by q, that would be a qsyk model. It is now a one parameter family of models. And a second comment is that a particular case where q is infinite, then this is Liouville theory. It doesn't, it is, we probably all know, probably several different ways of obtaining Liouville theory from various directions, from string, and other way. But this is maybe, this adds to your list of getting Liouville theory from somewhere else. This is Liouville theory. It's not very visible, but if you just, if you have that power q and you replace this by one minus q phi, introduce a new field phi, then this becomes e to the minus phi. That part I have proven, that's the part of Liouville theory. All you have to prove is that the trace log will give you the kinetic term of Liouville theory in that limit, q equal infinity. So, but in view of time, I won't be demonstrating that, but that would be the, the competition of, the competition of the proof. So, the statement is it is Liouville theory when, when q is infinite. So, but to continue with this three parameterization symmetry, which will in this problem play a fairly big role, which is what I said here. And now let's see what role does it play. First of all, it will relate to a problem, which is known always as a zero-mode problem every time there is a symmetry. And you are shifting by a background psi zero, so that you can transform that solution into a new solution. There is a zero-goalstone mode. The wave function of that zero mode is just the derivative of that shift, or transformed solution by that function f of t. That's a continuous symmetry. In this case here is, here's what you get. The solution for the background was very simple. It was just t1 minus t2 to the power one-half. When it is transformed, it is just f of t1 minus f of t2 and that derivative which you have here. So, if you take a derivative and Fourier transform, you get this Bessel function of t1 minus t2. This will be a wave function of a particular mode in the small fluctuation spectrum of this problem, which would be problematic if at the outset the propagator will blow up precisely at that value of the conformal dimension. But we also know how to treat such problems. Such problems are characteristic of solitons, or more generally, brains. They are treated so that this symmetry coordinate becomes a dynamical variable. Again, that was also called collective variable. I am not sure why Sakita and me called everything collective, because maybe inspiration with the success of the famous model of Soviet economy. But that is also collective coordinate, but not the same thing as the collective field. And here is how that evaluation goes of what would be the dynamics of that symmetry coordinate. It will come from that extra term which I referred to, which I dropped, the one which breaks conformal symmetry. If you transform that term, this does depend on f. There was there a source, because this was by local, and that term was local. So, it is that term, that breaking term in the action, which will capture that coordinate f. And now follows an evaluation which I skipped, and that is actually fairly non-trivial. It's much more complex than in the case of solitons, where we could evaluate things easily. The point is you have to evaluate things in this strong coupling limit. So, there is regularization and there is renormalization, which you have to understand. And we spend a lot of time agonizing over that with Kenta until we understood it. So, after that strong coupling regularization and renormalization, we were able to evaluate this quantity and obtain the action for that. So, similar evaluation was done by Kitawa and Su. So, we were happy with that, but an earlier evaluation was done by Maldusena and Stamford. However, they were working in a linearized approximation. So, the action which you get is the Schwarzian action, which is right here. It is non-linear enough, highly non-linear enough. It's well, the Schwarzian term is well known in conformal field theory. That's how the energy momentum tends to transform, but it doesn't mean it just follows from, this is not conformal field theory. This is a totally different subject. And that's the form of this dynamical collective coordinate for that symmetry-Golstow mode of the reparameterization symmetry. It's one-dimensional, which it was. It is one over J, which shows why we needed the strong coupling, the correction in strong coupling to evaluate it. That's proportional to the action. There is a parameter alpha, which is actually the strength of coupling of this action and the answer is while we were able to show this very convincingly, that coupling is still not known. So, it is known only numerically. As I said, Maldusena and Stamford evaluated this in a quadratic approximation by linearizing this. And in that case, they were able to nail down the value of the coupling numerically. There is an outstanding question even in this simple model to find that coupling analytically. That is the price for that. So, but that is the action of this collective coordinate. I should just quickly state that for just things which follow that the same dynamics comes from ADS-2 in a dilaton gravity of Jaquiv-Titelborn kind, which is written here, where you vary the dilaton field and you get just the equation R plus 2 equals 0. So, it's a topological gravity in two-dimension. It doesn't have any degrees of freedom, but it does have a boundary degree of freedom as always. And that boundary degree of freedom can be shown to be governed by this very same action with 1 over G, which was N here