 Okay, thank you. First of all, I'd like to thank the organizers for inviting me to this very stimulating workshop. I will be talking about a paper that appeared a few months ago with Vijay Brahmanyan, Bartek Cek and Gabor Sharusi. And I will first give you the very quick version and I'll go through it more slowly. So the strongly coupled D1-D5 CFT is a microscopic model for black holes in string theory. And this model is expected to have chaotic dynamics. We will study, instead, it's the integrable limit of this model, very quickly coupled. And in that limit, we know precisely the operators that create microstates of the lightest black holes in the theory. In those microstates, we will compute two-point functions of light probes, normalized in a certain way, which I'll mention later. And it was known already that those two-point functions exhibit some universal early-time decay followed by sporadic behavior at late times. And that's what we would like to understand. So in order to understand that late times sporadic behavior, we will propose some time averaging to be applied to these two-point functions. And to first test that method, we will apply it to random matrix theory. And we will show that the progressive time average that we will use actually smooths what is known as a spectral form factor in that model, giving essentially the same result as the ensemble average that people usually use. So this progressive time averaging can be used also on systems which do not have an ensemble to average over, so we will use it for the system of interest. And if we do that, we will find that this early-time decay will be followed by a dip, then an increase, which is called a ramp, and then a plateau. And this structure is in remarkable qualitative agreement with a recent discussion of the SYK model, but we will also comment on significant quantitative differences and similarities between these models. So that was a short version, let me now go through it a bit more slowly. So the fact that black holes and chaos are discussed in the same talk is no accident. Black holes have been known to be thermal objects for a long time, and it's also been known for a long time that what underlies thermal behavior is chaos. So there's several manifestations of this chaos. One is that it governs the relaxation to thermal equilibrium when you perturb a system. And a second aspect of chaos is that there is strong sensitivity to initial conditions. So in classical models, the sensitivity is typically captured by Lyapunov exponents. This is a simple example where you can imagine Q to be the position of some particle, and the dependence of a position at some later time on the position on earlier time is given by these Poisson brackets, and this typically diverges exponentially where lambda is the Lyapunov exponent. Now in quantum mechanics, it's more complicated to discuss chaos. The most naive notion could be that you consider states similar if they have a large in-product. But the problem is that the in-product of two states is preserved under unitary time evolution, so it will never change, that will not give an analog of this classical notion of chaos we have here. A better notion is available in a semi-classical approximation, where you can consider states to be similar if they discuss, for instance, two blobs in phase space which are close to each other, even though they don't significantly overlap. And to quantify that, you replace these Poisson brackets by a commutator, and you consider, for instance, the following observable where you take such a commutator, so I generalize Q and P to some more general operators, and then I multiply it by the Hermitian conjugate to get rid of the phases, and it turns out that this is an interesting object to study in this context. So where does this come from other than from this analogy? Well, this operator W of T, you can view as describing the growth or the spreading of an operator. Consider, for instance, some chain of quantum spins, and consider V and W at some time zero as acting on two different spins, so they will commute with each other initially. Then as W is evolved with this Hamiltonian in a chaotic system, you will find that its support will spread, it will become a product of sigma matrices acting on many spins at the same time, and at some point, an operator at this other location will start feeling it, and in a chaotic system, this will grow exponentially. So this is what one typically finds, and my presentation here is based on a paper by Polchinski from a few years ago. You typically find that this function f of t, so this commutator squared, grows exponentially for some time, and then at a later time, it will saturate just like in thermal equilibrium. So if you perturb the system, it will eventually reach thermal equilibrium again. In the context of ADSCFT, where thermal systems are related often to black holes, these exponential growth and saturation have counter parts. So first of all, this exponential saturation, which is governed by what is known as Rewell resonances or poles of retarded two-point functions, those correspond to the well-known cause-in-normal modes of black holes, which describe how a black hole will relax when you perturb it, and which, as I'm sure everybody knows here, have been found in the recent gravitational wave measurements. The initial transient growth, which is the analog of the Lyapunov growth I discussed before, that's due to another exponential that appears in black hole physics, namely if tau is the proper time of some in-falling objects and t is the short-shell time corresponding to it, then those are related by an exponential factor like this, where beta is the inverse Hawking temperature. And well, in the recent last few years, there's this precise correspondence that the Lyapunov exponents in systems due to black