 Third day of the school. So as usual in the morning session, we will have two lectures with a half hour break and then there will be the formal discussion and so we can start with the Julian's course on chaos, please Okay, it's a good morning. And as you of course have heard this is the last lecture in this series of lectures And so I'm going to be here all of today But I'm going not to be here for the rest of the school. So if you have any questions that you would like to ask in private You should try and catch me today. I'll be happy to talk of course Okay, so this last lecture I I called it the Chaos bootstrap, but that's only going to be the last a half of it So let me say so let's talk about something that we call ETH matrix models and finally Chaos bootstrap But to set up the the lecture I want to give you a different perspective on some of the things that we've already encountered And a perspective that is actually quite old So this is a way of understanding how matrices build space-time and I'm Still going to restrict on two dimensions and we will build our way to three dimensions and perhaps higher Using similar ideas in this last part. So this this is ideas that Happened in the 80s. I think associated with names like Kazakov grows McDowell More but there is a very nice Douglas more there is a nice review by more and Ginsberg or Ginsberg and more I guess Which also I will supply to the organizers to put on the website So the point here is that one way of constructing What should be the path integral of two-dimensional gravity? Including a sum of a topologies Okay, one way of defining this is By actually starting with a matrix model So it's it is going to be a certain limit of a matrix model. So I'm going to integrate over a Distribution of matrices. So something that we would have called a random matrix And I'm going to give this some potential Trace v of m and What I'm this limit I will have to describe a little bit But people refer to this as a double scaling limit So roughly speaking if M so M M would be Let's say an N by N Random matrix Okay, and it is random in the sense that I've specified a distribution of matrices from which I am sampling and There is going to be some some coupling parameters and in particular there will be some some For example cubic coupling G3 coupling I Will write down an Example of such a potential and so this double scaling limit will take and the size of the matrix to infinity and at the same time scale some some appropriate coupling to a critical value and The idea of this double scaling limit and I will explain this in a in a second is that it in some sense produces a Continuum version of a two-dimensional geometry and the matrix integral will then implement something like this path in goal in the sum of a topologies So for example e.g V of this matrix Could be or Well trace V of M could be something like trace one-half M squared plus trace well G of G3 Trace M cubed maybe divided by three But you could have other polynomials here and they equally make sense So now the idea is the following So you start by looking at Feynman diagrams of such models G3 G3 star will be some critical coupling Well G3 is the coupling and that appears in the matrix model and G3 star is a particular value of that coupling Which tunes this model to a critical point No, you have to take them at the same time. This is why they call it double scaling So as you take the number at the size of the matrix to infinity You take the the coupling constant to a critical value have to do this in a coordinated way such that a certain fine ratio of A combination of these quantities remains finite in this double scaling limit, but I will write a little bit more on that So so the Feynman diagrams of this of Such a model in Tuft double line notation Okay, they will consist of things like Propagator Which just comes from the quadratic term I I primed J J primed and it would be some Delta I I primed Delta J J primed Okay, and the idea is that every matrix of course has two indices So like I write like M ij and so this is like a propagator Between matrices and the indices run like this and I have to identify them in this way And then in this cubic case, I have vertices Which basically are Where three of these lines meet in a triangle and I have something like I J Sorry J K K I and this will be the vertex with strength G3 But the point I want to make is not to calculate individual diagrams but to point out that Using this to of counting or using using these ideas these diagrams Can be classified by topology meaning in this case again the Euler character and at the level of Drawings you might get things like Let's say you contribute you want to you you want to compute things like All Geometries all diagrams that have one boundary so then the the leading order topology will be something like the disk So like we had yesterday, so there's a boundary and this is like just the surface of this Okay, so this is really one way as we did yesterday the disk plus Actually turns out sub leading in this end you will have some diagram Which is the disk with a handle and so on but the point that I want to make is that if you look at an individual diagram in each of these topological sectors then the point is that they will be made up of a Bunch of these double lines which meet at these cubic vertices, so I hope I can draw this a little bit So one example might be Something like this Cubic lines Okay, etc Okay, let's say and so on and The so so so this surface here just represents the topology of this diagram, but what I want to point out is that you can actually convince yourself that if you Take something like Maybe I want to continue this up here not to be in the way to some sort of like Dual graph of this so you know Triangulate basically this discreet graph here what you find is that the the matrix model graphs Define a discrete triangulation of this surface Okay, which I have which I have drawn in red and of course the idea