 All right, thank you. Thank you for the invitation. It's great to be here. So the talk will be on SYK, and there'll be two parts. So the first will be an overview of SYK. And the second will be on the title, which is the bulk dual, the ADS dual of SYK. And the second part will mostly follow this paper. Is this board visible? So SYK, there's an incredibly broad community of people working on it from different fields. So let me list those fields. And then as a way of introduction, I'll try to describe why the model is of interest to all these different communities. So string theory, ADS CFT, QFT. Some of these, there's not much of a difference, but in any case, quantum gravity, condensed matter, quantum chaos, and black hole information. So I'll try to say why each of these groups, or some people in each of these groups, are interested in SYK. So that's why K model. It was introduced by Ketayev in a series of talks in 2015 at KITP. And his interest was, in fact, black hole information. And the model is related to a model, Sachs-Divinier, that's why it's considered in the 90s from the condensed matter viewpoint. So the model is quantum field theory. In fact, there was a much earlier group started this model in 1970, 1971. This was in the field of nuclear physics to body-random interaction model. Wachigas, French, and Womko. It was for certain that that's why K model. Exactly, as it says, in the era of Sachs-Divinier in 2016. And they are cited only by us and everybody. So this is quantum field theory in 0 plus 1 dimensions. So the Lagrangian consists. So there are n-mironophermions, n is large, chi-i detail chi. This is the kinetic term. And then there's an interaction. So this is just the kinetic term. And this is a Q-body interaction. So Q can be any even number. 4 is a typical case to consider. But one fixes Q and can talk about Q-body SYK. So n, the number of fermions is very large, much bigger than 1. So these are mironophermions, and they anti-commute. You can think of chi as just a creation operator plus an annihilation operator. And these couplings, so here I have Q fermions and a coupling with Q indices. I went through IQ, which each range from 1 to n. And these couplings are drawn from a Gaussian distribution with variance that scales as j squared in an appropriate power of n, which you'll see why this power is chosen. So this is a Q-body Gaussian random and all-to-all interactions. So there's no notion of locality. In this interaction, one picks any Q fermions out of the n, and they interact with each other with this coupling j, which is drawn from this Gaussian distribution. Every question is scaling with n is really important. If you choose a differently, change is really the outcome. Yeah. The scaling with n is chosen so that the two-point function of fermions, that the correction to the two-point function of fermions from the interaction is something of order 1. If you chose this power to be larger, then it would just be trivial. Or more precisely, probably, if you chose a different power than a different, maybe a different class of diagrams would dominate. But this power is important. So how important is that they are my run fermions? Or we can take usual terms? You can take Dirac. That's fine, too. Kitayev just took, it doesn't matter. You could take Dirac, and then you would put half of the Q over 2 creation operators, C dagger, and half C. Yes? But then if you take, say, half C dagger at C, then you conserve a number of particles. Yes. But here you do not. That's right. Here you don't. That means it's somewhat easier or less easier. Yeah, one can repeat all the calculations with a Dirac case. It's not so different. That's correct. You can also write each chi. A Dirac fermion is two my run fermions. So you could just write chi as C plus C dagger. And then rewrite this. And this would be a sum of Dirac Lagrangians, which don't conserve particles. And you have contributions with more C daggers. Yeah, that's right. But the difficulty is the same. Yeah, there's no difference. Kittayev prefers my run fermions, and Satya preferred Dirac. So that's the model. The surprising feature of the model is that it's solvable. It's solvable perturbatively in 1 over n. But by solvable, I mean that correlation functions can, in principle, and in practice, the correlation functions are determined by integral equations. So for instance, working at leading order in 1 over n, one can write down an integral equation for the fermion two point function, which you can then solve numerically. Or in the UV and IR, one can write down an analytic solution, which is already much more than you can do for a general theory with a Q-body interaction. So that's the first feature that it's solvable, which is obviously very useful. The second feature, which makes this model interesting, is that it's a CFT in infrared. It's actually nearly a CFT in infrared. I'll describe later what I mean by nearly. For most of the purposes of this talk, one can just say it's a CFT in infrared. So the infrared is the same as strong coupling, strong coupling being J. And in the infrared, the fermion dimension, we call that delta, is given by 1 over Q. So the two point function of fermions in the infrared, chi i of 0, chi i of tau is sine tau over tau to the 2 delta. So it's the two point function you expect for a CFT. With scaling dimension delta, which is 1 over Q. So for Q equals 4, this is 1 quarter. In the UV, the fermions, they're my own fermions, so they're dimensionless. So one can see the anomalous dimension is 1 over Q. And for finite Q, that's an order 1 number. So this is a large change in the dimension as you flow from the UV to the IR. Any questions about the definition of the model? It's solvable only in this sense. You cannot state that the statistics of level like random matrix theory or something like