 It's a pleasure to open this school. So I'm going to be lecturing about the conformal bootstrap. Thanks to those who filled out the questionnaire, it was very helpful to understand your level and not to repeat things that everyone knows. So it would be nice if we could jump kind of head on into the subject and then understand the details. So what's the conformal bootstrap? So the conformal bootstrap is an approach to CFTs in D dimensions, so D Euclidean dimensions. So D can be equal to two, that's already interesting, but mostly in these lectures I'm going to consider D larger than or equal to three. But also in D equal to two, there remain many things to be done. So what are CFTs? Why are CFTs important? It's, let me say a few things about that. So when we do quantum field theory in general, then most of the time we are dealing with a massive quantum field theory. If you have massive quantum field theory, then you have particles of a certain mass m which you can scatter also if you are in Minkowski's signature. But if you are in Euclidean signature, then the way you detect a massive quantum field theory, you look at the correlation functions and the correlation functions, they decay exponentially. They go as e to the minus r over some distance, which is called the correlation length. So these are called massive or alternatively, they also sometimes called gap. And the mass of the particle is called the mass gap and so on. But then sometimes you deal, sometimes you end up dealing with the theory which is massless or gapless. I'm gonna write, I'm gonna write like. So massless or gapless. So this happens when, so the mass goes to zero or the correlation length goes to infinity. So when this happens, then the correlation functions in such theory scale as one over r to some power. So this is clearly a different behavior. So typically in order to, I'm going to give some examples. So typically in order to reach this behavior, you have to fine tune some parameters. You have to fine tune some parameters in the original theory. So for most choices of the parameters, you're gonna have a gap theory. And for some very specific choices, you will have the gapless theory. And so the two point functions, they scale like one over r to some power, but not only the two point function. So all correlation functions, they exhibit power law behavior. So when you rescale all the distances by some factor, then the correlation function also rescales by some factor. And this is called scaling variance. And as most of you know, this scaling variance in a relativistic quantum field theory is almost always, so generically it gets enhanced to conformal variance. Conformal variance, and so you get a CFT. Any questions about that? And so, you know, so CFTs, they form a subclass of all quantum field theories. Which has an additional symmetry. So in addition to rotation variance and to Poincare, it has scaling variance and moreover conformal variance. So it is still a special subclass, yes. So there was a question, what are the specific conditions for scaling variance enhanced to conformal variance? So there is no full understanding of this question. So some conditions help, for example, the existence of the local stress tensor operator is very important for this. Sometimes unitarity helps and so on. But there is no, so there is some understanding, but there is no, you would have to go like case by case basis. But generically, this happens. I was not planning to talk about this because most of the people said that they are familiar with this question. But I can maybe mention later on or we can discuss in private if you're interested. So what I said is that the CFTs are special. So it's, you know, if something is special, you are interested in this. But also, if you understand CFTs, then you can go back and try to understand the massive theories. So CFTs, they are kind of, the reason why, one second reason why they're important is that they are kind of assigned posts. In the space of all quantum theories. What I mean by that is that if you have a CFT, you have a one CFT, then you can perturb it by a relevant operator and you can start an RG flow, which can bring you to another CFT. Okay, so this is the first CFT is called CFTUV. Another CFT is called CFTIR. So it can be, or it can be nothing. So if this RG flow gets you a massive theory, then this CFTIR is just nothing. And so you can start with the, if you start first by understanding CFTs, then you can, in the second step, understand the RG flows connecting the CFTs. And understand these RG flows would be understanding already all quantum field theories. So that's why CFTs are a good, if your goal in the end is to understand the roads, the road map of all quantum field theories, the space of all quantum field theories, then CFTs is a good starting point. To understand CFTs is a good starting point. So these are theoretical reasons for the importance of CFTs, but also there is experimental reason for the importance. So CFTs describe phase transitions, second order phase transitions in real materials. So that's okay, this is a high energy school, so I'm not sure how much you're interested in this, but you should be interested. I