holes precisely correspond to essentially 2 pi times the temperature of the black hole. So roughly what's the timescale for it to get to this saturation point? Well, that will depend on the details of the system. So for instance, if you have a very large system, you can imagine starting to perturb it at some position, and then this, if you have local interactions, these perturbations will spread ballistically, and while the bigger your system is, the longer you will have to wait until the operators cover the whole system. I mean, is it, for black holes, is it believed to be an order in the scrambling time or some much longer time? Well, so typically the scrambling, or let's say, I think it's related to scrambling time, so typically if you have some, so for instance, suppose you have some system of matrices, as you would have in many ADSEFT models, then you would have fast scrambling within the matrix degrees of freedom, and then you would have ballistics spreading because of the local interactions in space, and there it depends on how big the space is until it saturates. Okay, so it's been conjectured that this Lyapunov exponent that you find in black holes in Einstein gravity is some universal upper bound you can have, that's a so-called chaos bound, which is therefore saturated by black holes. And in fact, the law has become that whenever you see a system that saturates the chaos bound, you think, aha, this system may be dual to black holes, and that's in part how the interest, the recent interest in the SYK model emerged, but I'll mention that later. Now the main focus on this talk will not be on the Lyapunov growth, but it will be on this Ruel saturation, which you also find in two-point functions. So I took a four-point function here because that's the smallest object that exhibits this Lyapunov growth, but this saturation you would also have in two-point functions, so I will actually focus on two-point functions in the remainder of this talk. But the thing I'll be interested in is that this saturation is not quite true, as already was mentioned yesterday in the introductory talks, because quantum mechanically, if you have some system with discrete energy levels, this decay cannot continue indefinitely. And there's several probes that have been proposed to study this lack of monotonous decay. So the first is two-point functions. So here I take a two-point function in some thermal state, and if you expand it in some energy basis, you get the following structure. And at sufficiently early times, you can coarse-grain over these energy levels. Their discreteness will not matter. And if you do that, you find a monotonous decay, which is often exponential. So that's the quasi-normal mode decay that I referred to before. If you look at late times, however, then these various phases will be sort of independent, so you will have some sum over lots of random phases, and this will give rise to erratic oscillations. And this happens for this thermal two-point function I wrote down here. It would also happen for two-point functions in pure states. It would be extremely similar. These phase factors would still be the same, and those are what governs these erratic fluctuations. It's just what multiplies them. That would be different depending on what state you work in. But really, the important thing here is the energy spectrum, not the precise state you choose. So a somewhat simpler diagnostic than a two-point function has also been studied in this context. So it's called the spectral form factor, and the idea is that you start with the partition function, and then you analytically continue the inverse temperature, and then you introduce some real time, so to say. And if you look at this object and you expand it, you get again a structure with these phase factors. And while you can study this object as a function of time, so why is it useful? Well, first of all, it's somewhat simpler than two-point functions. Also, you can study it in theories where you're only given a Hamiltonian and you're not given operators, which is a completely generic, quite primitive quantity. It only depends on the energy levels, not on any choice of operators or states you might want to make. So let's look at this quantity and let's first look at what happens at long times. So I told you already it will have these erratical oscillations, but these oscillations will be around some average, some long-time average value, which you can easily compute. So if you take this average over infinite interval, then clearly the contributions where these phases are non-trivial will average out, and you will only get contributions from degenerate energy levels. So here I summed over the energies and E is the degeneracy of the energy level E. So for instance, in a situation where there are no degeneracies, all the NEs are one, this long-time average is just a partition function with twice the inverse temperature, and you can easily check, for instance, in the examples of CFTs, that this long-time average is much smaller than the initial value at time equals 0 of this quantity. So I gave here the result for a CFT and you see that it's suppressed by some exponential of the entropy of the system. So at late times there's erratic fluctuations around some tiny value, typically. So this spectral form factor was recently studied in the context of the SYK model, so it's been introduced in much more detail already. Let me be very brief. SYK is a quantum mechanical model with this Hamiltonian, and the size are N-maiorana fermions. The couplings J are independently drawn from some Gaussian distribution with some width that is set by some coupling J here. The Hilbert space of this, the dimension of the Hilbert space is 2 to the power N over 2, so it's exponential in this number of maiorana fermions. And I guess Ketayov noticed that this SYK