that. So it's a statement typical for quantum chaos. Yes, good. So as I'll describe this model, as Kitayev noticed, is maximally chaotic in a sense of quantum chaos. I'll define. One could then ask the question you just asked, which is how does it compare to a definition of quantum chaos that was studied in the 90s of level statistics? And that's been explored in some papers. Excuse me. But meta-analytically, you cannot prove that. Yes, it is true. Analytically, at finite Q, I do not know how to compute the density of states. So by solvable, I just, I mean what I wrote. So here's one story, how level statistics is another. They are not related. In the Anderson localization regime, density is not changed when you go through Anderson transition, quantum statistics of levels is changed from Poisson to Wigner-Deis. Yes, yes, yes. Yes, it would be interesting to analytically see if Wigner-Deis is satisfied. The only thing I know how to solve is the correlation functions, which is the common definition of solve. Yes? Excuse me. Like, for instance, the two point function of fermions, where is the average of the disorder? It's already been taken. So all quantities are after disorder average. So if you don't like disorder, there are multiple ways around it. The simplest one is just to think of that j just has a, it's just a free field with a two-point function that's a number, and just proceed doing field theory as you would normally, and don't include any quantum corrections to j. So just draw a free propagator for j and proceed, and then you can forget about the disorder. There are other ways around it. So let me, let me, so it's interesting to compare how SYK relates to other large n models. So let me, let me just, is there an eraser? I can erase this. So there are roughly two, there are two notable classes of large n models, one that I'll call hard and one that I'll call easy. So on the hard side, we have matrix models. Matrix models, some at leading 1 and 1 over, and there's some planar diagrams, things like this. And except for special cases, it is generally hard to sum all planar diagrams. If you have a single matrix then in zero dimension, then it's easy. Otherwise, it is generally hard. So an example of a matrix model is n equals 4 super Yang-Mills. And the bulk dual to this CFT is, of course, string theory. So it's string theory, that's the bulk. And in the bulk at large, so in n equals 4 super Yang-Mills, one can change the tuft coupling. And at large tuft coupling, there's a large gap. There's a large gap, so there are a few light operators and the rest have very large dimension, which translate into the statement that the bulk dual has a few light fields and the rest are very massive. So just what you get with string theory. Right. So in this case, the bulk theory is familiar. But the boundary theory is not that familiar. And it's supersymmetric and it's hard to solve. So that's one case. Another well-known case is vector models, those sum bubble diagrams, for instance, the ON vector model. And the critical ON model, that's a CFT. And so one can ask what its bulk dual is. And its bulk dual has a tower of massless particles. And the theory in the bulk is higher spin vacillia theory. So in these two cases, they're on opposite sides of the board because they're very different. Both the theory, here we have a free ON model. Here we have n equals 4 super Yang-Mills. This is a strong tuft coupling. And the theories are very different. The boundary theories and the bulk theories are also very different. Here one has string theory with a large gap. So a few light particles, the rest are very massive. Here one has higher spin vacillia theory where there's simply a tower of massless particles with spins changing even integers. And here you mean three dimensions? Three or more or other dimensions. Yes, the canonical case is ADS-4 or CFT-3. But it's been extended in some ways to other dimensions. So those are two cases. So SYK, I think it's fair to say, sits in the middle. So the diagrams, the SYK sums are melon diagrams. They look like this. I'm only drawing the fermion lines. So this is SYK. So SYK can be said to be a new large n model. That's truly different from either this or that case. So it sums melon diagrams as opposed to bubbles or planar. And so now we have that the boundary theory is SYK. The bulk theory, one thing that is known is that there is now a tower of massive particles. I'll describe that in more detail shortly. So there's a tower of particles and their masses are roughly spaced by even integers, roughly. So it's different from this case. In that case, I guess it's more similar to this case. There's no large gap. But they are massive. And the theory, what the bulk theory is, that's dual test, why SYK still remains an open question. And in the second half of the talk, I'll describe progress that's been made towards answering this question. Unlike these, well, in this case, there was a brain construction from which, Maldessayna knew that n equals 4 super Yang-Mills is the dual of string theory. In this case, Klobunov-Polyakov, there was Vasiliyov theory around. And so it was natural to say that it's the dual of ON. Whereas here, there isn't any obvious candidate for the bulk dual. At least not one that's mentioned. Actually, let me go here. So in terms of symmetries, after you do the disorder average, which is all quantities, will be after disorder averaging, SYK has an ON symmetry. Yes? So in the case of string theory, I mean, we know that the one over an expansion is a genus explanation. Yes. We have something similar for SYK, where it can sort of be identified. Not that I'm aware of. What? Good. So in the matrix case, as was known from Tuft in the 70s, already in the 70s, it was believed that matrix theories are string theories because the expansion is in one over n