think it's very beautiful. So take a material. Material is a dirty thing, so it's some sort of crystal or lattice. You can think of it as some sort of lattice. And on each point of this lattice, there is an atom which has some degrees of freedom, spins and so on. So there's no continuum limit here, and there is no rotational variance on the scale of the lattice. But now suppose you change the temperature in such a way that this material approaches the second order phase transition. So the second order phase transition, the correlation length goes to infinity, and this means that all the microscopic details, all the dirty and interesting microscopic details of this material become less and less important. So the thing which becomes important in the limit of large collision length are the symmetry that this material possesses, and basically that's it. So this is the phenomenon of universality. And so at the phase transition point, the correlation functions of this microscopic theory are going to be described, are going to be universal, and are going to be described by a CFT. And one example, so this is going to connect nicely, so example is the three-dimensional easing model. Well actually, not just three-dimensional, but easing model. So if you take, so everybody knows what the easing model is. No? Well easing model is a microscopic model for ferromagnetism. So you take a lattice in d-dimensions, and then on each point of the lattice you put plus or minus. You put a variable Si which takes values plus or minus, and then you write the interaction energy as a sum of pairwise nearest neighbor interactions. So you have Si, Sj over all nearest neighbor pairs of these variables which are called spins, easing spins, with a minus sign to get a ferromagnetic interaction. So the interaction favors energetically where all these spins point in the same direction. And then, so there is still a parameter. So this is a classical statistical mechanics problem. So there is a temperature, and then if the temperature is very large, then basically these variables are going to be randomly pointing in all directions. If the temperature is low, then you will have ordering. So you'll have all spins pointing in one direction. And then there's going to be some critical temperature, T equal to C, where there's going to be a critical point separating these two phases. And this critical point is described by a CFT. So this is true in D equal two and D equal three dimensions. And actually if you know the two-dimensional CFT as many of you do, then in D equal two, you know the CFT. In D equal two, this CFT which describes the critical point of the two-dimensional easing model is the M34 minimal model CFT, dimensional easing model. So in fact, this CFT sometimes called the easing model CFT, but it's not the easing model because it's only the easing model describes the behavior at any temperature, while this CFT describes the behavior only at the critical temperature. So this CFT, it's an exactly solvable CFT, you know, we know all about it. And it contains, you know, it encapsulates all the long distance behavior of this lattice model. So if you now take a different lattice, instead of taking square lattice, you take a triangular lattice, then the microscopic model changes, but the CFT describing the critical point. So the critical temperature changes, but the CFT describing the critical point stays the same. So this is the universality. So that's very, very beautiful. So where should we go from that? And so when you have a CFT, you have various local operators. For example, this two-dimensional CFT has local operator sigma and epsilon. And the numbers, the most important numbers that you would like to know are the dimensions of the separators. So delta sigma and delta epsilon. So these dimensions, they determine the behavior of the correlation functions and large distances. So sigma of zero sigma of r goes as one over r to the power two delta sigma and analogously, analogously epsilon. And, you know, once you solve the CFT, you know that in 2D, the dimension of sigma is equal to 1.8 and the dimension of epsilon is equal to exactly one. But now let's go to the three dimensions. Let's go to the three-dimensional case. So again, you know, you will have the easing model, you will have a critical temperature, you will have a CFT which describes this critical temperature. So what do we know about the CFT? And until recently, people did not really take this perspective. So there was a very different way in thinking about the two-dimensional case where you have the CFT and everything is beautiful and then the three-dimensional case. So the three-dimensional case, everything was considered dirty and CFT is not gonna help you for reasons that I'm going to explain why there were these prejudices. And so until recently, in 3D, people were taken a very different approach to understanding the properties of the critical point in the easing model. And the way people were approaching this problem was through the Lando-Ginsburg Effective Theory, which is actually nothing but the lambda phi to the fourth theory. So they were considering the theory of scalar field in three dimensions. You consider