would be that also locally if you look here You have the set you have the same you have the same sort of idea that you know You actually draw a bunch of triangles that make the surface But of course here you have to take into account the topology So I don't want to draw the global picture But you have the same thing that all of the Feynman diagrams Of this matrix model which are in this topological sector define a discrete triangulation of Now surfaces that have a topology with one genus in here one boundary this this is just one example by the way and the power of n and is also Related to the Euler character the relative power of n is related to the Euler character. So in a in a In a standard n to infinite limit you get only the planar contributions Any to plane a limit You get the leading diagrams will be the ones of the simplest topology Which will have the least suppression in terms of the 1 over n powers And that's a topological statement here But what the idea is of this double scaling limit is to actually look at a slightly different a slightly different way of making the size of this matrix big and the point is that This double scaling limit so it it it it it tries to get the continuum limit of the Discrete approximation and retains coherent contributions of the non-trivial topologies and so roughly speaking again the Mathematical or the physical idea behind this is that if you only Take into account if you find a limit which takes into account graphs and Where the number of triangles goes to infinity? While you can scale the area per triangle to zero so to retain a finite limit of the total area Then this triangulation with curvature concentrated on the edges gives you a continuum metric and more over You retain the fact that you have summed over topologies Okay, now and this in other words So what you do is then you you take this double scaling limit and it means in this case something like as I said n goes to infinity G3 goes to G3 critical such that and divided by G3 minus G3 I call it star To some appropriate power which depends on critical exponents is finite This is what you might call in our previous language e to the s zero Okay, so that you end up getting a continuous theory of two-dimensional Euclidean geometries Where subsequent powers in topology you work this out are Suppressed by this topology counting parameter e to the s zero, but e to the s zero is something that came from scaling two parameters To infinity in such a way that there that their Caution remains finite and so and what I want to say here is that this is a different way of sort of associating matrices with geometry or quantum geometry, but here M is A mathematical device Okay, this is nothing but This is nothing but a clever way of Of constructing if you want this path and ago and I encourage you to read this very beautiful review And which which does a great job and also better job at drawing these diagrams I'm in a computer aided way But what can I can I ask a question? Yeah, so what happens if you go away from the scaling limits on the left-hand side of the equation? And But you don't have these parameters on the left-hand side. I mean I'm defining the this By the limit of the right-hand side you're supposed to reproduce a Path integral over to these gravities for any potential be I mean that's easy to important Well, that's that's a good point. So first of all So there is some sort of universality here because you tune these potentials to critical points And so there is a certain class of behaviors That that comes out which is not Well, it depends I think on the power P that you retain here and it has to do with the kind of Gravitational theory then you you recover in the continuum limit but one of the Ames of the game today will be that well, we will change the perspective Which is what I want to say here is that what we have seen in some sense already Is a new perspective namely that M is actually a physical thing? M is H Has to do with the Hamiltonian Hamiltonian of Quantum chaotic system, okay, and we have already seen this explicitly for For the case of JT gravity right in the sense that well, I I sort of showed you this somewhat indirectly but one way of interpreting the work of of SSS of such anchor and Stanford is they wrote down a matrix model for JT gravity But this matrix model you really integrate over the Hamiltonian. We call it the boundary Hamiltonian But let's just say you integrate over a Hamiltonian. So M no longer is just is no longer a device that allows us through this Very beautiful story to Construct a path to go via a continuum limit of some discrete triangular the triangulation But it's actually has some some some meaning in terms of a physical Hamiltonian and of a quantum chaotic system So in that in that sense Also together with what we've seen in these lectures so far. It's true that on the level of a slogan. It's true that these sort of Quantum We call my maybe level statistics or level correlations Okay, build up spacetime and that's a sort of a philosophical perspective on this but what I would like to do is now I would like to Take the opposite root in some sense and I would like to try to build up models of chaotic systems whose level correlations build up spacetime and That perhaps allow us to go beyond some of the Models that have been studied as I said since the 80s and in particular I want to I will show you how to get some story that Looks a lot like this, but in two-dimensional CFT is related to three-dimensional gravity or and I should say that This this is related to work which I already Mentioned with Colt Schmeyer Muhammad Sunoff Sorry, Jeffress has an alphabetically preferred Name and myself as well as work with Alex Berlin Jan de Boer Daniel Jeffress Pranjal Nayak and myself, okay, so Firstly I want to get back to this idea of The eigenstate thermalization hypothesis We're going to use now the insights that we have from random matrix theory in the chaotic sense and from Quantum