with the genus, which is like a string expansion. And so one could ask, is there an argument in SYK that this is some theory of extended objects? An argument like Tufts. I'm unaware of such an argument. So there has been work on papers understanding the one over an expansion, but in the context of tensor models. But I don't think it's anything that simple. Well, I'm not sure. So because it has an ON symmetry after disorder averaging, it's, by the usual rules of ADS CFT, the fields that are due to something in the bulk are due to the singlets under ON. So the bilinear singlets are schematically of the form. So this is analogous to in the ON model. And that duality, the bulk, one considers the ON singlets. So we'll consider them here too. So ON will be this operator. And these are the things that are due to fields in the bulk, phi n. And these have dimension, which I'll call hn. And the fields, the bulk fields due to ON have masses by the usual ADS CFT dictionary, which is set by the dimensions. So if we were weak coupling, then the dimensions of these operators would just be 2n plus 1. And so from this formula, you see that the masses in the bulk would be approximately 2n and as an integer. So that's the statement I was making, that in the bulk, there is a tower of massive particles. So these dimensions are not 2n plus 1, because if we're interested in strong coupling, they're an order 1 shift away from 2n plus 1. And I'll say what they are. So how would you speculate if you speak about massive particles, but how would you speculate with the statement that there are no cosy particles in the passive type? Good. This is the bulk theory. This is not in SYK. So there is a dictionary, the ADS CFT dictionary, by which we will construct the bulk theory dual to SYK. So it's like in the usual case of strongly-coupled CFTs, there are no quasi-particles, everything strong coupling. And indeed, when the boundary is strongly coupled, the bulk is weakly-coupled. So it's because there are no quasi-particles that we can, one could say, there will be a nice bulk, perhaps. OK, so now let me go through this checklist of, so let me list why SYK is of interest for all these different reasons. So the first one, string theory. So as I said, there is no natural candidate for the bulk dual. And so if it is reasonable that the dual of SYK is perhaps a new theory of extended objects. So that's why it's interesting. From this point of view, from the point of view of ADS CFT, it is a simple model of holography. Indeed, the title of Katayev's talk was a simple model of holography. And so one could hope that using SYK, because it is solvable, one can understand in more detail how holography works, how the CFT degrees of freedom organize themselves into a bulk. This is, of course, a question that's been well thought about over 20 years with partial success. One could say the success is partial, perhaps, because the theories that have realistic bulk duals, like string theory, are very hard to solve. Whereas something like the O and free O1 model, the bulk dual is Basilev, which is very hard to understand. And so since that's why K is in between, it's less easy to solve than On, but less difficult to solve than n equals 4. One could hope that it's an appropriate range where neither the boundary nor the bulk are sufficiently or too difficult. Yes? Why can't you get this power by just doing some KK reduction? So for example, like in this paper by Ebbetsky and Das, if you just take ADS two times another direction, and you just solve the wave equation there with some funny potential, you can always hook up any spectrum of part of KK. You can, correct. The Cuba couplings will be completely off. We gauge the whatever. Yes? Is there something that has to look at singlets or just? No. No, it's, here it's being forced by hand that we consider singlets. It's unlike, so in the case of that case, there one can introduce a term, Simon's term, then take the term, Simon's coupling zero, da, da, da. Here, we're just introducing it by hand, but it seems like a reasonable thing to do. Otherwise there would be too many particles. Yeah, yeah, otherwise there'd be too many particles. So that's why we say singlets. I guess one could, one could gauge the On, because it's zero plus one dimensions, it doesn't really make any difference. So you could gauge the On and then just restrict the singlets, but no one forces, unlike a real gauge theory, you're not forced to consider singlets. So you could try to consider non-singlets and ask whether they're dual too. Yes? Since you put it in this context, so the matrix models and the vector models, we know how to study in various dimensions, and you're gonna study this S y k in zero plus one, but since you're building a bulk, is it, I mean. The bulk will be. What would it be? Can you make any progress in other than zero plus one? A theme sort of melonic, but perhaps because. Yes, good. So the bulk in this case will be ADS two. It would be extremely useful if there was a theory in higher dimensions, that's not zero plus one, which only summed melonic graphs, but I'm unaware of such a theory. Because the other two models are easiest to study, not in third one dimension directions. I mean the bulk of dual of these things. Yes, ADS two has some features which are not present in higher dimensions. Currently there is only S y k summing melan diagrams. Currently there's only S y k and that's in zero plus one. One can consider higher dimensional versions, but they're not the same. There is nothing. There's nothing. They will not have the same symmetries. S y k is the only nice one. So this is in fact, so this is why it's interesting from the QFT point of view that it's a new large N model, which is more involved in the ON model. Summing melons is more difficult than summing bubbles. Summing bubbles is