 Thank you, thank you very much for the invitation and happy birthday Samsung So I'm one of the probably two people here from Trinity College Dublin Where I came to get to know you more in the recent years that I've been there And of course I have learned a lot from you and I also Probably owe it to you a lot in terms of support especially in solving our two-body problem so Without let me also thank the organizers for giving me the opportunity to speak and apologize that I'm not going to My topic is not on neither or no normal is not on the gravity Although maybe some of the structure could have something to do could have some elements of integrability per se so a So the title is CFT and black holes and it may be much more catchy than what we actually Going to see we are not going to see much of black holes here But we're going to see some elements of them or let's say we start our baby steps in this direction so So where does this whole talk fits into so? It starts from the holographic principle the ADC 50 correspondence that tells us that we have some quantum field theories and they are have equivalent description in terms of a gravity in ADS space and given the recent Advances in techniques in conformal field theories for which some of the people here are Experts and pioneered them We would like to use them to deconstruct in a sense the holographic principle and learn more about gravity So what kind of questions we would like to ask in general is what type of theories have gravity description? What restrictions you can impose by physical consistency conditions and more specifically about black holes? Can we learn something by studying conformal field theories and here I'm merely thinking of higher dimensional conformal field theories? And of course several more questions So so in this talk we are going to discuss what we call holographic conformal field theories and by that I I don't mean theories that have an explicit holographic dual But theories that have certain characteristics that we come to acknowledge as such that give us an equivalent local gravitational description in terms of Einstein gravity actually So these are large and or large C where C is the parameter that defines the two-point function of the stress energy tensor in the CFT Conformal field theories which also have an infinite gap in the spectrum of operators which have of primary operators Which have spin higher than two? so there's been a lot of progress in studying such theories and one of the Perhaps basic work that sparked this whole initiative this whole approach was the work by Hemsker, Penedones, Polchinsky and Sully where it was first shown or depending on your level of You know rigorous your desire of rigorousness in proof They have shown that by studying the crossing equation that there is a one-to-one correspondence between local couplings in gravity and solutions to the crossing equation or peak coefficients, so furthermore several Several works followed that try to use consistency conditions and principles like unitarity or causality which Showed that basically this the couplings that are not present in Einstein gravity lead to several various inconsistencies and and So so these are some of the things that historically have brought As into into where we are today and we would like to go beyond that and study many things But at the same time now we are using we are we are using this all these techniques in order For example one approach would be to study try to compute Feynman diagrams in gravity by using conformal field theory techniques and of course there are many many other topics But the topic that as you as I alluded to it would be from the title that interests me are black hole and black hole physics Obviously, this is quite Further from where we stand at the moment and if we would like to understand them Well, the first thing that we want to do is perhaps see how in some sense emerge When we study conformal field theory, so we would like to see some elements of black hole physics that show up in conformal field theory correlation functions So one standard approach to study to study that to start for example looking at this would be to study thermal correlators because we know from our from the holographic principle and the idea safety correspondence that when we want to study CFT at finite temperature we can equivalently look in the large n and large coupling limit in the for the theory in ADS farce of black hole So and we can study correlation functions from there. However, this There has been Some approach in this some work in this direction, but the work that I will present here follows a slightly different route It uses the fact that We can look of the theory who can understand possibly several aspects of the same physics by looking at different correlation functions Since we have we are interested in theories at large n or large central charge We expect this I understand this in some sense like a thermodynamic limit for our system where we have a large number of degrees of freedom and In that sense we can may try to address a system by studying correlation functions Which involve two heavy operators that will effectively produce for us our thermal Backgrounds so these operators are termed heavy because in the limit where we send the central charge to infinity They're conformal dimension also scales the same way and the ratio delta heavy over C is roughly fixed so in the dual gravitational description the parameter that we are going to see a lot and its equivalent basically to delta heavy over C is the ratio of the Horizon radius to the ADS radius that's in four dimension specifically this equation since this will be mainly the focus of the stock although there are results beyond that and you can get to this relation through