that complies with that formula and some other numbers. So, then this was shown by Polchinski and Armheri, who independently earlier studied some details of this topological model. So, it was very easy to put these two things together. You just ask the question where does a Schwarzian action appears. And then we conclude that the Goldstone mode or the SYK model just correspond to 2D Jaquiv-Titelborn gravity in ADS-2. So, this is now a result. This is a statement and great. We can now carry on. The next step, obviously, is to study fluctuations and ask the question, what is the full spectrum of the model? So, that's the next step. And that procedure again is... So, in fact, this identifying the precise Schwarzian was slightly non-trivial. And from the fact that even the coefficient is only known numerically, you might assume that it actually fairly non-trivial. But the next step sounds trivial because I will concentrate in that critical region of conformal theory. Only two terms. Trace log. Derivative of that is psi in the matrix sense inverse. And the second term was just psi to the power 4. So, this is the equation of motion you have to solve with. That's psi 0, which I already used previously. You can generalize this to finite temperature, which is of interest to studying black holes. That is also one of those reparameterizations by that reparameterization symmetry to finite temperature. But to study the spectrum, you just take one more derivative with respect to... And I didn't try it with respect to psi t3 t4. So, now you get a kernel between... For this by local t1 t2 and t3 t4. I will not take a derivative. Everyone knows how to do that. So, you get a kernel which depends on psi and you plug this in. And that's your kernel. That's a very non-local kernel. But it is very easy to diagonalize it to find a wave function, which will diagonalize that kernel. Because of... We are in the conformal limit. There is an SL2 symmetry, which is the dilatations plus conformal symmetry plus translations. But now, what is operational on that by local is just the by local SL2 R, meaning you have to copy for translations d1 plus d2. And from dilatations t1 d1 t2 d2. So, it's a very simple implementation of the very simple one d conformal symmetry and then the k. If you compute that, you get the very simple answer. The chasm here of this SL2 is t1 minus t2 d1 t2. You can think of this as some kind of basic Laplacian in this by local space. It will not be d1 squared plus d2 squared or something like that. It is this chasm here of SL2, which is the Laplacian in by local space. You'll make some further comments on that Laplacian that will have a space-time interpretation. And the eigenfunctions are the conformal... related to conformal blocks. There are general three-point functions with an arbitrarily conformal weight h. Those functions look like that. There will be an eigenfunctions of the chasm here that is well known in any dimension. So, you use that for one dimension. You do a Fourier transform maybe. So, there will be two quantum numbers. We are in two dimension. The quantum numbers will be h and omega. If you do a Fourier transform of what I did before, you essentially get a plain wave e to the i omega t1 plus t2. That's the center of mass of the by local. And this is t1 minus t2 and the function z, which is a linear combination of Bessel function. It would be the wave function in ADS except for this second term. The second term comes with a fairly non-trivial coefficients. Because of that, on-shell, when you now ask what are the on-shell values for this conformal weight mu, you get an equation. This equation will actually have not one solution which you might have expected because we basically would want one scalar field in two dimension. But it has, this is a transcendental equation. We are known to everyone who teaches quantum mechanics. It's a transcendental equation which has infinitely many solutions. Those are the, that's the spectrum of the SYK. There will be an infinite number of states. And if you now ask the question that by local kernel, how does, how should I picture that? Here is how you should picture that. It is really an infinite product of Laplaceians in ADS II with growing masses. Because if those growing masses are precisely on this phone, then they, this is what, what agrees with that infinite spectrum. So, so this probably tells you that I, I, I began saying we'll be studying a simple model of a simple model. This actually is not such a simple model at all. This is a fairly complex, you know, it is in one dimension. But it, for example, comes with this kind of non-local, extremely non-local Laplacian. So that, but that is the spectrum. That's the answer which just comes from evaluation of that quadratic eigenvalue problem which, which was so easy to write down. Okay. And this interacts with this gravity degree of freedom which is the Schwarzian. So we have some matter interacting with the Schwarzian degree of freedom. I should maybe tell you where I'm going by studying this model slowly and through the last several years. We are trying, I, I forgot to tell you, we do not know what the dual is in this case. It is not like the beautiful case you had previously, where the dual was stated very clearly. It's, it's 10 dimensional super-sync theory. In this case, we have no idea what the dual is. We know what the CFT theory is. We would like to know what the dual is. We know this is a nice theory to study black holes and do maybe possibly non-perturbative physics. And we are maybe trying to figure out the