this Lagrangian and you already feel that you are doing great because now you can do perturbative computations in this theory. You can compute correlation functions perturbatively in lambda. And you can see, and you might think that you would compute these correlation functions as a function of, so mass squared, you keep say lambda fixed and m squared is your parameter that you vary. And you see, if m squared is negative, then you are going to be in the phase where the symmetry is going to be spontaneously broken. If m squared is positive, then you are going to be in the phase where the wave of the field phi is going to be zero. And then somewhere in between, there's going to be the critical point. And you might think that you will be able to find this critical point using quantum field theory. The difficulty with this approach is that this is a strongly coupled problem. So looking for, so lambda in 3D, so in 3D, lambda is massive. So lambda has dimension of mass. And so the critical point is located at lambda over m or the one. And so it's a strongly coupled point. You cannot easily reach this point in perturbation theory. And so people invented various tricks to get around this problem. So one trick is to work not in 3D, but in four minus epsilon dimensions. So in four minus epsilon dimensions, this coupling lambda is very, very weakly relevant. And so you can do perturbation theory. You can reach this critical point in perturbation theory. And then you take this four minus epsilon dimensional result and then you extrapolate it to epsilon equal to one. So this is called the epsilon expansion. It was invented by Wilson and Fisher. And numerically, at least for low order synapses, and this was numerically successful. So it was given results in agreement with experiment for the critical point of this theory. But as you see, it's a very different, it's a very philosophically, a very different approach. So there was this big dichotomy where in three dimensions, you had to do some field theory computation. While in 2D, it was this beautiful safety. So recently, more recently, this imbalance has been restored. So now, and this is what I would like to talk about. So now it's known how to compute in 3D using CFT. So these parameters, delta sigma and delta epsilon, we can now compute also in 3D, not using this field theory the tour, it's a detour because why is this a detour? Because conformal field theory, as I said, is universal. So it does not depend on the microscopic physics. If you take the easing model or if you take this lambda phi to the fourth theory, the critical point is the same. So you should ask yourself, so if this critical point is the same, why should you solve the critical point by flowing to it from some microscopic Lagrangian? So this is really an unnecessary, it seems like an unnecessary detour. So it would be nice to find a method which focuses directly on the critical point on the CFT and solves for these parameters, delta sigma and delta epsilon and other numbers directly from CFT. And so in 2D, it was known how to do this. And in 3D, it's been discovered relatively recently. And so if you do these computations, then this is not just some abstract method, it's really a method which allows you to do computations and very, very precise computations. So the advantage of this is that, okay, it gives you a new perspective on the problem. It is more general because before, the only CFTs that people could consider were the CFTs which had some microscopic Lagrangian description or a lattice model. But there are also some CFTs which do not have Lagrangian description, which are important for string theory. And so now, those CFTs were not accessible, but now they're accessible, so applicable also to CFTs without Lagrangians, like the 2,0 6D theory. And it's very precise. So this method, it turns out to be extremely precise. And for example, just to give you the example of its precision, so the value of delta sigma is now known using these methods with six digits, five digits to be precise. This is the current world record, which I think in a paper which will appear this week or maybe next week is going to be improved by one more digit. So this, by the way, is much better than whatever you can do using this field theory approach. Because this field theory approach, given it's a strongly coupled problem, at some point you run out in difficulties with summing Feynman diagrams because perturbation theory diverges while here there's no this difficulty. This method is non perturbed. So I think it would be nice if at the end of this, if at the end of this lecture course, you would get some idea about how do we get these numbers out using this method. So that's going to be the point of my lectures. But maybe any questions about the motivation and, okay, so yes, so the question is if I hear it well, if ADS CFT can do better. Well, I'm not sure I understand the question because ADS CFT only, you know, in ADS CFT you only do computations for large gen theories and here there's no large gen. Well, there's no, let's put it this, let me put it this way. There's no computation of this problem using ADS CFT. And my opinion is impossible because of the