the ADS if the correspondence So this will be an important parameter for us here So what is the correlators that will That we will focus on in this talk is basically a correlation for the four-point correlation function the first perhaps Non-trivial step we can take we can look at two heavy operators and two light operators And they're all going to be scalar operators for simplicity and we expect that this will be more or less equivalent to a thermal CFT two-point function of light scalar operators so We know from experience from our holographic duality that in fact this object is cannot be analytically computed beyond for example cases in ADS 3 and So you would wonder how much mileage can we get how far can we go by looking perhaps at the unequivalent description in CFT and Well, what we can do so far is approach the problem in several Kinematical regimes that make it a bit more tractable and these are mainly two so far that we have looked at One of them is denoted by regi or a corner limit and the other is the light cone limit So let me say a few words about the first that are unfortunately. I'm not have time to discuss in this talk So in this limit, this is basically the technical definition of the limit let's say in terms of the CFT correlation function Which implies that you have to do a certain analytic continuation You're always looking Minkowski and correlator and you want to take both of zines the bar going to 1 which correspond to the conformal cross ratios with this ratio fixed So in this limit what we can do is we can we expect to get from this four-point correlation function or rather from this equivalent basically the scattering phase shift of a particle That is affected as traveling along anal geodesic by the presence of the black hole Which we expect to be like a particle of a very large mass so this is physically the limit that one is interested in there and this is a Work that we have been doing with Yin-Chen Andrei Parnachev, Robin Carlson and Peter Tadich two of our students and there is also a Contemporary work also by Fitzpatrick, Huang Li and Li as well So Unfortunately, I will not have time to discuss this. There are a lot of subtleties there Mainly in how one can define The classical phase shift from the gravitational description, but what our focus to it is in a much Simpler case in the what is called the light con limit where in terms of the cross ratios We have both zines the bar smaller than one the The two operators are such that they are the distances space like and we gradually approach the point where the distance between them becomes null and in this case there was There is like a wealth of results and it seems like a lot of information we can extract from this correlator in that limit Unfortunately, this doesn't have a clear let's say scattering picture interpretation that you would like to So so this is basically the outline of a talk There'll be some safety basics, but rather probably I will skip them in this audience We don't really need any safety basics and start from defining what is a holographic CFT What are basic our assumptions and then how we'll try to address the computation of this heavy heavy light light correlator So let me skip all this Generalities This is the crossing equation. Okay, and let me go to the holographic CFTs so To examine a CFTs, which have a holographic description as we said at the beginning we defined them in a sense We we will assume that they have a stress energy tensor operator and that there exists also to large Parameters in our system. We have a large number of degrees of freedom Here I call it n effectively is the square root of the central charge of the The parameter that sits in the two-point function of the stress energy tensor and that there's also this characteristic scale that separates the spectrum of operators of spin two and higher for primaries and Lore so in general we call we use the terminology of the Gauge theories, but here we don't have in mind any specific Lagrangian for our CFT so we will think of a single trace primaries like a set of primaries that We are given in our theory and can have spin up to two with Celebrated stress energy tensor and then double trace and multi-trace primaries are composite Operators that we can build from this original building blocks that our theory has So in general this would be There the rules that we are going to be interested in so when we have this is of order one in the large central charge a large number of degrees of freedom limit and These are like m2 double trace operators operate operate schematical primaries That can be composite primaries of this form where both operators are the same now when we have two scalar operators say one with m2, but m2 is Depends on to well, that's probably on two different operators than the correlation function behaves like one over n squared when The composite operator is made out of these two operators here that we have then it's of order one And in general when we see the stress tensor in the middle with two some other light operators that behaves like one over n So these are enough for what we're going to discuss here And we generally will also assume that the conformal dimension of the light operators is Not an integer because there are complications that show up in that case so what we're going to try to do is we're going to consider this correlator and We'll try to study the crossing equation Order by order in this parameter mu that we defined and in the light cone limit We're in this definition corresponds to Z bar one minus Z bar Z bar going to one So note that in effect what this does is Allow us to study what we would call the stress tensor sector of the correlator We'll see that a little bit more explicitly later And this is because effectively