dual. So at this point, you have this highly non-polynomial action and the Schwarzian degree of freedom which is really two-dimensional gravity. That is a 3D picture. As I said, that formula which identified the masses that was reminiscent of something. If you study, if you study a quantum mechanical problem in the third dimension, so that has something to do with certain version of Kaluzak line, but a very peculiar version of Kaluzak line, you can understand that infinitely polynomial non-polynomial Lagrangian or Laplacian from going from a standard scalar field theory in three dimension. And this is what we have kind of figured out with Das and Kenta. And this works very simply. So you are in three dimension, gravity coupled to a scalar in three dimension, two are ADS, and the third is that Kaluzak line dimension which will have to give us some desirable result once we evaluate what's going on in this third dimension. So we obviously will fit things. We will say that that extra dimension will be compact and at y equal to 0 there will be a delta function spike which I didn't write, but in that case the spectrum is given by that equation from quantum mechanics of momentum being decided by a transcendental equation. So that spectrum looks like coming from the third dimension, not of spacetime, and the test for that is really to evaluate that propagator in third dimension. And if you evaluate the propagator in third dimension and take it at y equals 0, you get this answer. Sum over those m's with the right values for p m's, the right conformal weights, and with the right wave function which I featured before. So I never wrote down the propagator for the SYK model because I knew I didn't want to repeat it writing it down. So here comes the same propagator from third dimension where this is how we go down to the SYK. Obviously all the other degrees of freedom are being integrated out in that Lagrangian that I wrote. So this might look that we are out of the woods, but in a way the interesting stuff is just beginning. So let me prepare you that this will be, I hope not an endless situation of new things in the trivial model, but nevertheless there will be some, there is a further question. So number three, we understood the matter sector, we understood the gravity degrees of freedom, but here is a question of spacetime. And this is in the answer is honor, spacetime. So if you just make the combination T1, T2 and T1 minus T2 that's trivial, but that casimir which I mentioned is this form. That's the sitter casimir, the sitter Laplacian, or the anti the sitter Laplacian. In two dimension it is always SO12 or SO21, you cannot distinguish which one is it, is it the sitter, is it anti the sitter? Because so that's the question now. So obviously you ask the question is it the sitter which would be maybe even more interesting if this theory gave the sitter space, but it will not be, so first of all there is this comment on the question is it the sitter or anti the sitter? You say let's look at those wave functions Z which had that linear combination with that strange factor between two Bessel functions and those who know the sitter wave functions know that there is an infinite number of possibilities in the sitter called alpha vacua, they all lead to different type of excitations. This is a particular alpha vacuum, you just specify the value alpha and it's very simple in terms of new and the well-known the sitter wave functions in the alpha vacuum will agree precisely with this formula. So you might say then okay now maybe this is the end, this is the sitter and you are in particular alpha vacuum and that's what you are, that's what the S-Y-K is studying. These are the alpha vacua that have a charge at the anti-bord? That's right, yeah the standard alpha vacua which you just opened the books of the sitter, the paper by Alan who first wrote it down, so looks fine, but there is a problem and I'm sorry but there is a problem. The problem is the problem of an I which goes as follows. I should say and someone should have noticed it or because we were studying the partition function that was an Euclidean problem, we studied the trace of e to the minus beta h maybe at infinite time, but nevertheless it was an Euclidean S-Y-K model that I was starting with and studying all the time. Nevertheless it seems to have something to do with the spectrum is that of a Lorentzian problem. So I don't think you have ever seen anything like this that you started out with studying a Euclidean problem and looked at the fluctuations and you encountered a Lorentzian Laplacian. There was no mistake in the algebra so that's just a fact. So this collective field theory sorry was like this, it came with minus that collective action, it was based on Euclidean logic or deriving it, but even though it was Euclidean the small fluctuation spectrum looks Lorentzian. You either say this is some kind of contradiction maybe this theory is sick because for a Lorentzian theory we would have matter coupled to gravity much alike head but there would be an I and then all the endpoint function that I is not irrelevant that is there for a reason so then simply you would say you will not be able to claim agreement of this with anything like that that I is an obstruction unless you agree to study Lorentzian Laplacians without an I in front of the action and that you will not agree but so that's the problem at this point in time of making a statement what is the space time. We were thinking between the sitter and anti the sitter the point is that none of them is really a possible