almost trivial. And as we just said, it would be very nice if there was a higher dimensional version that only some melons, but there isn't really anything yet. It would also be nice if there was a limit in which summing melons corresponded to something physical. So if you took a lambda phi to the fourth area or something and summed only melons, would that correspond to something physical? As far as I know, the answer is no, but it would be nice if it was yes. Excuse me. Why the dimensions enters in the arcane counting? Just Lagrangian, that's in zero plus one dimensions. I understand, but why the graphs? It seems that the N counting of a graph depends on the number of dimensions. And if you take the number of dimensions, why the graphs are not the same, the leading graphs? If you, well, you say just replace the, it will no longer, so one can simply write the same Schringer-Daisen equation for the two-point function, but just make it in two dimensions rather than one and assume the ansatz that in the infrared one has a fixed point. You will get an inconsistency, so that would be an incorrect. At every point, but the selection of graphs seems to be the main thing that depends. That's correct. That's, that's correct. Yes. About something physical. Actually something very close to this is when you do fermions in a high dimension, this one you get EMF two. Ah, yes. Which is, well, it's either melons or melons for something else, but it's very close to this kind of approximation. So that's physical, if you accept large D as physical. Yes, large D is like, good, so large D is like the fact that we're using all-to-all interactions. So that's a very generic feature. The thing that's special is summing only melon diagrams. So in fact, so this is now we get to point number four, quantum gravity. So in fact, summing melon diagrams is something that some of you have been doing for years before SYK in the context of tensor models. So in tensor models, those have been known for a while now that one sums melon diagrams. So tensor models are a reasonable thing to consider because it's known that matrix models are discretizations of surfaces. So one could wonder from which you build geometry, one could wonder if tensor models are the natural thing to build higher dimensional geometry. So from that perspective, it's reasonable to consider tensor models and those some melons, which is what SYK is summing. From the condensed matter viewpoint, so as was mentioned, SYK does not have quasi particles and it has been an open problem to describe non-Fermi-liquid strange metals. And so SYK can be viewed as a model of this from which one could try to understand its transport properties and explain properties of strange metals that there is resistivity scales with a temperature unlike temperature squared like for Fermi-liquids and so on. Next, quantum chaos. So in the 60s, Larkin and Avchennikov pointed out that there's a natural analog of the classical Lyapunov exponent to the quantum case, the quantum Lyapunov exponent, which is defined as the growth exponent of an out of time order four point function. There are other definitions of quantum chaos, but one can use this one. It's reasonable for reasons I won't get into. In any case, this definition of quantum chaos, unlike classical chaos, has a bound. So there's an upper bound on the quantum Lyapunov exponent and as Gitaev recognized, SYK saturates this bound in the infrared at strong coupling. So it's maximally chaotic. Finally, black hole information. So the reason that it's interesting that it's maximally chaotic if one is not interested in chaos by itself is the fact that black holes, if one computes the same quantity of the out of time order four point function of black hole background in Einstein gravity, one also finds the Lyapunov exponent as maximal. This was initially some of the motivation that SYK is a good model for black hole. To be more precise, the bulk in two dimensions, there's no Einstein gravity, one varies the metric, one just gets identically zero. And the bulk gravitational sector of SYK is your key title of bone gravity. So dilaton gravity. So there is a Lagrangian is phi r plus two plus l matter. So phi is the dilaton. So one introduces and two is the cosmological constant. So it's different. If the dilaton were constant, this would be like the Einstein Hilbert action. But this JT model was introduced as a model of gravity in two dimensions because Einstein gravity is trivial. And in fact, it naturally arises when it dimensionally reduces from higher dimensions. In any case, if one computes, if one forms a black hole in dilaton gravity and computes the Lyapunov exponent, this out of time order four point function, then one finds that it's saturated in the same way that it was saturated in SYK. And indeed the gravitational, so to speak, gravitational sector of SYK, that portion of the bulk is this action. The remaining open question is what is the action for the matter for this entire tower of massive particles? In any case, so in this context of dilaton gravity, one can pose the information paradox and try to resolve it. It's unclear how helpful SYK is to this because the information paradox is a bulk problem, so one would need to state, so it's unclear what CFT quantity one should compute in order to resolve it, but if one could formulate what quantity that is, then in SYK one could compute it, perhaps. The dilaton, yes, partially. So I wasn't going to describe this too much, but the reason I said, so you'll notice that this is not ADS-2 if it was because this is not quite ADS-2, which is consistent with what I said that SYK was only nearly CFT-1, so there's one operator in the spectrum that breaks conformal invariance in the infrared due to reparameterization invariance. This is related to the fact that there's dilaton gravity. I