chaos in the ETH sense to build up matrix models which go beyond These kind of ideas and in ways that you will see the obvious joke of course with this with this kind of stuff Is that if you ever get approached again at a party by someone says are you walking theoretical physics? I have this idea that we live in the matrix and you can say yes We do Well if we live in 2D or 3D So in this new perspective The planar limit means that you are taking the dimension of the space to infinity. I mean, I'm not taking the planar limit I'm taking this double scaling limit Okay, but in particular and going to infinity means that you are including all the energy levels Yeah, that's okay, that that's a good point. So This double scaling limit is something slightly Sneaky I would say So what what it really means is that you go and you write down a theory which takes into account only levels Which come from the edge of the spectrum? but So what you need to do is you need to scale the energies to be very close to the edge of the spectrum But you also want to work in a limit such that you retain an infinite number of energy levels close to the spectrum but then in the In the actual double scaling limit the coupling that you get this coupling here e to the s0 remains finite and From the perspective that we had previously developed This was counting effectively the dimension of our Hilbert space so and I would say effectively you end up with a system that behaves as if it had a finite Hilbert space That includes also all these non trivial topological corrections But I think that What you point out is that it's Not intuitive a little bit subtle to see how that works If you just were staring at your matrix because I agree that you take them the size of it to infinity mass and energy in general utility Buildup space times and in the JT machinery quantum level correlations build up space times. Can I compare this to? So first of all, I would say that that's one that is one Interesting interpretation of the work of SSS so they wrote down Doubles well, they wrote down again I said this before they never wrote down the matrix model and that's partly by the way because of this double scaling It's very inconvenient to write down a potential directly in this double scale limit. So they they Skirt at that issue by writing down these Recursion relations and pointing out that the JT recursion relations and the matrix model recursion relations are the same But just because of some technological necessity As part of this work and we actually wrote down a non double-scaled matrix model Which has admittedly a rather ugly potential and we defined a double scaling limit Which which recovers the the physics of the SSS model. So so to to say it So so having said that and I think that in the JT case there is an exact story Where we can think of it that way and because the Hamiltonian the matrix that comes up here is a Hamiltonian I would say that this is an exactly is a true statement in JT gravity Okay So this Vertices will have different will correspond to different diagrams depending upon how I contract them Yeah, yeah, but you sum over all diagrams and the point is that by summing over all diagrams, however So okay, let's let's just say it first So you sum over all diagrams and each diagram is sort of like a different metric if you want in a given topological class It's a discrete version it's a discrete version of a metric So by summing over all diagrams you are performing an a sum over the different metrics that you can put on the surface now This double scaling limit again Does it in such a way that what you retain are diagrams that that all are Continuously so this this becomes infinitely fine this mesh and you but then you still sum over all diagrams in this class right with infinite number of vertices and So that becomes an integral over all the metrics that you can put on the surface So that's precisely a part of the idea if you want Yeah, I don't know actually do you Okay so So what you wrote before is like a larger and limit on the right and Some path integral with a metric on it on the left So is this large and limit related to large and limit that you have in the holographic correspondence and I Mean I want to say yes, but It's not completely obvious to be honest with you Because you can also take tough limits which are more obviously this thing that you do in In holography So I'm not sure it's It's so clear that you should you should say like this, but The fact that you take large n That you sort of have a large Local number of degrees of freedom is I think in the same spirit Yeah, I mean, you know to the extent that we can think of Jt and and it's a sss dual as a holographic duality I mean so in that context, so it's exactly the holographic limit that you take okay, but in higher dimensions I don't I don't want to make such statements Okay, but let's let's let's get let's crack on a bit so So I want to now actually define these eth matrix models and Random geometry, okay, so the first thing that We need to do is we need to now revisit this this eth answers and maybe I'll remind you of it So what we had was we had I O J and This was going to be supposed to be equal to or this is equal to some smooth function of the average energy of the eigenstates i and j plus a fluctuation contribution F e omega Rij and Rij I was saying is a random matrix and In many treatments of eth you will see the statement that this matrix should be Gaussian But in fact, it is easy Just to say to see or it may be not easy to see but in hindsight it is clear that Rij Should be a non-Gaussian Distributed random matrix, okay, so And and this actually I should say again We are you in some of these papers, but it's also related to a very nice way of thinking about eth Which has come up in recent years in the statistical physics community and in particular the Paris group of Fouigny and Corchan And various collaborators, for example Sylvia Papallari comes to mind and