the parameter mu counts how many stress tensors appear within our correlation functions So whatever I will say today can apply more generally So it's what we're going to discuss is the stress tensor sector in a sense of this correlation function But the same sector is true and valid this in terms of OP coefficients and so on and so forth for a correlation function of all light operators so So what is the method we said we study by crossing so we'll study the contributions in both What we call T and S channel and just to set the notation here by T channel I mean when I bring the operators that identical to each other together and by S channel when I bring the operators that are not identical to each other together so Let's start from the T channel, which is a little bit easier But that's where the stress tensor sector sits and will be interested in it So if we expand it if we if you use the OP expansion to write our correlation function in terms of T channel We we have this G which denotes the blocks and P the product of OP coefficients of our operators And here by S and T I denote the spin and the twist of the operators that contribute in this generically infinite sum So now we are we are saying that we are going to study the correlator in some kinematic regime This kinematic regime is the light-con limit and in this case we see that the specific blocks Behave like this what I mean that they behave is that this is the leading term in an expansion in one minus The bar of the conformal blocks that we have so this leading term basically tells us that that the most important contribution to this infinite sum of For for the correlation function comes from operators of the smallest twist Also note here that what f is is basically something like a product of a hyper geometric function That depends on twist in the spin because we are going to see a lot of this F function later So So let's start our analysis step by step. We start from the first step, which is like the identity operator This is what gives us the disconnected correlation function Correlator where we have like product of two-point functions of heavy and light Here we don't see any dependence in mu in this parameter that we introduced now Let's see what happens at the next step. So now in Next we have to discuss what would be After twist zero, what is the lowestest operator and because of the unitarity conditions that That primary operators satisfying conformal field theory this leaves us to consider conserved currents and perhaps some some extra scalars beyond those so I Thinking of like a most generic situation we can have like a most generic safety We're going to consider a stress tensor and assume that there are no other Conserved currents in our theory. So in this case because of a word identity We can see that the product of will be Coefficient that our company the conformal block in this infinite sum Come are proportional to this parameter mu that I discussed from the beginning of the stock and will be the main An important aspect of this so now you can say, okay Definitely, we can have the stress tensor We can think of a theory that doesn't have other symmetries no other concept currents But what about other scalar operators in general if there are no symmetries There wouldn't be any reason for other scalar operators Even if they have lower twist than the stress energy tensor to have an OP eco efficient that scales with mu So that's why I discussed before that we are going to effectively focus on the stress tensor sector of the correlator But of course since we are setting the rules of the game. We can also say we assume that there is no other Contribution, so let's see what happens later basically what what we can see is that this product that the correlator admits an expansion in powers of mu in the t-channel and That goes also for the OP coefficient and what exactly do we have so at order mu Let's say we focus on the stress tensor sector, which comes with a factor of mu at order mu squared What we see is that we can make composites of two stress tensors and several derivatives in between You can obviously think of other composites where you have where you can also take Have some contraction so that Laplacian shows up or contracts several of the indices of one stress tensor with the other But in this case the twist is higher than this object without contraction So from the class of this multiple how I will call the multi-stress tensor operators that we have these This correspond to the minimum twist multi-stress tensor operators they will So now we see that the parameter mu will accompany the number of mu that we have tells us how many stress tensor seed within a composite operator that we have and Also given that we're looking at the light con limit We are allowed to focus on a specific set of all these composite operators However, still this does not simplify a lot of our work. Let's look at order mu squared So before at order mu you we could say, okay, we had just one operator That was giving us the lowest twist maybe some scalar or we can say that there will be coefficients in our theory Zero we can throw them away, but at order mu squared here. We see that we have Basically an infinite sum of operators because no many no matter how many derivatives We put here still the twist remains to be equal to four What's the actual reason and why why you are not you are giving me just the straighter rate of no love lessons or something Exactly what I said. So if you if you introduce you'll get higher conformal dimension and therefore the twist will be Higher so it won't be a lowest twist operator No Actually, we'll see in our recent work Maybe at the end I will say a few words that we can consider those as well Which expansion are you doing? Are you doing expansion