can give a more detailed answer afterwards, but simplify the case information log entropy. Again? Five value of five on your horizon, if horizon horizon. Yes. The case information about- Yes, yes. So it should have an analog in your type. Yes. Yes, another feature of SYK is that there is a large ground state entropy, but there's a large, at infinite end, there's a large ground state entropy. Which looks analogous to extremum black holes having entropy. Okay, so that's the end of part one. Any questions before I move on to part two? More questions? No, go ahead. Yes? Is there a relation between the operator that breaks conformal invariance and the diapunov exponent? Yes. The entire feature of maximal chaos is due to this breaking of conformal invariance. One would have naively thought that the diapunov exponent is a property of the four point function, so the entire spectrum of the theory, all these operators should enter, but because of this breaking of conformal variance, there's one that dominates, and so it's that one that determines. So it's a universal, the maximal chaos is probably a universal feature of CFT-1s, of nearly CFT-1s. It's not something that's special to SYK. Which in fact, in retrospect, is not so surprising because black holes in Einstein gravity, they all saturate the chaos bound, and we know there are many different theories that give Einstein gravity in the low energy limit. So Einstein gravity is universal in that sense, and so this analog in two dimensions is also somewhat universal. So if you somehow suppress the contribution of this operator, what will be the four point function? Do you have no diapunov exponent? It would, but it would be. So the correction, the correction to the diapunov exponent if you move slightly away from strong coupling, one could ask what that is. That's in fact, larger than the bound, and the coefficient is of the wrong sign, but since it's a subleading correction, it doesn't matter. Do you understand that you are saying that the main property of SYK is not to saturate, to be maximally chaotic, because that is sort of general, but it's to be solvable or what? Okay, if it wasn't solvable, then we wouldn't know what the diapunov exponent is. No, it's, so in this model, in this bulk theory of dilatone gravity, if one computes the diapunov exponent, it's maximal. And this is the theory that one would consider for gravity in two dimensions. So from that perspective, it's reasonable that the bound is saturated for this reason that it's nearly conformal invariant. So do you think any nearly conformal invariant theory in one dimension should have the calculation of yield? Yes, it will have, yes. I mean, yes, to the same extent that one would say that any large and CFT in higher dimensions is due to something in the bulk. That is something that we could not call gravity. Yes, correct, correct, correct. Yeah, normally we would say it's gravity if that large and CFT has a large gap. Then we'd say gravity, yes. Okay, let me move on to part two. But for no energy excitations above ground state, what do you expect? That they are exponentially close to the ground state in the number of particles or it's not required or do you expect that random, ignore Dyson statistics or that type will be very, very early excitations above the ground state or it's not required? I would think that the model should have ignore Dyson statistics, but I think that's been studied but I'm not sure what the answer is, so let me look at that. One, it's known that it has, from 71, it's known that it has ignore Dyson statistics in the middle of this thing. Yeah, there's a question about low and- Oh, yes, yes, I see what you're asking. The question, low energy, that is really then, I'm not sure. Yeah, let me just say I'm not sure. No, no, I meant to leave that one up. Oh, and it's true, I have no mechanism of taking it down. I've got any more questions and I don't know how to respond. So constructing the bulk tool. So as we said, the bulk has a tower of massive particles and so we would like to find their interactions. So the cubic coupling, squirted couplings, et cetera, of this entire tower. So the idea is going to be that we'll compute a two K point correlation function of fermions in S, Y, K, and then using the ADS-CFT dictionary, this will tell us about a K body interaction in the bulk, the coefficient of this term. So the larger that K is, the more suppressed in one over N, these coefficients are. So they're N to the K minus two over two suppressed. So by computing correlation functions in S, Y, K, we can order by order in perturbation theory in principle reconstruct the bulk Lagrangian. So that's the idea we will. So the program is we will compute these correlation functions, then construct the bulk Lagrangian, and then hope that at the end, one can give some interpretation of this bulk Lagrangian, give some, hopefully, in terms of some simple theory of extended objects, which will naturally reproduce those couplings that we will find. It's unclear if that would work, but we can at least find the couplings, yes? Is this a lot of expansion of the effective Lagrangian by local things? No. I will simply use, so within S, Y, K, we will compute correlation functions of these O's and then use the ADS-CFT dictionary. So we're not going to try to rewrite the Lagrangian in some ways so that it looks like a bulk, we will simply apply the rules of ADS-CFT. So to get started with the four point function, so the four point function of fermions, so one draws all the Feynman diagrams that contribute a leading order in one over N, so it's the sum of these diagrams, each line is really the full propagator, so it includes the sum over all the Mellon diagrams. So this is the fermion four point function. So this is known what this is. So once one brings two kais together, two kais, so