model saturates the chaos bound, which motivates one to view it as a model for black holes. So if you compute a spectral form factor, you get a red line like this, so it starts at the initial value, then it decays, so this would be the analog of causing normal multi-K, but then this doesn't continue forever. At some point this erratic oscillations kick in, and one remarkable thing here is that the late-time average is higher than the values that are typically reached at some earlier time. So to study that more quantitatively, it's convenient to average over the ensemble, so you average over all the possible draws of this coupling you could get, and when you do that, you get this solid line here, and that's clearly a curve one can analyze. And what one sees is that this initial slope ends at the dip, after which there is a ramp, and there is saturation at this plateau at the level of ensemble-average quantities. Can you see those without ensemble-averaging also? Can you see these features without ensemble-averaging? I mean, from the red curve, it looks like the domestic rampato is in general... Well, that depends on the details, like for some parameters. Like in this case, I guess you could argue... Well, in this case, you could argue, you could see it, I guess. There might be other cases where it's more difficult to see, but in any case, if you want to study it more quantitatively, it's clearly useful to extract from this some smooth curve that you can then try to quantify. Do you average Z or log Z? It's not a quench, but you're nailed disorders. So these authors took the simplest prescription. Also, I think they averaged, like, numerators and denominators independently, but then they claimed in the numerical computations that if you took some other prescription, it would give essentially the same result. So I don't think it's crucial for anything that will be sent here. Okay, so the motivation of these authors was that by understanding these deviations from the quasi-normal mode decay, one can hopefully learn something about quantum gravity. So if one understands the bulk yield of this kind of behavior, it would tell us about the way gravity breaks down. Not perturbatively. So what they focused on was the similarities with random matrix behavior. So let me briefly review some relevant aspects here about random matrix theory before applying it to this SYK context. So a central conjecture in this context is that if you take a quantum chaotic system, which you can think of as a quantum system with a classical limit that is chaotic, then the spectral statistics of this system will be described by random matrix theory, by which I mean more specifically the Gaussian ensembles, which go by Gaussian unitary, Gaussian orthogonal Gaussian symplectic. So I gave one example of a partition function in one of these. So when one studies these systems with given energy spectra, which may be random or may be not random, depending on the context, the first step is to define some mean eigenvalue density. So if you have some disordered system with ensemble averages that's straightforward to define, you just look at a certain position in energy space and you take the average over all the couplings in your ensemble and that will be your mean density there. If you have only a single Hamiltonian without disordered to average over, what you do is coarse-graining. So you look at a certain position in energy space and you take an average density in a certain interval around it and that will give you the average. And then the statistics of eigenvalue that is relevant here is strictly speaking, one means the statistics of the unfolded spectrum and by that one means that one changes the, well rescales the spectrum so that the average value is everywhere the same, is everywhere one. And relative to that, you can look at the position of individual energy eigenvalues and what their statistical properties are. For instance, one extreme case would be that they're all equally spaced. Another extreme case would be that they're looked as if they were randomly inserted and that would of course be a completely different picture. So two important properties are first-level repulsion. So as we know from quantum mechanical perturbation theory, energy levels tend to repel each other and this is an effect that is important at short distances in energy space. So it tells you that in a chaotic system you're unlikely to find two energy levels that are very close to each other. A second aspect is spectral rigidity and that describes repulsion between energy levels at bigger distances in energy than just neighboring energy levels. For instance, what spectral rigidity tells you is that in quantum chaotic systems typically the actual number of levels in a certain interval will be very close to the average number of levels in an interval of that size which would not be the case if these energy levels were randomly inserted. So some intuition. Why is this level repulsion? Why does one expect it to be important in chaotic systems and not in integrable systems, for instance? Well, in integrable systems if you look at a space-space distribution of solutions, the solutions will be on invariant tori. So you could expect that different solutions will be on different invariant tori and they will not really overlap in space so that they won't feel each other, so to say. Whereas in chaotic systems because the solutions will explore the whole phase-space that will be sensitive to the fact that there's other solutions that will feel each other and therefore they will repel each other. So that's the intuition behind these properties of random matrices and therefore of quantum chaotic systems. So we had the spectral form factor numerically for SYK so let's see what it looks like for random matrix theory. In blue, I plotted the spectral form factor