they have actually worked out a story that is Largely equivalent to what I'm going to tell you about but they use a very different perspective They don't use the perspective of these eth matrix models, but and they Show this very nice hierarchy of non-Gaussian corrections to this Rij so So one thing is that for example if you if you had only Random matrices of Gaussian statistics then something like and OFTO OFTO So it calculated using eth So something like an out-of-time order correlation function Okay, would always factorize exactly into two-point functions immediately and that Cannot be possible because it's in direct contradiction Is in contradiction with having a non-trivial Lyapunov exponent with this idea that you have a non-zero Quantum Lyapunov exponent because the quantum Lyapunov exponent comes precisely from the connected contribution to the four-point out-of-time order correlation function So it factorizes it has no connected contribution So it would never be able to describe something like the Lyapunov behavior. So just to tie it back to And something I said in the beginning in general however This factorization Leads to many sort of contradictions with obviously Necessary physical requirements for example one way that we have seen the necessity also to add The Non-Gaussian parts of the statistics is that when we first actually the first the first project that we set ourselves here Was to add an actual couple a matter field to JT gravity and see if we can write down a matrix model description of the same spirit as SSS and what you find is that if you just retain the Gaussian statistics of this matrix then the Four-point functions of that theory will not be crossing invariant which is in direct contradiction with the gravity theory So if you just take JT plus a matter field tells you so for example, it also contradicts things like crossing relations in Let's say ADS CFT type matrix models Okay, and I will get back to this to what the end of the lecture and we will look at this in much more detail but for example in order to in order to In order to address this OTOC thing you can say that there should be a connected contraction of these are matrices which of the form are Ij rjk Rkl Rli Connected Okay, which is suppressed with three powers of this Macro-canonical entropy which is now at the average of four energies So you have an energy one associated to this energy two associated to this three and four associated here times some continuous function g4 of those four energies Okay, and if you add this you can show that for example this function g4 encodes The Lyapunov exponent of the theory Okay, so now two two remarks one remark is that This function g4 is actually just a generalization To if you want four-point matrix elements of this f f even though I've written it here as a one-point function I've already said it before f is sort of like the fluctuation f times r ij is the fluctuation So it pertains really to two-point functions and this guy pertains to four-point functions and in fact as I said Funi and Kurchan have written down a whole hierarchy And In our way, so have we as you as you shall see Which basically generalizes this to higher point moments so in other words you define higher moments of the distribution of r by Feeding some data and the theories such as for example the fact that it has no trivial Lyapunov behavior Or that it satisfies cross crossing relations, etc so the other comment is that this is Seemingly highly suppressed But in fact it still contributes at order one to correlation functions and The reason is that whenever you use this in a correlation function in a meaningful way So for example in the four-point OTO You actually have to sum over the Hilbert space many times And in fact you always have to sum enough times such that you just undo this factor because you know Every e to the minus s is sort of worth one sum over the Hilbert space So in general you can actually say that if you were to do something like an endpoint connected correlation Okay, this should be something like e to the minus n minus 1s of the average entropy times some smooth function gn of the n arguments so so in other words So one could say from from this energy perspective if you work in the energy basis Then eth gets corrected or the Gaussian eth gets corrected by very small non-Gaussian corrections That are more and more increasingly small in the entropy But in fact from a physical perspective They contribute to correlation functions of physical interest at leading order So this is the sense which you might say that it's actually strongly non-Gaussian so this structure here Yeah, this g4 for example encodes the Lyapunov exponent Was the converse true to the Lyapunov exponent? Imply exactly what you four should be No, I don't think that's true, but what it does do is it gives you well There are there are arguments for example for this bound on the Lyapunov exponent that come from sort of common sense bounds on this g4 But you could also turn that around saying that the bound of the Lyapunov exponent would imply some common sense bounds of this g4 But the statement that it's fully determined. I think it's incorrect and in fact That that's a general that's a general thing that we will need to face which is that even when we talk about eth, right? This is a statement Which is to some extent? Kinematic we don't specify what the function f is or what the function always that's supposed to be theory specific That's fine because this is a generally still a predictive framework It predicts Thermalization to whatever the sum of value will be and then there'll be fluctuation relations with whatever the two-point function is So from a statistical mechanics point of view that is fine But from our perspective now we would like to have a way of determining what these functions should be because they actually determine our matrix model And that's something where I need to input some more ideas Before before we get