like one or in one over C? It's not here the commuter. No in one minus the bar in light come It's the smallest parameter. It's even smaller than one over C one over See ah, no, no first. So first I take a C large And then I take mu and then I take one minus Z But then we'll see that basically I can define a parameter mu times one minus that bar which controls basically the summation If you want to expand the new that presumably You can even do it away from the light come because the expansion parameters mu times one minus it What's the actual spot? Yeah, the point is what contributes so the light come tells me that I have lost twist Maybe did I answer the question? Yeah, maybe you're right so So let's focus on order mu squared and try to see how we can re-sum basically this infinite contributions But in order to re-sum them we have to know what are the OP coefficients for these Operators we know that they come with an overall factor of mu squared but at the time Now the way I present the story is basically historically in the way actually we computed that now We have many ways of computing this OP coefficients At the time we didn't know this OP coefficients The only thing that existed in the literature were just a few of them like a handful of them computed in gravity by some similar expansion of the two-point function and But what we knew was that they all seem to have a structure of this form where this is the the conformal dimension of the light operator and this numbers a, b and c depend obviously on the spin of the operator that we consider since we have Operators of the same twist but different spin so the question was what were these functions? Could we basically these parameters can we write a generic formula for them so that we can perhaps? Afterwards be able to sum this infinite contribution and see what it gives us a order mu squared in our correlator And then whether we can actually evaluate the sums So once we have that the sums will be over some f functions, which are just some hyper geometric so We were actually able to find this OP coefficients With a little bit of guesswork and a little bit of looking at the other channel at the s channel behavior and the first thing that Helped us was the fact that we knew that in gravity when we have a large Delta light now Afterwards we can tell our delta light. We see that we can compute the two-point function of that this operator Remember, it's a two-point function at a thermal background. We can compute it through the black hole calculation by some geodesic Computation so this is the geodesics a computation you can do some regularization that doesn't that's not important But let's suppose now that we this is the result in ideas Let's suppose now that we expand in the parameter mu that Has the mass of the black hole inherent in it the first term will be the usual one That has information from the stress tensor sector from the stress tensor itself and the second term We see that the term that is the highest with delta L Comes basically from the first from the leading term But because of the exponentiation So that basically tells us that whatever this AS parameter is is such that when we do the sum What we get should be the light cone block. Let's say of the stress tensor squared And that's and looking at identities for hypergeometric function You can explicitly find what a s has to be in terms of the speed is this a guess or Well by now it's us. It's a checked with Laurentian version formula is checked with cross Channel calculation so it's now we can extract it. I'm just going through let's say the way we found it And at the beginning we found it a little bit with a bit of guesswork. Let's say But under which assumptions because you said are you aiming for gravity or using great is an input? So he saw here I had to so I didn't have this opaque coefficients and I would want ideally to have them or compute them in the CFT and get this result that will a give me gravity now I didn't have that so I tried to guess go my way around it with whatever results I had now However, we have tons of means to compute them and I can say that we can compute them From CFT alone using for example the Laurentian version formula So so that's what we use then we also use some information from the s channel computation and Let me just not bore you with the details of this computation just want to say that What what we see in the s channel is just all these double trace operators that are made out of heavy and light And the only thing that happens at each order in mu is that we get anomalous dimensions and no peak coefficients of these operators corrected compared to their mean field theory result So one can go ahead and try to study and it was useful for us at the time to focus on the small Region on top of the light cone limit because there we had a little bit of moral control and allowed us to basically guess The other parameters that sit in the opaque coefficients. How does this work? You usually look at the At you compute the correlator again in the s channel at order mu Assuming you have some corrections in opaque coefficients anomalous dimensions this you don't know but you can extract them from the stress tensor contribution in the t channel and then when you're going out order mu squared everything that you're interested in comes at the terms of Log squared and log where these pieces are the leading corrections at small z and these pieces are completely Determined by the previous order data So once we had the result we could actually also check with the Regel limit because we had explicit results there this would be the large impact parameter region of the regi limit and Also, it would be inverted in terms of how the limits are taken usually in the regi limit You first take