answer it must be the answer actually we know it probably it no it must be Euclidean anti the sitter we are studying an Euclidean problem standard ADS CFT would say it's an Euclidean anti the sitter so you better exhibit Euclidean anti the sitter it's possible to do that that is a transform and I should say that in this by local theory and it especially becomes visible in higher dimension in this d equal one things are a little bit misleading but in higher x1 x2 is a by local that's not really a space time of any kind it's just an anomaly that in d equal one that Kazimierz was so simple that it looked like the sitter but that was just a special case of of d equal one if you study things in high dimension you would not even see anything recognizable for the for the Kazimierz of the by local it would not be a recognizable space time so you wouldn't we would immediately know that to go to the space time probably there is some transformation the simplest point is maybe to do a coordinate transformation between this by local to maybe some physical space time it's actually close to that but not a coordinate transformation it will be a coordinate transformation in momentum space it will be a momentum that the next simple possibility either I will make change of coordinates or I will maybe make change of momentum if it's anything more complicated than that then I'm in trouble if I have to do canonical transformation which which involve both coordinates and momentum and here is the answer of that transform in fact it's a very well known transform it's called the radon transform it looks like as follows you transform for this by local space here you remember eta and t to an euclidean ads space and you might have an euclidean ads wave function and you will get our wave function which is the by local wave function the radon transform has a you know even greater use than the membrane matrix duality because it's so everywhere in the CAT scan so so hence you still have a way to go to achieve such a such an important as Mr Radon it is it is a transform which is which is just obviously it's a transform which allows for recovering the image in in CAT scans why does appear why why does it appear here I don't know but I know how to understand it from our discussion namely we and this will be true in other higher dimension this is not only for one dimension such map is required because the conformal group is realized differently here with the in the by local we had two copies of just the standard CFT realization of the conformal group but in the dual ads theory we have standard curd space ads realizations of the conformal group so it's a very simple question why don't you just take the generators of SL of realization of the conformal group on your by local which are here this is the dilatation t1 p1 t2 p2 p1 p2 and the conformal t1 squared p1 p2 and you go to euclidean ads tau p tau plus z pz that all standard and this is how the analog of conformal k looks in ads space so you say just make a identification that this should agree with each other and figure out what transformation you have to make between the by local realization of the conformal group and the actual space time realization of the conformal group these equations that is a nice nice I guess advanced group theory problem in the class to ask that question because there is an answer and remarkably the answer involves something which is very obvious that this is a center of mass and then pz that's the extra dimension of ad s this is a formula for what z is actually that's z in our case is never holographic we even though I use the term by local holography because I am by local everything is being recovered in particular z or its conjugate is being recovered from from from from from the cft coordinates but it's a momentum change the change in momentum space it's not a change in coordinate space it's change in a momentum space and if you evaluate the kernel of this kind of momentum kernel very simple in this case you that's one new way to get the red on transform that's not how red on got it but this would be our way because that's how that's how we met the problem in this case so it's a change of coordinates in momentum space and if you fully transform that in these four variables then you you get that red on transform so okay then the red on transform will take us to the Euclidean ad s space and you might in good this is maybe the end of this we finally are happy we have we are now our theory is now in Euclidean ad s everything is consistent almost true there is still a kind of some very unusual feature going on in this case which will have some relevant consequences which I hope I'll be able to transmit I know I probably I probably yeah yeah I probably kept all my science rate or at least some quantities were positive oh very much that's the next issue you you spotted the issue okay so let's go to that issue okay it transforms at the ceter laplacian into an Euclidean the ceter laplacian if I concentrate not on the time but space and just maybe introduce exponentiation of variables y and the ceter this this is what is being transformed it's a Louisville type potential minus into plus so so and that that is very visible why this happened you know this is the consequence the fact that I'm transforming the Euclidean to a Minkowski and problem you are transforming a potential which is which looks like but before anyone gets negative I should say that's what the redon transform does it transforms a problem and you all go for examinations so you trust the redon transform so it