there okay, so I'm just to just to Finish it up. So this this structure here is something that people refer to as extended We and others as extended or generalized eth and I think There is one person who deserves to be mentioned here So I think the first time that this was announced was in a paper With my former graduate student Manuel Vielma, but as I said, I think there is a very beautiful Systematic work also by this group I'm on this hierarchy So right so now let me see Okay So, okay, so the point is now that One further one further point is that we one can package this This information as well as the information on the spectrum in a two matrix model and this is what We would like to call something like an eth matrix model So so you have something like this eth, which will be now the integral over a random matrix H and A second random matrix which should be this matrix Oh, but we can rescale things and maybe by abuse of notation We we can we can call the second random matrix Oh, and that's of course with I on sort of the holographic application because we always like to call operators. Oh So we have one random matrix that takes care of the spectrum That's the H type random matrix and we have one random matrix which will take which takes care of the matrix elements of an operator and We have some some measure for DH and DO And so we could for example a priori put a flat measure here and then just say that the details are in a potential of H and O Which is a single trace potential that is made up of words of H and O And if you go to the energy basis, so you diagonalize the matrix H You get essentially a potential Which has functions of energies? Okay integrated over the probability distribution of energies where it's implied by the H integral and the DO with its own Gaussianities implies This hierarchy of correlations. Okay, so this eth matrix model it Incodes exactly the same information as this hierarchy of of of let's say eth functions Okay, and but it encodes them in terms of a spectral integral and an integral over what stands in for the eth Hierarchy of correlation functions. So this is just a different way of packaging the information And it becomes in this case a two matrix model Okay, and what is interesting Something that we haven't explored much But let's say you wanted to for example study this case for a number of operators Then this will become a multi matrix model and those things in their interpretation as generating a probability distribution for matrix elements of O and O's and H They also are related to what mathematicians called free probability theory, but what I want to spend Now the remaining time on Is really the question is so so that let me just say that this part now of this lecture you should view as Something in statistical physics in the same spirit as eth Which as I said on the very generic condition captures the physics of quantum chaotic thermalizing systems, but it is intentionally also Widely open it doesn't specify exactly the theory that we're working with I mean that is also a feature not a bug. This is like a framework not a single theory but I promised you that we get back to some Holography and in particular higher dimensional versions that will connect to this Quantum level correlations build-up space time. So the question that I want to answer now is What determines this potential I mean beyond beyond as I said this overall framework because There is you know, we've determined the general structure of the potential But we don't know what are actually these functions that go in there and so on and so forth So So the question is How to determine the potential okay? And so here well, I Would like to go almost directly to this 3d case, but I do want to make the statement so One thing that One can do is so one example is this Jt plus matter so in the sense that you actually couple another matter field, which is not the dilaton D phi squared, let's say plus m squared phi squared So this gives you some kind of Jt plus matter Two matrix model and I well I will obviously supply this reference on the website There's a there's a fairly long Literature on this And what we do is basically we implement We determine the potential by implementing certain constraints which determine the potential v in terms of Something that's called the 6j symbol and the 6j symbol will come up now as well in higher dimensions So I just wanted to to point that out and point you towards this paper And of course so as I as I said I'm around so if you want to know more about this you can also ask me but So let me go directly to the case of ADS-3 So let me talk about something that actually is a is a tensor model. I mean, it's a random multi-indexed object, okay, so a Tensa model for random two-dimensional CFT and Therefore 3d gravity and that's also, you know, because we're not going to talk about actual CFT is this is sort of This is sort of a Idea of how to implement the chaotic version of the bootstrap so So So so one of so now I also want to make it some physical perspective on this perhaps from from holographic perspective, so the the idea is that You know, we have this immensely complicated Hilbert space that describes the black hole which we have seen is exponentially dense However a Semiclassical observer doesn't see it doesn't see the details of this Hilbert space So however in quantum chaotic systems as we have argued many times, you know It's as if this kind of set of states behaves like a random Like a random distribution So the thing is that we do is so let's say a Semiclassical observer that don't have access to the details of the black hole microstates but They have Access to some information So what you can do is you can fix everything that you know so fix Fix the low-energy data But you know so for example dimensions of low-line operators and then average over everything else but subject to of course to constraints which we have Basically assembled in terms of what I would have what I would