for a single realization of a random matrix and okay, you recognize the structure from the SYK plots I showed before. And again, we can average over the random matrices to get some smooth curve which has the same structure of a slope, a dip, a ramp, and a plateau. Now in random matrix theory, you can look more closely where the various features come from. Well, first of all, the fact that for different realizations you will have the same initial decay. That's called self-averaging. So this initial slope is self-averaging. It will not depend on the specific realization. And the details of this slope turn out to be determined by the mean energy density I mentioned before. So this mean energy density is not meant to be universal across quantum chaotic systems so that will be specific for any given system. But then the other features, particularly this ramp and this plateau, they are determined by the fluctuations around this mean energy density so by the spectral statistics of energy eigenvalues. If you look at the very late times, well, it's clear that there you will only be sensitive to the smallest energy differences because the others have already averaged out by that time. So clearly for very late times, it's important what neighboring energy values do. So this will be determined by the level repulsion properties. The ramp here is determined by the property of spectral rigidity so that's sensitive to what happens at bigger distances in the space of energy eigenvalues. The question is that Heisenberg time, is it round seven where you have this change of the two curves? The change from the second to third by the suppose that he's a Heisenberg time in the first level space? Yeah, I don't know exactly where it should be but yeah, it should roughly, does anyone know what Heisenberg time would be in this context? Generally it depends on energy because it just stays with this energy of excitation and it's a big pain. Yeah, so this time is exponential in the entropy which I guess is probably the Heisenberg time, right? The inverse should be proportionate to the matrix dimension. Right. So which is four, which would be... So this time will be set by the inverse of the smallest energy difference so if the total range of energies is some fixed number then you have exponential s of those. So this... Yeah, so I... Yeah. What about the position of the deep? Yeah, so this one can analyze. I mean, is it going to be exponentially entropy in time or could we... Okay, this will be exponential of the entropy. I think this will be... Okay, I don't remember for... How long is the ramp? Well, it's parametrically long so that's important and it will be exponentially in the entropy. So this will go like exponential s and this will be... I think it was exponential of some smaller power, with a smaller coefficient. But they're both exponentially long. So there is a parametrically long ramp here. I don't remember the precise coefficients. I suppose in mesoscopic fields that people speak of soulless energy and when the system becomes classically ergodic and the random matrix level statistics that means also the region of spectral religion is valid up to times... up to energy scales to the soulless energy. Okay. Of the timescales we had beyond this time but for random matrixes should be a short time because you are immediately ergodic. But if you take, say, a bent matrix model or this ordered system then you might have short times that's regular and you have to wait longer times then that might change. I think this might be dependent on the model and for the SYK model then you may have a difference between these... maybe a bit different because there's more sparse the SYK model compared to random matrix. Yeah. So, okay, in the numerical computations people have done for SYK and random matrices I was going to come back to it later but the claim is that from here onwards it looks identical, essentially. Then the details of the initial decay they are somewhat different also although it's difficult to see in the plots because that depends on the details. For instance, for random matrix theory this mean energy distribution is this Wigner semicircle and then the rate of decay at the beginning is set by these sharp edges so if that is different in other model it will be different initially but what is supposed to be the universal part for quantum chaotic systems is basically what happens from the deep onwards and indeed for that people find good agreement. That's what is behind these numerical computations. Why is it more useful to consider the spectrum from factor rather than directly studying the distribution value values? It takes a number of variants it's also interesting that you mentioned where you say the number of fluctuations of numbers of level in interval if you have anchor-related levels it grows linearly in a random matrix case like log n and in complicated system initially like log up to some energy solid energy and then somewhat different and the growing somewhat different is what's the left size specific of the system. If you imagine take a bent matrix model where you can have different regions localization or diffusive behavior you might tune this first energy scale with the parameters I believe that means it's a region where you have for short time some diffusive dynamics and then it becomes ergodic and the diffusive dynamic would be the left part I would suppose. Let me add that in the context of SYK people have also looked at other things namely not the spectral form factor but the short distance spectral properties of the system I'll have it on the next slide I think I'll come back to that but I think one motivation to look at this is that it's a proxy for the two-point function and this two-point function I presume in other models one may hope is easier to compute than actually diagonalizing the