this is kai, kai. One gets a sum of these ONs with some OP coefficients Cn. So the dimensions of these are Hn, as I said before. Which is 2N plus one plus two delta plus some correction, which is not necessarily small, but of order one. So two delta plus, so from looking at this operator, the dimension one expects is delta, delta from here and here, two N plus one from the derivative, plus some anomalous dimension coming from the fact that we're in the infrared, which I labeled by epsilon. So one derives these precise dimensions by summing all these Feynman diagrams. Can the epsilon are computable? Yes, they're not. You just sum these diagrams and you get it. And these are just ladder diagrams. So the four point function of Bayesian integral equation, so it's trivial to write down what that equation is and then it's somewhat non-trivial to solve it. Yes. And from this one determines the dimensions and the OP coefficients. So more explicitly these ONs, they behave like CFT two point functions. With dimension HN. So ON ON goes like one over tau one two to the two HN. So as we said, the bulk, so now we're going to apply the ADS-CFT dictionary that the dual of these operators, each of these operators is a bulk field. And so Feyn is a scalar field of this mass. So one could draw the corresponding bulk wind diagram of this two point function. So it's just, this thing, this is the bulk. And so from this computation we learn that the bulk Lagrangian has this tower of scalars and ranges from two to infinity where the masses are set by these dimensions, which we know. So at infinite, at leading order one over N, this is the bulk Lagrangian. And so the next question is, what are the three point functions of the O's? Which we'll then tell us about the cubic couplings in the bulk. The four point function doesn't need to also have a part of a continuum spectrum. No. So what was the one half plus IS? Yes, so in the computation of the four point function, so the four point function is trivial in the sense that it's just solving that integral equation, the difficult part of solving that equation. So one writes, one rewrites. So one needs to diagonalize this matrix. One could say it's a matrix, this kernel. And one chooses a complete set of eigenvectors. And that complete set is what you're thinking of these dimensions, H is one half plus IS and two N. But then one needs to do that integral and sum of one half plus IS and two N. After doing that, then one gets a sum of conformal blocks and those conformal blocks are at the physical dimensions which are these. So this is in fact analogous to higher dimensional CFTs. Though slightly different. In even dimensions, one can write the four point function as a sum of conformal blocks or one can write it as an integral of one half plus IS. In odd dimensions, the representation one needs, if one is going to use this basis as one half plus IS plus the two Ns. So that's slightly unfamiliar. But at the end, it's just sum of conformal blocks with these physical dimensions. So are these going to be operative if we take the OP of two Ns? Are you only going to get ONs or you might get one bit more. Good, ON, OM, CNMK, plus the double trace. So we can write this, this term is squared of N and then here there's a one plus one over N, a double trace, ON, OM, P. This notation means double trace plus one over N, double trace of some different OR, OS, Q. So one would like to compute all these coefficients so right now I'll compute this one. It's the leading one and one over squared of N. This one just comes from free field with contractions. So there's no cubic coupling that's in versus dilaton that only shows up as four for order or? There, good. I'm restricting to, I will not be saying anything else about how, so there, I will just be talking about the matter sector and the couplings among themselves and not discuss how they couple to the dilaton. That's in principle computable as well. Though all of, so the reason we need the dilaton is because we can't work at SYK in the deep infrared. We have to move slightly away. One can keep moving away and then there'll be more and more terms involving the dilaton couple to other stuff but I'm not so interested in that so I won't be computing it. But is there, is there not a cubic coupling? Just dilaton cubed? Of dilaton couple to two. Yes, I think there is. And it's in principle computable though I will not compute it. Okay, oh dear, okay. So the thing I will compute is this coefficient, the C and MK. So this involves computing the six point function of the fermions. So the Feynman diagrams look like this. So there are two classes of diagrams, one class are planar and the other class are not planar so they're a little more difficult to draw. So this is the six point function. So this involves basically three four point functions. Those are the four point functions glued together. So that's what I did here. So I glued them in a way which is planar and here three four point functions come together to line. I'm drawing all my pictures are for q equals four but the equations are for general q. q equals four meaning four body interactions. Good, so one can again sum these diagrams and from this compute the six point function and then from that extract the three point function of the bilinears. So this is non trivial to compute but we computed it. So this determines, so these two things give this three point function. It's a conformal three point function so its functional form is fixed by conformal variance but there's a number, well a number of, depending on n one n two n three which one has to determine and that's determined by summing these diagrams and this in turn using the ADS CFT dictionary determines the cubic coupling. So in the bulk, in the bulk there's just one Feynman diagram, Feynman, which is this one, Feynman-Witton