have now called, you know chaos Universality and now also we will put in things like So because we describe a CFT we will put in something like crossing symmetry Don't worry. I will write some equations for this Okay Crossing symmetry So for example modular crossing, let's say and four-point crossing on the sphere a generalization to CFT of this week We're sort of the maximum ignorance distributions that were compatible with what you know about the system namely for example It is a Hermitian Hamiltonian that has time reversal invariance All other information so then you get out the Wigner distributions The outland CERN bar distributions you can get by putting in more, you know, but discrete symmetries Now the question is how would you do that with a CFT namely a theory that that has Interesting symmetry constraints that relate different parts of the spectrum It can't just be a completely random matrix So what we write down is we basically will write down a probability distribution on the data of a CFT Subject to what we know namely some low-energy data Chaos universality and then these non trivial virus or a crossing constraints And what will remain will be a very interesting model Which is like a random tensor model, but of course that's in the same spirit as a random matrix model Which will have an interesting interpretation and it's it's going to be of course of this type, right? I wouldn't have wanted saying this before So that's a bit of philosophy. So now let me try and Write some equations put down some equations to see so Yeah, I guess I have a bit more time So you see so a CFT is Related to the set of conformal dimensions delta and the OPE coefficients see IJK Which are basically three-point functions. So these are The operator dimensions. These are the conformal dimensions and these are the OPE coefficients and They're basically three-point functions By the way, a three-point function defines a triangle Now of course that these are the eigenvalues So maybe we just should think of them as the eigenvalues of the dilation operator and then what we have is we have a Matrix and we have a tensor. However, this data doesn't define a good CFT unless It satisfies further constraints Okay, and so those constraints so so this This is what One would refer to a CFT data and now again to to Pay homage to people who also have thought about this the idea of treating all or some of the CFT data as random objects Well, it has I think first really been announced by debour by belong and debour Yes, I think again alphabetically. Yep and also by Chandra Collier And Maloney and Hartman But what I'm what I'm about to say here Is you know a different perspective and of course it goes it goes beyond what has been described in these papers So So this is CFT data, but CFT data becomes a real CFT once you imply once you impose bootstrap constraints The problem is in this program of of defining averages is that Typically CFTs if you just have an exact CFT You they don't allow that they're very rigid objects. They don't allow to just Average over a large space of just CFTs even if you want to do so They don't have like a large manifold typically of of say marginal directions that preserve CFT in variants That you can integrate over so you can so what you can do instead is you can integrate over this Ensemble of data Subject to certain constraints which which will write down Yeah, sorry I had a question So I guess in the second point what you also want to say is that the CFT actually has an OPE know because if you really yeah, okay, but so if I Is it obvious that if I make these things random that an OPE should Exist in the first place because if I now start making some gigantic OPE with a huge sum over a huge amount of random things I could imagine that things get more complicated Yeah, I think that such so what actually is the Status of an individual member of this ensemble Is very much an open question So for example, they will not be CFTs But they will be Approximately CFTs now that question That one would ask then do they sort of approximately have an OPE Okay What might be even more? Scary in some sense. I mean, I'm I'm okay with it because you will see that the model that we Get in the end just looks so nice kind of must be right But if you ask what is the individual member like is it even a quantum field theory? Because a quantum field theory, you know is something that has like locality properties. You can write it down in terms of Lagrangian density and I mean all the things that we learned from wine by spoke for example, but the fact that The CFT data Plus the existence of it on OPE plus the bootstrap gives you actually a well-defined quantum field theory is a very non-trivial fact and I don't know if this is true if if the data only satisfies these constraints approximately Like so, yeah So those are those are very deep questions I think that would be certainly interesting to To to investigate so this this of course Opens a lot of these interesting questions to investigate But yes, so that so the individual element of this ensemble is an object that we should be That we should scrutinize heavily, but let me now write down This ensemble and again, let's try and let's try and be be charmed by the answer Maybe close our eyes in the middle a little bit Or say this opens very interesting questions, which we need to which we need to further investigate So let me let me actually focus on one of these bootstrap constraints in particular the Crossing constraints that comes from we're a sorrow. So what we're going to do is we're going to write some local Crossing constraint So and the crossing equation I'm thinking of is something like You take CI one I to P C star I 3 I for P times the conformal block P of Z I've told them P of Z so Let me first write the equation and this Is the expansion so this is the S channel expansion Of a four-point function where I'm fusing operators. Let's say like this summing over P One two three four