Hamiltonian to look at this D1, D5 system well okay so here is this some aspects of the comparison I will not go into the details but just to try and give you a flavor of the kind of tests people have done so I think this was this unfolded nearest neighbor level spacing I mentioned before I think this was first studied in this first paper here all the plot this is from the second paper so here you can see what you expect based on random matrix theory so those are the two solid lines you see here I gave two different ensembles because it turns out that the SYK model will correspond to one of those three ensembles depending on the value of N mod 8 so they appear all three and it has to do with discrete symmetries of the model that depend on the precise value of N and then so you can test whether SYK is indeed well captured by random matrices and you can see that for a value where you expect agreement with GUE these dots lie exactly on the curve and similarly for this value N equals 32 where you expect agreement with GUE so this works beautifully similarly for the spectral form factor there's also some subtle differences depending on N mod 8 so depending on which of these three ensembles you expect to get either some smooth saturation or some sharp corner I guess this is one or some kink here and this again also matches very precisely with what people find numerically in the SYK model so there's quite a detailed agreement between what people find numerically in SYK and what is expected based on the random matrix behavior that is expected there was also work earlier in numerical work we have a short cut yes thank you by SYK you always mean Q equal 4 yes that's the specific have anybody tried to change it I'm not sure in this kind of computations I don't know if anyone else knows I'm not aware of other values of Q for this computation I've been doing if you take for example Q equal 6 sparsity then you will become much stronger closer to full random matrix that means the larger the Q the stronger the closer I had the opposite feeling that Q equal 2 would look like random matrix no no Q equal 4 Q equal 4 is the case which is non-trivial in the sense the matrix is sparse because you have many zero elements if you have Q equal 6 at fixed size of the Hilbert space you have less zero elements and have more independent random numbers and you are much closer probably you have the same statistics but scale up to which for large energies up to which random matrix series will become larger probably yeah but okay I'm not really an expert I mean my expectation from what I've seen here is that well since these various SWI k-molds are supposed to be maximally chaotic I would expect that from the time where the behavior starts being determined by the spectral statistics that from that time onward it should agree with whatever random matrix ensemble is relevant for it where exactly the dip will be on details and on this initial decay which I'm not sure is that easy to anticipate but okay okay so now let's move to the main part of the talk which is a string theory model for black holes so the D1D5 CFT well the same CFT in which Stromiger and Waffer did their counting in the black hole microstate which will also prominently present in your stock this morning so I won't say too much about it it's the CFT is a marginal deformation of a sigma model on permutation a symmetric orbit fold so here's n powers of the four torres divided by the symmetric group if you choose not to deform the sigma model then you get the orbifold limit which you can think over the zero coupling version of the model where it's integrable so it's highly non-chaotic the strongly deformed version is where the appropriate description is in terms of black holes in gravity and there it is chaotic we're not strong enough to access this so we thought as a first step let's study these two point functions in a model where we can easily compute things where actually the two-point function had been computed before and see what we get in this context so we took the integrable limits and we took the microstates of the lightest black holes in the theory so those are the so-called Ramon-Ramon ground states we studied two-point functions of what we call graviton operators so they represent small deformations of the volume of the T4 for instance and then we normalize this two-point function by the same two-point function in vacuum and by vacuum here I mean the vacuum in the NS sector that's to get rid of certain divergences that you get in this integrable model like if you have perturbations that go around the circle they will periodically meet each other so to kill off those we normalize by the same two-point function in vacuum this object was actually computed more than 10 years ago and this is what people had found now you see if can say whether you see some structure in this or not I guess it depends on anything is structured so the black hole is just going on forever so the fact that it's starting at all that should be structured anyway so we found this curve in this paper okay this is a model with a well-defined Hamiltonian, no disorder whatsoever so what you do with this you can stare at it and then the idea was well what if we try to do some time average instead of ensemble average and okay you can play around with it a bit and then it turns out that the most promising thing would be to use a time window not a fixed one but one that grows with time in proportion to the time and well I can already anticipate that this gives a nice smooth curve if you do this but I can also imagine that you might think I'm cheating a little bit or I'm not really motivating why I'm doing this it's not like you're giving some disorder system and you average over disorder because that's what you're doing in systems like that so to try and motivate the thing before I show