sigma to zero and then Allow for fixed row and then a large in pop parameter. You take roll large Here we have first zooming in some sense in the light cone and then take sigma to zero However for several terms you can see exact matching So now let's go suppose we have these numbers I don't write them as complicated, but we have them and I can tell you we can compute them in Thousands of ways by now so we can establish we can do a purely safety calculation. Let's do the summation We can do the summation using other identities for hyper geometric functions that unfortunately We had to derive ourselves because although they're simple They didn't exist in the literature and what we get is the following result So this factor comes from the light cone expansion and accompanist factors of mu squared This overall term is just the disconnected correlator and this is the result that we see at order mu squared So we see that we have three terms Each of them depends on these functions f which are proportional to hyper geometrics and in front of them There are parameters that depend on delta light Obviously, this is the highest term that comes from stress tensor squared So what is the interesting observation here? It's like a trivial observation But however, it gives us a lot of mileage It's that every square this three plus three is six two plus four is six one plus five is six So what does this mean? So we thought okay Let's see. Would that mean something? Let's go to order mu cube. Let's now use some answers for the result of the correlator and Using this answer extract this coefficients So let's assume that at order mu cube the contribution of the stress tensor sector looks like this so it contains all the possible products of three f functions that we can have and There's some should go to one because to nine because we have F3 cube and F3 Remember is always the conformal block of the stress tensor sector in the light cone limit So we use this answer and by solving the crossing equation We were able actually to see that it's true and that and determine explicitly this coefficients However, there was a slight disappointment in the fact that we noticed that this sort of Hypergeometry functions are not really a true basis of something some of them were not independent from the other and You would say how on earth would you think of such a thing? and The reason we thought of that is because we actually had By chance observed a similar thing happening in two dimensions So in two dimensions the heavy heavy light block is known it exponentiates. There is a very clear simple form and In that case you can expand this in terms of said and you see the following That's now notice the function that sits in the exponent not the full correlator although for that is the same as well And what you see is indeed that you have a similar structure, but now of course You have the function if the hypergeometric Corresponds to F2 because this is what gives you the stress tensor So having this in mind is what led us to conjecture That something like that will happen in four dimensions and actually higher and Here I'm just writing the conjecture that the stress tensor sector of this heavy heavy light light correlator, and I would perhaps Remove the heavy heavy light light is just the stress tensor sector of any four-point correlator in the large and large Delta gap limit Should be given in terms should be expanded in powers of mu and at each order in mu to the k You will have a sum of products of f functions Where the there will be a k such functions that at order mu k and their sum should be equal to 233 times k What do you think Z bar to 1? So this is what I mean by that is that this is the leading term in the correlator the one that corresponds to the light con limit because I have neglected contributions of Multiple stress tensors that have higher twist. So this is exactly new Yeah, as a sum it will be exact in mew if I could re-sum this. Yeah, it would be exact in you and Of course, that's what you what you would actually hope for given that you know that something like that happens in two dimensions But unfortunately, although I was hoping that I would have to report on that. We don't have a finite We don't have a sum yet So what we have shown since then is that we can consistently use a such an answer to I don't know how Whatever higher order you want to check I mean you can do it Computational computer and you see up to order mu k and this answer satisfies the crossing equation is a consistent result And from that you can extract O p coefficients at higher orders in mu For these multi-stress tensor operators that you can then also cross check with the ones you can get from the Lorentzian version formula so what else Now actually we have checked that and we have also determined of the Coefficients and we established that at least to each order up to mu to the six that we can compute This correlation function this sector exponentiates in the same sense that it happens for the virazoro case So what I mean is that the structure in terms of a functions that show up in the whole correlator persists in the exponent in the exponent of an exponential and I can write my exponent in this form so maybe I will finish probably a little bit earlier, so The question is basically what underlies the structure and Unfortunately, we don't know yet. It would be much help to understand why Something like that can be even done in two dimensions. Does it mean something? one Point is then the question is can we resum the series and as I said unfortunately, we're not able to have a result as far But what we thought of obviously was the following Beyond the pure virazoro. We also have higher spin cases with w3 w4 and higher algebra algebras and for w3 The corresponding global