is this problem into that problem they are very different there are only scattering states here reflection so that Louisville wall and there are scattering states here but also bound states here in fact an infinitely many so you are transforming a problem with bound states to a problem with not not you but redon is transforming a problem with bound states into no bound states that has a simple consequence which I will state and then I wrap it up at least what we have learned if you take a normalized wave function in one of the spaces Euclidean you do not get a normalized wave function in the other space you get a lag factor the lag factor is this ratio of gamma functions those gamma functions have poles precisely where those bound states are so it is something we would expect you you say there must be some way that that this problem here is equivalent to this problem but there must be some extra accounting for the fact that there can be bound states so there like like this I call lag factors but there are factors between the two wave functions and then this is the final result so if I take the bilocal propagator which I had some over those wave functions but now I am in Euclidean space the only price I pay for this Euclidean space is this extra factors because that that's what the and that that those extra factors have those poles because I transformed one problem with bound states into a problem with no bound states so this is the proposed answer then of what the propagator of this theory is in maybe what we think is a physical space meaning Euclidean it's Euclidean at least but it has this lag factors okay maybe then that that has a lot of physics in it in addition to those poles which are here those are the solution to that I already discussed those are those massive states there are now poles which are also there so there is an extra set of states at least this is what one is one is predicting and that's what I said here now and this is my chance to mention a matrix if you think that this story is maybe confusing this what I did here I just repeated something which I probably gave a lecture on you know in 1990 in an identical way with with with the identical you know just just a slight change of slight change of terms at that time the duality which we studied was between a c equal one matrix model and in that case the dual was known identified from first numerical studies the 2d non-critical string theory this is really a predecessor of ads cft and has been studied at that time in detail with precisely this sequence of you know statements I don't I'm not sure yes how much time I have oh then I certainly don't have time to go to two then three and higher dimension this is still one I'm still stuck in one dimension but let me then complete the one dimensional story because maybe I will just illuminate this statement so that there is some connection to a matrix model after all that I use the word spacetime sufficiently many times so let me then switch to the matrix so that matrix model was was this model and with the potential in fact the potential was a simple upside down harmonic oscillator potential so in that sense it's a very solvable theory on the outset and the collective field which is the analog of this by local was the distribution of eigen values which then led this is the density and led to two dimensions so it is again d equal one and the collective description is in terms of d equal two in that case it was even simpler to do I will not write down the the the filter which governs the large n and those stringer Dyson equations which which I mentioned but after you linearize everything and evaluate quadratic collective Laplacian it just gets to be this this one so it's a standard Laplacian in two dimension so you immediately jump at the statement oh this is just a massless scalar field that's been zero scalar in two dimension and the 2d string which this is supposed to be describing does have such a state it's only the center of mass of this it's a closed string but open or closed and that would have been a tachyon that's the scalar that's the would be tachyon in other dimension but if you remember string theory for non critical strings that mass is equal to two two minus d so it would be tachyon in twenty six but in d equal two is precisely massless so this is a winner this model that's why it describes two distinct theory but that's kind of not enough and that's where the long story start along the same line which I was proceeding before you say but this I know cannot be the the actual space time time and some other this is now Lorentz invariant the two denon critical string theory involves uh you know the second dimension is not a c equal one conformal field but the Louisville Louisville Louisville conformal field and then the Laplacian would be at best at that you know this would be the Laplacian but there is a transformation between this Laplacian and the one I wrote down and then if you work out the transformation you get again an extra factor it is the same story that you you you transform now to this Louisville description which is the right space time but you get a factor and this is the factor this is the pole which is that massless scalar of 2d string but an infinite number of poles this I should say that even though I discovered or with Schumann does we discovered this contribution actually this contribution which was discovered by others is much more important that lack factor is actually much more important than the main excitation which was the tachyon because it's the lack factor which