Propagating the The the block P here and Z is the cross ratio Z is the cross ratio conformal cross ratio and then this should be the same as the expansion in The other channel Which is if I fuse I 1 with I 3 and I 2 with I 4 and I contract into the crust Versailles block F Q 1 minus Z and this would be This fusion sum of a Q 1 2 3 4 Q propagates here So this should be this and that's the famous bootstrap equation and I'm f of P Z is the S channel Versailles block and F Actually, we could put an S here and FT of Q 1 minus Z is the T channel version of it Okay, and we could We could have written SS and TT here So this equation has to be satisfied For the set of conformal dimensions delta and OPE coefficients here that you OPE coefficients Cijk that define a true CFT Now Let's do the following. Okay. Well. No, first of all, let's let's let's do one manipulation on this equation Okay, I probably will need about 10 more minutes. Okay Okay, so So So what we're going what we're going to do is we're going to use this so-called inversion formula of ponzo and Techner so Techner inversion Namely that I can write the T channel block P 1 minus C as a linear superposition of the Of the S channel block and the thing that goes in here is this this this famous inversion kernel. So P T P S And PT PS. Okay. These are sort of PT PS Is related to the conformal dimension conformal dimension of of the operators and What I'm using here is what people call because in this business people Use this so-called level notation. Okay, so there's some parametrization of what are the conformal dimensions and so on and so forth But you can write just a technical question. Yeah, this is a face and if you are the same functions, right? No They are the same functions, but The things that you sum over are not the same. So what this does is it tells you what are the S channel States That you need to sum over in order to get a given T channel conformal block, but evaluate it of course at 1 minus Z Yeah, yeah, but so the arguments are different, but yeah, and the functions the same. Yeah Functions are the same, right? Okay. So then what we can do is we can write What we can do is we can write the crossing equation basically as the equation Zero should be equal to the sum of CI1 CI2 P C star I3 I4 P Actually, let me call it Q Delta P Q minus Delta P. What did I call it here? Let me Minus P s Minus and I'm summing over Q only see I One I4 Q C star I 2 I3 Q Times and now we actually have to take the holomorphic square of this S Q and the holomorphic square because we have to invert the block and It's and it's conjugate and this thing here is a famous object It's a 6j symbol of Virasoro. So this depends on S Q and in fact, I didn't write this but it also depends on I I1 I2 I3 I4 I didn't write this because I didn't want to put the labels everywhere Okay, and by the way, this is also maybe in response to your question right the block also depends on the external states so this should be labeled by I1 I3 I2 I4 and This one should be labeled in the same order now, but but I I1 I2 I3 I4 column wise okay, and so this inversion kernel also Dependent always on I1 I2 I3 I4 here, but if I write so many indices Yeah, it starts getting a bit annoying, but this is what is called the 6j symbol of Virasoro Or if you want the Poisson-Teschner inversion kernel so now is One key move that I need to do then I will have written down the model and I will tell you how this relates to to Discrete geometry, and then I think we'll wrap it up because I'm already threatening to go too much over time Okay, so now So now the idea is that Let us think as this as some equation, which is of the form m I1 I2 I3 I4 is equal to zero For external so so these are the external states For the external states I1 up to I4 and one way that I can Impose this on my matrix model of the CFT data is by simply saying that I integrate over my delta Ij's and Delta C Ij case with an a priori flat measure as we did before but I add e to the minus Some large number times M I1 I2 I3 I4 Squared So summing over all of the indices of course and I integrate over DPS Which is the open index that I need to integrate over and A is now for now is a large coefficient Okay, what is this saying? This is saying that the Predominant contribution to this random matrix distribution comes from comes from Solutions that satisfy this crossing equation Approximately where this approximate satisfaction has to do with the size of this coefficient Okay, but I am sampling over Elements of the ensemble which don't exactly satisfy them But then now we can look at the moments of the distribution of the crossing equation and see to what extent it satisfied Okay, now this is by the way Is there a reason why not imposing a delta function with a Lagrange multiplier? Why I think I think you could also do it with I think you could also do it with a Lagrange multiplier You won't do you don't want to however at this point impose it exactly right because if you impose it exactly You're back to having something like a CFT at least at the level of this virus or a crossing constraint Yes, because yes because you don't know how to average over exact CFTs, you know, they don't actually have You know the generic CFT won't have a flat direction that you can actually average over Right, you think that the crossing equation would localize on some point Not on some region that you can actually average over Perhaps there are many points and you know, maybe maybe there are many points So then that's that's the question actually and if we write down this model And there are the other potential terms which come from as I said modular crossing invariance Okay, which I'm not writing now view of time, but if you take all the limits Naively that these coefficients go to infinity you think you land on solutions to the bootstrap equations exact solutions, okay, but Unlike in the usual bootstrap you would still sum over every single solution So that might be for