block is f3 is this function that is now in four dimensions the global block of the stress tensor is the one of the spin three current in In two dimensions, so the question is Is it perhaps that if you try to resum? In that case, you know people have also computed their analog of the virazoro vacuum block for the w3 and From there you can take a limit where you scale the spin three charge with a central charge and that allows you to have only The spin three charge at order mu and then combinations of that at higher orders But what happens is that there is a term missing? So it's not the answer that we want is very close. So let me just show you at order Mu squared the W3 block will have a term that goes like f3 squared and a term that goes like f2 f4 But does not have a term that goes like f1 f5 Another piece of information is that this term is crucial to get the scattering phase shift correctly so now another Question is what happens when Delta light is an integer? I didn't say anything specific to that But some of you may have noticed that this OPE coefficients that are wrote schematically have a Terminator that it's equal to Delta L minus 2 and you wouldn't expect obviously to have any divergence here What happens is that when the light scalar has dimension equal to 2? There are composite scalar operators which contribute to the same order in the light con limit with a Multistress tensor operator and you have to solve some mixing problem in order to determine the OPE coefficients So that has not been done yet Now another question is whether you can go beyond the light con limit So what I mean by that is we computed the leading term in the expansion of Z bar going to 1 What about the sub leading or sub sub leading or how does it work? What you have to do is add now consider other operators that contribute together with the lowest list operator and These operators are the ones that you can have Contractions either between derivatives or between the stress tensors themselves and they obviously increase the twist So this is work that it's going to show up soon We were able to actually consider such contributions find the corresponding OPE coefficients in several sub leading orders in the slide con limit and what we determine is that It's okay. Maybe I should make a parenthesis here so one interesting aspect of this whole discussion is that you we were able to compute the stress tensor OPE coefficients exactly and They didn't depend on the details of the theory. For example, there was never any occurrence of Other parameters that normally appear in the three-point function of the stress of the stress tensor like The a conformal anomaly coefficient. So that could well have been Included in this but somehow the leading light com the multi stress tensor sector is independent of that and it's the same for Results for all the cases however, what you see as you go to sub leading orders you see these terms infiltrate and possibly spoiling This sort of universality as we can call it and what what is really Interesting here is that you can basically Determine and prove that everything is universal all this contribution Except for those that have total spin equal to zero or two Remember that in the t-channel only even spin contributes first So from spin four and higher all this multi-trace Multi-stress tensor sector is fixed and universal but spin zero and spin four contributions are not So and then so up to now I have only focused on the stress tensor sector of the correlator Obviously, this is a program and wants us to push it further and maybe after possibly we are able to re-sum this Consider Check if we can consider quasi normal modes or go beyond the large see address Generally physics close to the horizon That is possible probably due to the fact that we have like two competing limits that we can look at both the regi limit and the light Con limits we can extract results. So thank you very much for attention I understood that this approach you have that works in principle for any dimension. Ah, yeah, sorry I forgot to mention this. So we have checked several even dimensionality cases And they and they work but in all dimensionality CFT is that let's say that would correspond to ideas for Things were not as simple So we have we have we have tried to Consistent see get an answer but not yet. So I don't say so the problem is that for example one immediate the issue is that This these blocks are this is hypergeometrics in even dimensions that give us the block are Basic algebraic functions. That's not true in the other case And then it's also the fact that you have possibly to consider half integer values and different mixings and that Makes things a little bit complicated Thank you for the question Yes Can you with these techniques? Do the kind of thing that I'm at tea in chapter the Veneziano did which is ultra energy gravitational scattering of light states But the ultra energy large s makes a large mass and therefore Yeah, that is a little bit further away from the moment, but that's One thing that we are interested in doing so this so this stress tensor sector our Contributions are different orders in one over and basically so forget me It's like one over and squared one over and four So that they effectively tells us that we have a sector of several diagrams that could contribute in the process And we can possibly re-sum that So once that is done, then I could say that maybe we can approach this problem If the kinematic regime allows us to concentrate just on the we probably need some extra scalar contributions That's my feeling but Yes, you didn't mention the comparison with ideas Can you mention it if it works? Yeah, so so There's one to one You've got the same infinite sound Piper Joe metrics in it. Yes. Yes. Thank you