tells you this is string theory otherwise it could be just a scalar field in two dimension you say why 2d string theory the lack factor and this is now the correct answer this was written down by I should give credit to those people whom I learned this from at that we learned this from that time it's more cyberg and staudacher it was a very nice work they identify those the the the propagator of this 2d matrix model but now it looks like a propagator in Louisville theory but it has an infinite number of extra contribution at this value speak while I am those are called discrete states that imaginary values of momenta those discrete states were confirmed by polyacuff because 2d string theory even though it should have no degrees of freedom if you are very very careful as polyacuff was you will analyze for example here is just you can do the exercise for spin one this is momentum this is a shift due to Louisville the background charge for the non-critical string and this would be the the equation of motion for example s equal one this would be the gauge condition to Lawrence Lawrence gauge condition then you know to remove the other degree of freedom you would use an on-shell gauge transformation and then 2 minus 2 is 0 so you conclude that the two-dimensional gauge theory has no degrees of freedom but polyacuff noticed that when p is equal to minus q for that one special value of momentum there is no chance to use this remedial gauge transformation and therefore there is a physical state at p equal minus 2 that's the first discrete state that's the origin of these discrete states that's also the origin of schwarzen degree of freedom so then you start analyzing the string to all other degree all other higher levels and you find a perfect agreement with where this pulse of the slack factors are so then you have a complete agreement with 2d string theory and you you at that point are very happy so in this case we do not have in this case this is in a way a prediction of what we accomplished with kenta shumidas we are claim the following that this simple one-dimensional field strongly correlated theory in is equal to some gravity in two or three dimensions it has this soft mode called the graviton it has an infinite sequence of matter fields which are easy to find but those can be also understood for one higher dimension which by calzacline but it also has an infinite sequence of discrete states which I just featured so all three are then that's that's according to what we believe at this point would be the complete spectrum of this theory and then it's anyone's guess it's a it's again a challenge and the answer is not known what is this theory we we have a proposal we have something which we we kind of think that is the spectrum and the question is what is the theory so in this case the dual is not known for the higher dimensional case the dual is known it's vasilev's highest high spin theory it's only in this case where the dual is not known and it's a challenge to find thanks thank you thanks very much for interesting talk I will not mention what I asked you to talk about but it's very interesting but okay I told you this is what you I mentioned space time yeah quite interesting for me personally also it's the first power point I see from you oh I say okay that's thanks thanks to canta really are questions or comments is there any natural regularization of the theory that would allow to go from the weekly to the strongly correlated theory in a in a way you that parameter j it's up it's your it's still under your control you can study the weak coupling theory and that could be an exact normalization that there could be and in fact you know from the way kit I wrote his paper you might think that he might he's discussing that but I didn't see it in his paper and that's in a way something to discuss I I don't think that was done ah that's a good point okay so and that this can be analyzed Louville only takes only keeps the that h equal to the the lowest of those scalars you know there was a sequence of scalars I call them PM and the lowest one actually is the Louisville and the others the couple you can analyze that in the propagator the residue of the pole is proportional to one over q for the others except for one so that's why you get a nice theory that describes what you expect there is one scalar field the Louisville that's precisely the lowest mode of those scalars so it is a very nice nice situation the others the couple and is there a nice I mean apart from it when you had the transformation with the total momentum that's right so some other interpretation or it just comes out like this it is the fact that so again you realize the conformal group by by additively here you know basically so you have l1 plus l2 for all generators but the space time description should be realizing the conformal in ads space as it is easily done as as a curve space those two don't look the same so you can ask the question can I make them equal by a change of coordinates the answer is no by a change of code so you give up on the change of coordinates then you try change of momentum well which is the next simplest possibility and yes it works it's remarkable that it works it's just a curious fact I don't know but deep answer why it is momentum space that it it works it's a simple mathematical fact which but that's also the answer for why your cats can exist that's the red on transform okay we have now the shortest break of our conference okay there is an additional talk a quarter past five so we just rush do something thank you very much