you the result I will test this proposal in the well-studied context of random matrix theory where people have been taking ensemble averages forever can you average over the black hole my hostess that doesn't help so if you take an average you will again find the same curve that looks pretty much the same except the details of this noise maybe the slations no no they won't no because that's essentially what I said at the beginning I think so this is for example an average in a thermal state and you have the same sum with these random phases so the only thing that is different is that you have these matrix elements as opposed to things that depend on the precise microstate but these phases are still the same so the details of this erratic behavior will be slightly different but it will look the same to the naked eye so that doesn't help so you really if you want to do ensemble average you need a way to average over different spectra but that we don't have because we have a well-defined Hamiltonian so was this an exact calculation of the two-point function or was this an approximation regarding the distribution of strings in the order from the... this was computed in a typical microstate in a typical Ramon-Ramon ground state the two-charge system or the two-charge system the typical of the three-charge system that's true, yes this has square root of n strings or square root of n strings yes indeed yeah so if you took a... oops okay in practice this is done in some ensemble this actual calculation not just one state right so it's okay so technically to do it you take typical microstate but then it's technically slightly easier to do this in some grand cononical ensemble but those are technical details I don't think that will affect what you see in the picture here yes what does play the role of the energy all these states have the same energy right? yes but you were pointing out about it phases e to the i so this system is slightly different yes okay do I have the formula here well okay so you if you compute this two-point function I don't know if I have it more explicitly later on but you can again it will be the excitation spectra that matter because if you look at a two-point function you have to insert complete sets of states so in that way all the energy differences or some of the energy differences in the spectrum will show up in this two-point function I probably have some more formulas later anyway let's first discuss it at this level so so here is the spectral form factor for a single realization of random matrix and let's ask instead of doing the ensemble average that people usually do could we have done some time average instead so here's the first attempt take some here's four first attempts take some time window with 10, 60, 110, 160 and what you quickly find is that you're squeezed because if you take the time average to be small the time window to be small then things go reasonably well at the beginning reasonably well but you fail to average out this late time oscillations remember that this is a logarithmic scale on the plot so you don't average things out here if you take a very big time window then you're doing a better job at this time average but you're completely butchering this deep structure over here so you're squeezed between wanting to capture the early times and the late times so that's where this idea of progressive time averaging came in where you say let me take a fixed time window but in the logarithm of the time not in the time itself and if you do that here's what you find so this works beautifully and it's what was the difference here? here I take a fixed window delta t and in the next one I will take delta t to be proportional to t so in other words it will be a fixed window on this logarithmic scale so this is for the spectrum for factor not for the color that's true yes so I'm comparing spectral form factors of one with correlators for the other system but it doesn't really make a difference for this kind of considerations you understand analytically why this works? we have some heuristic arguments but not an extremely precise one so we convince ourselves that there was some analytic reason to trust this because you know the variance in the spectrum and so on in the random matrix theory so we have some we have some argument at the level of the nearest neighbor distance so for that we have some analytic argument and show that it sort of works as you would expect but I don't claim to have a solid understanding of that I mean we just convinced ourselves using that but then the evidence we present is in the numerical one that's where I think it's quite convincing at least at this level so you see it basically indistinguishable from the ensemble average in terms of how accurately you get a smooth curve okay so with this encouragement from the test on random matrix theory we applied it to this string theory or black holes in quotation marks because again we're not in the gravitational regime we're in the regime where the system is weakly coupled in the field theory so if you do this progressive time averaging you find these blue dots and there it's obvious that there is this structure of slope deep ramp and plateau okay we did it for various values and it works so one lesson from this is that this deep and ramp structure which is present in random matrix theory and so okay at least the way I read some of the earlier papers was that well because of quantum chaos you expect this structure like this the implication does not go the other way around so if you don't have chaos and we definitely don't in our integral limit you can still have this structure with a ramp and a plateau so that's I think the main lesson but also because we are very far from being chaotic in the regime we work in there must be important quantitative differences and indeed if you look