diagram. So the reason I say this determines that is because one simply assumes the bulk cube that one has a bulk cubic term of this form uses ADS CFT to correspond to compute the CFT three point function one gets from this bulk Lagrangian and then compares that with the three point function one gets in SYK and that sets this coefficient. So in this way we're deriving the bulk dual of SYK. Could you just say with, see do you compute it in the same manner of this conformal blocks and then? Good, so. So we insert three four point functions, and then one has to do three integrals over these and so we do those integrals. It amounts to, well these points are taken to be close so the conformal blocks are very simple. One can, the hyper geometric function is just one but it amounts to, then one has to do what is effectively a four loop integral which is a conformal four point integral which is difficult to do. But physically this is very simple one just glues these together. Yeah. So let me tell you the answer. What are the structures responsible for this particular range of distribution of the stream? None. The dictionary is you compute the bulk boundary dictionary maps the CFT three point function into the bulk cubic coupling. I don't see any way in the bulk to distinguish which Feynman diagrams. So some all Feynman diagrams are that and then the dictionary maps that some. You can't map. I don't know of a way to ask the question what is the bulk dual of a single Feynman diagram. Though that would be an interesting question. I don't think it would be anything reasonable. Yes. I'm a little bit confused about, so you mentioned that there are many, there are many ground states, to the national ground states, right? No. Or approximate the rest in the large delimit. Yes. So the question then is, when we compute these correlators, on which state are we computing them? The ground state. And we have between my questions, whether it's clear how to distinguish the one over the corrections coming from the interactions between the particles from one over the corrections that you may have by considering either the ensemble of ground states or particular microstates. I don't know. So I'm at zero temperature in the vacuum. There's a unique ground state. I just compute in that. But there are many ground states, right? No. In the large delimit at least, there's accumulation of states. At finite temperature. At finite temperature. But do you mean by ground state at finite temperature? Yeah. I'm getting a little confused. I just, I just compute. So this is Euclidean computation, right? Yeah, but that, I mean, yeah, everything's Euclidean. It doesn't matter. It's the same. Lorentz and Euclidean were in one dimension. It's the same thing. Well, you have the ground state entropy, right? In this model. Yeah. So there are many ground states or in some approximate sense. Yes, yes, yes. So then the question is, these correlators are computed on which of them? What's the difference? Well, I imagine that these correlators should be approximately the same on all of the ground states. Yes, good. But I'm worried about the one-over-end corrections statement and how those one-over-end corrections are distinguished from the one-over-end corrections. Yeah, yeah. Okay, I see the question. I'm not sure. I think it doesn't, I'm not, okay. Let me, let me state the answer to what this lambda looks like. So the, this lambda, it's useful to think of these two contributions, two classes of Feynman diagrams contributing to lambda. So, so this one is in fact easier to calculate and its form is somewhat simple. It's some product of gamma functions of the NIs. So, so lambda is, we can write this as lambda and one and two and three, one plus lambda and one. So this comes from, this one comes from these diagrams and this comes, comes from these diagrams. So this one is, is simple and for large Ni it decays. This one, this, this one coming from these diagrams is complicated. We can only evaluate it at large Q and it grows exponentially, it grows exponentially for large Ni. So, so the couplings, the couplings, these couplings for, for the, the self coupling of a very massive field grows exponentially with the mass, which seems surprising and this actual function is complicated. Okay, so that's the answer for, for SPI K. So one would like to understand this answer better. So then we did the following computation. We considered free field theory, really generalized free field theory and so compute, we compute O and one, O and two, O and three in, in generalized free field theory. So, so just take those, those operators there. That's actually heuristic. They're more complicated because they need to be primaries and just do with contractions and compute these, these three point functions. So we do that calculation and the answer is in fact, extremely similar to the answer one gets from, from these Feynman diagrams in SPI K. The answer in free field theory is similar to the answer in SPI K. In fact, in fact, one can write lambda and one and two and three as a simple function and one and two, three lambda and one. Can you explain what generalized is? I will in a minute. Let me, let me just say this. So if, if one considers the bulk dual of this generalized free field theory and computes their lambdas, then these lambdas are related by a simple function to, to, to a piece of the dominant piece of, of, this piece of, of the lambdas coming from SPI K where this function goes to one for large Ni. So in other words, the couplings, the couplings of the highly massive fields among themselves in the bulk dual of SPI K it approaches the couplings in, in the bulk dual of a generalized free field theory at large mass, which at first seems surprising. So by generalized free field, I meant take the, the two point function chi I to be one over sine tau