example a naive large end limit of this of this model Okay, now let me write the potential. So this is just one part of the potential. So so we have now some V Which we could call four point and if I write it out it is of the following form Okay, it's of the form. Let me just Write it once and then I can draw some pictures. I one I two P C star I Q C star I Two I3 Q times the six J symbol of verisaur and it's P Q one two three four and this sum is a sum prime Because I sum over all Only the heavy operators in the black hole spectrum, which if you're heading for pure gravity You would say that You have just the identity high-energy states So the average here is only over those high-energy Micros over states which have H H bar greater than C minus 1 over 24 to be technical Now what is this? Okay, so We'd be nice to have four colors now And that's the last thing I want to say that's okay. Yeah, very good Hopefully they are distinguishable. Yes, okay, so and we have See in blue see in red Oh See in green. Sorry. This was purple and see in blue And what we do is we now draw a triangle for each of these OPE coefficients and we label the edges So each edge has one verisaur representation on it. So this one will have a representation one two and P and it shares an edge with the red triangle So let's do it like this. It shares the edge P This shares an edge With the green triangle Okay, I will write the labels later and then then there is a blue triangle here That's the last face of this tetrahedron Okay, so well, I guess I might mess this up But so we have the blue one was one two P so the red one has P and Then I have three four the index four is shared with the green one. So this one must be then four right for and then two for This was the green one so the remaining index on the green one is the index Q Okay, and and you can fill in the other indices on the back triangle because if I draw them I'm just going to make a dog's breakfast of this. So this is now a vertex of a tensor model and this vertex is a tetrahedron however This model also contains a quadratic term in C which basically comes from the identity propagating Making two of these OPE coefficients to be equal to one and this gives a particular contraction of the 6j symbol So I also have a propagator in this model Which glues two triangles? Okay, so it would glue the edges Okay, maybe glue like this like this and this glue these two triangles and this propagator comes out to be exactly the formula that So these people wrote down a Gaussian model Okay, so in our in our language that Gaussian term would be the propagator and they called that their prop Gaussian thing which we call the propagator the C0 formula The CCC0 formula because again these Leoville People that worked out many of the details of Leoville theory as we use it today Ponceau and Teschner they use that notation So in other words the graphs of this will be three simplices That glue Triangles edges and tetrahedron along So along vertices of tetrahedron So this is actually what people call a version of simplicity or 3d gravity But what is different to what people have done in the past with simplicity or 3d gravity? So maybe one name a bullet off And also people like Jan Ambir and Renate Loll who have worked on this, okay So this is now that the discrete version of a three-dimensional Tetra had tetrahedronization of three-dimensional Euclidean manifolds and in the same sense that I started with the matrix this tensor model integrates over all configurations and Of course now the idea is that you take some double scaling limit in the same sense to to get actually some Continuum passing all of three-dimensional Euclidean quantum gravity. So I'll stop here because I'm waiting one minute over time Thanks for your attention There will be more time After lunch for questions, but perhaps if there are some urgent questions now Quick question you are imposing the crossing symmetry, but you are not imposing modular invariance, right? I am I just told you that there are also terms from modular invariance Which I'm not actually writing down in view of time, but one one consequence this will have for example So you can do this you can play a similar game. There is another inversion kind of and The identity part of that which here gave us the propagator. It imposes that on average the spectrum is Cardi So the desert constraints are on the conformal dimension, right? I Don't think I mean there are constraints on the conformal dimensions But they don't constrain individual conformal dimensions But for example the density of conformal dimensions asymptotically behaves like Cardi Not all possible set of conformal dimension would work in this in this integral because from the constraints you've wrote It seems you are integrating over all possible conformal dimensions I'm not sure I completely understand your point. Maybe we should talk offline I mean as I said there are constraints, but those are the usual constraints that you get from a modularity Like Cardi and so on Maybe we can take one more question before the break and he You want to take a infinity or is it like? Well, that's that's the So we we don't yet understand everything about these classes of models But the idea that we have at the moment is that you want to take some kind of double scaling limits So you can't just take it naively to infinity So what actually you you do is that? And this is again very parallel to what is in this rather large work Thanks to some very powerful collaborators So this is like a half double-scaled version of the model that I wrote here So what you need to do is you need to step back and you need to actually Write a properly non double-scaled version which involves Q deforming these objects and then Once well, we do this very explicitly in this paper. We can talk about it Quantum's a J symbol is already Q deform to wanna additionally Q deform it. Yeah Okay, so maybe let's thank now Julian for his beautiful series of lectures