more closely the plateau we find this much higher than it would have been in a chaotic regime and it forms much earlier than it would have done in a chaotic regime and that's easy to understand let me come back to it a little bit later so first let me just mention that okay this was all numerics but we did come up with some analytic approximations to the ramp and the plateau so here I won't give you the details or even the formulas just to give you some flavor these dots are the numerical results and then these step functions that we have are the analytic rough approximations that we came up with and they match reasonably well and then maybe more to our surprise then we added to this analytic approximation to the ramp we added the two-point function in the black hole so just a semi-classical approximation and then we added we simply added to two results so we added what you get from the black hole to our analytic approximation to the ramp and the plateau and when you add it you get a green curve and the dotted black curve is the result of our numerics and their spot on so I think it's safe to say that the combination of the early time decay so to say and the analytic understanding we have of the ramp and the plateau match the data very well so your initial decay is exponential in this case it's not because we're at zero temperature so it's power go here the comment I was going to make is that there was this paper by Dyer and Compton where they were approximating the CFT partition function by different black hole saddle points so in the end they're getting some sort of polynomial decay can you comment on of course zero temperature? I think the zero temperature is the difference so they find different time scales than we do but I think it's explained by the fact that we're at zero temperature plus the zero temperature there's no credence you don't have all these images which Dyer has so when you're at a finite temperature then you have all these images when you're at a finite temperature but in zero temperature you don't have any of these so that's the only thing they have so it's a good polynomial so then the deep time is what comes from this? so so the deep time I think it was square root of entropy and the plateau time was entropy in our case so it's again parametrically long but remember that in the chaotic case we were talking about exponentials and also the similarly the height of the plateau is very different it's much bigger for us than it would have been in chaotic cases but that's fine we understand why that's the case and I'll come back to it in a minute so let me tell you a little bit more about this CFT in order to comment on the qualitative differences on the quantitative differences sorry so CFT comes from the near-horizon limit of type 2B strings on a compactified on a 5-taurus which are split into a circle and a 4-taurus I have some 1-brains and 5-brains wrapped in the way indicated so a string theory on this near-horizon geometry then corresponds to this orbital CFT and if you look at the states of this orbital CFT they fall in different twisted sectors and any twisted sector but you have n strands of string and these strands of string can recombine in various ways the only constraint is that the total number has to be n so here I gave an example of 5 strands 1 short string and then 2 strings with length 2 and the Ramon ground states are created by twist operators the twist operators will indicate how the strands recombine there's order something type root of n exponential root n of them and they all have the same energy but they have different excitation spectra that will be important so we studied the 2-point function of a bosonic non-twists operator that's also important in a typical Ramon ground state and such untwisted operators they only pick up contributions from they only have non-zero matrix elements if the long strings on both sides of them have the same length that's because they don't contain twist operators so they cannot change the length of a long string and that's why you will only get contributions from energy differences of the form m divided by n where n is the length of a long string now if you were in a chaotic regime all these degeneracies would be broken the energy spacings would be exponentially small as opposed to only suppressed by 1 over n here and in that case you therefore would have a much lower plateau and it would be reached much later okay so let me first summarize by repeating what I said at the beginning so in the strongly coupled d1 d5 CFT we expect black holes and chaos we studied the zero coupling limits we looked at two-point functions of untwisted operators and we found this universal early decay followed by this erratic late-time behavior we motivated progressive time averaging by showing that it seems to work very well in random matrix theory we applied it to the system where we didn't have any ensemble average so we had to use some time average and then we found this deep ramp and plateau structure there's very good qualitative agreement with SYK and random matrices but because our model is integrable rather than chaotic there's also important quantitative differences so let me end there thank you do you expect the chaotic structure to appear when you go to strong gravity or when you go away from extremality because I can go from your place I can increase gravity so I can go to the region where gravity is important so I can go away from a different point but then because I still have 8 supersymmetries in this particular case I could still have an integrable structure even if gravity is important on the other hand if I increase temperature clearly I'm going to mess up all this stuff so there are two ways to approach a chaotic system do you approach it in both or do you expect it to appear in both examples or just in one or I