over tau to the two epsilon and then take epsilon to zero at the end. It's just because if you take sine tau then, then it's topological. So taking derivatives gives just delta function. So we make it. So, so there, there are two things, two things left to explain. The first is why, why we got, why the result of this difficult calculation in SPI K a piece of it, only a piece of it, but the one that's dominant at large mass, this one is so similar to generalized free field theory so that we've understood and that result will appear soon but I don't have time to explain it. The, the second question is what happens? So that was the three point function. So the cubic couplings in the bulk, the second question is, is what happens at the level of the quarter coupling? So then one should, so one should now compute the eight point function in SPI K and some of these diagrams. So this is again, conceptually easy but technically difficult to do these integrals and the doing this should be very useful because this will teach us about the, the, the quarter couplings. But, but in the bulk, there'll be many quarter couplings. So in the bulk one will have lambda n1 through n4, phi n1 through phi n4, but one will also have terms with derivatives on some of the fives. We each with their own coefficients. And so it will be very interesting to know how those higher derivative terms, how those coefficients compare to the lower derivative ones. If, if the higher derivative, so they'll also be terms with lots of derivatives, lots of derivatives acting on the fives. So the, an interesting question is how non-local the bulk theory is. So if it were the case that one just had this term with no terms with derivatives, then the bulk would be local at this level. That seems extremely unlikely. So there'll be higher derivative terms. And depending on how quickly their coefficients decay or not decay will indicate to what extent the bulk is local or non-local, which is, which is an interesting thing to ask because one might think that, well, we have a variant of the model which is conformal at all couplings. And so we can tune, that's the explanation of the first point. So I'll say it but won't write any equations. We can tune the, we can change the coupling from, from zero to strong coupling J and observe how, how the Cuba couplings change. And so it's interesting to ask, as you change from the theory being a free theory to being one that's, that's strongly coupled, how, how the bulk changes. Since one would think that the bulk should be more local, the, the stronger the coupling in, in the CFT. But I don't know the answer yet. So thank you. Okay. We have many questions during the talk. So I think it's just a few, please. So if I look at the looper's function or defective function. Yeah. I might identify the items and I don't know if their thing would correspond. Good. So the, the, what I was discussing was constructing the bulk Lagrangian, the tree level bulk Lagrangian. And then there are loop, indeed. Then there are loop corrections in the bulk. Those correspond to one over N corrections in, in the CFT. So in particular, if one considers the, so the C, let me go over here. So to leading order in one over N, the four point function of fermions was this. To leading order in one over N, but this has corrections which one over N corrections which go like this. So one could compute these corrections and that would correspond to these loop, loop diagrams in the bulk. So you can see can the diagrams of the fermions sort of you see the probability of the. You can, well the diagrams of the, the rules we're using, the fermions themselves don't correspond to anything. So from this I extract the, from this I would extract the, so from this I move these close together and this gives the loop correction to the two point function of the ONs. So the two fermions make an ON. So this is a loop correction to the ON. You said that the CLE is almost conformable variant. Yeah. But then you use J as a free parameter to change it, the series. That was my comment at the end. Good. So it would be, so David and I have, so David and I introduced to appear. So we call this CSYK. So this is like SYK, but the Lagrangian we change the, the kinetic term. And instead of making a chi detail chi, we make it by local. And the interaction term is the same, J I 1, I Q. So, so this is CSYK. And the statement is that this has a line of fixed points. Ah, sorry. Delta is one over Q. And this has a line of fixed points. So this is conformal, has SL2R symmetry at any value of J. Normally it's very hard to get a line of fixed points. This is relativistic or what? No, no, no. Well, one dimension. So what's the dimensionless parameter on this? J, J, the two point function of J. So, so J is again disordered with that two point function. Isn't that dimension free? Oh, yes. There J had dimension one. Here it's dimension less. This, J, exactly. Now chi has dimension delta. In other words, what we did was that in SYK, the fermion flows from dimension zero, which is what the kinetic term gave it the dimension one over Q, which is what the interaction wants it to have. We simply change the kinetic term so that it starts out with dimension one over Q and then always has dimension one over Q. And now, and now this, the dual of this, there is, so because we changed the kinetic term, there's no diffeomorphism invariant. There's no time reparamerization invariance and the bulk is not gravity. It's just field theory in a fixed background. Right. And so we, there's no dilaton in the bulk. So for the purposes of constructing the bulk theory, this makes it easier because, yes, for the purposes of studying the information paradox, it's not good because there's no gravity. Okay, is there one more question? Okay, if not, let's say Vladimir and all the speakers of today.