think the intuition is that in the dual gravitational description you have a black hole in Einstein gravity that you should saturate a chaos bound but this could be a stick or a zero, right? and actually the two-chart system if you have a macroscopic so you need an actual black hole so it's a macroscopic horizon this is in a sense a two-chart system so it doesn't have a black hole so in a sense you could still have gravity and you have to do it in a three-chart system to see the black hole and to see the poetic so in the three-chart system at strong coupling you should definitely have the chaotic behavior like whether in some intermediate things where you don't end up with the macroscopic that's why k is supposed to be equal zero so this point it's disordered and disordered by hand but there are relatively direct arguments for this out-of-time order four-point function how it gets translated to some scattering event which would typically happen near the horizon and that's how this universally Apunov exponent comes in so it's an extremum horizon okay right but it's an extremum horizon which is very different yeah okay right so okay I don't know we intuitively kind of expect that a supersymmetric zero-temperature thing would have a more rational spectrum of the kind of spectrum that we're talking about there so I'd expect something more like what we're showing here but that's just again it actually depends if you have a super stratum of funny shape you know you can have the most respect on the planet I mean you know the typical state in the world would be some super-temperature like you know a completely weird shape and you know in principle you have any spectrum in one just fine the chaos that is being discussed here it appears to be more molecular chaos a microscopic number of degrees of freedom rather than deterministic chaos that is associated to butterfly effects and typical so for instance this is an extremal black hole in the extremal limit even if you are chaotic would it be possible in this technology to probe whether a microscopic number of degrees of freedom are contributing to the processes here or rather a small number of degrees of freedom like what's happening in deterministic chaos are in fact contributing when you talk about Lawrence attractor it has a microscopic number of degrees of freedom but only three of them contribute to the dynamics of interest can something similar be seen here in some approximation I don't know the model for which we did specific computation was in a non-chaotic regime so I don't immediately know what would be the result if we were able to push it to the there are say N1 D1 brains and eventually a third charge but when we take microscopic values nevertheless a subset of each contributes to the degeneracies of the final I don't know I would expect in a very strongly coupled regime that is maybe not so obvious to keep track of this in the field theory so I think the simplest description in that case would be the gravitational one presumably maybe one last question the fact that you have to average over time and you cannot average over the microstates is that due to the fact that you have zero coupling essentially or do you think it's going to go no that's because one has fixed Hamiltonian so as soon as your spectrum of energy eigenvalues is fixed which it is for a given Hamiltonian then all these phase factors will be fixed and all you would change by going to a different states is the factors that multiply these phase factors but if these phase factors at late time start acting as random phases it doesn't matter what you multiply them with but then what is the physical interpretation of that it's a time it's a coarse-graining in time so I don't just you get this erratic behavior and you look for some way of getting out of this erratic thing some smooth curve of which you can then study try to study the properties either numerically or maybe even analytically so this is a proposal for what one can do when one is presented with some erratically oscillating curve like that and how to get something smooth out of it maybe a very short so as you said in the beginning the puzzle since Maldicinus paper has been that kind of black hole background the two-point function keeps decaying where it's an uniquantum system of wiggles so if one were to take one of these microstates like once discussed in previous talk and compute the two-point function what would it look like that way of times that's a question for Joseph I guess it wiggles it just goes in and then spends some time gets out and then spends more time gets out so you know if you look at the geodesic you know you have many geodesic that's the two-point function I just want to follow the two-point function at late times does it keep decaying strangely does it wiggles but does it wiggle in this erratic way I mean that would be surprising to get that from some classical gravity background it would be very surprising to have these erratic oscillations at late times because the mass gap on it so if you ask what's the energy what's the spectrum of energy above it you know let's suppose you have a two-chart system and you have a super tube which has an arbitrary shape an arbitrary shape if you do a billiard an arbitrary shape is going to be an erratic system it will not be regular at all so if it's an arbitrary shape super tube which is going to be the typical state in your system if you look at a billiard for example in this system and you know you look at all these other points they all be erratic there's no regularity in them only very few states are regular generally it would be very rare but this would be very interesting to see a plot in these microstates and compare that with the plot in random matrices there okay I propose we continue over the break after Frank's talk to keep so let's thank her Ben