large gen. There's no large gen here. It's a small n, n is equal to one. It is about two orders of magnitude better than Monte Carlo for this particular problem. No, but CFT is obtained from this theory. In the limit when you take, so to get CFT, you have to set M. The critical point is realized for some value of lambda and some value of mass. Let's keep lambda fixed and let's vary mass. And then for some value of the mass M equals M star, there's going to be a CFT. At this point, the moment you reach the critical point, you cannot vary the coupling anymore. So if you vary the coupling, then you detune. So it's no longer a CFT. So we can start then. So what we should then discuss is that, so I told you that there exists some non-perturbative way to think about CFTs. So let me tell you what are the ingredients. So CFT, non-perturbative data. So any CFT is going to be characterized in the first, so they're going to be considering CFT in flat space in this course. There are other in flat, infinite space. There are going to be no boundaries, no defects. So these things are also interesting, but we're not going to discuss that. So the only operators we are going to consider in this course are going to be the local operators. And any CFT will have a spectrum of local operators. So there are going to be some local operators, OI, OX, and these operators will have scaling dimension, delta I, and spin. So scaling dimension we already discussed. Scaling dimension is the parameter which tells you how the two-point function decays at infinity. The spin here means Lorentz spin or rotation group spin. So our CFT is going to have a symmetry group, SOD. So it's going to be, the full symmetry group is going to be SOD plus one comma one, which is the conformal group of the d-dimensional Euclidean space. And this contains SOD, the rotation group. And so the spin of the operator just tells you how this field transforms under the SOD. So they're going to be scalar operators, vector operators, tensor operators, terminal operators perhaps as well. And if you are in larger number of dimensions, then they can be also more complicated representations, mixed symmetry representations and so on. So to know the spectrum of the theory means to know all these deltas and else. So to solve the CFT in particular means that you should determine all these numbers. And there are infinitely many of them. So here there is a big difference between the two-dimensional case and the d-dimensional case. So in the two-dimensional case, in some theories, so-called rational theories of which this minimal model is an example, you have a finite number of primary operators. So here, for example, in this M34, there are only three primary operators, one sigma and epsilon, unit sigma and epsilon. So in d-dimensional CFT, there is going to be, so if d is larger than two, there's going to be also always infinitely many primaries. And as you remember, in two-dimensional CFT, there are two concepts. There is a primary field and there is a quasi-primary field, so in 2D. And quasi-primary is a field which transforms nicely under the global part of the Versailles algebra under the part which is generated by L0, L plus minus one. So this comment is only for those who understand 2D CFT. And each primary field in two-dimensions contains and has infinitely many quasi-primary descendants. So the number of quasi-primary fields in 2D is always infinite, but in d larger than two, we only have the global conformal group. There is no infinite-dimensional extension in larger dimensions. There's no analog of the Versailles algebra in d-dimensions. And so the primaries in high dimensions are like quasi-primaries in 2D and there's always going to be infinitely many of them. So you have to determine these infinitely many numbers. Of course, going to be a complicated task. And the only reason in present-wise is possible is that there is a certain phenomenon of decoupling, meaning that if you are interested in operators of low dimension, like for example, sigma and epsilon are operators to scalar operators which have the lowest scaling dimension. So if you are interested in these operators, then it turns out that the influence of the operators of high dimension. So it's going to be infinitely many operators. And this is going to be their spectrum. In practice, you are interested mostly, mostly interested in low dimensions because those operators are actually the most experimentally relevant. If you measure critical exponents, okay, I did not discuss critical exponents, but anyway, if you make experiments with these critical points, then these are the operators which are mostly interesting. So it turns out that all these other operators of which are infinitely many, you cannot just throw them out as we will see. But there is a certain decoupling, meaning that as you go higher and higher in dimension, then the influence of those operators on what happens to the low dimension operators becomes smaller and smaller. And so this is what makes computation feasible. Questions? Yeah, that's a very good question. So in two-dimensional, so there was a question whether there is an assumption that the spectrum is discrete. So in two-dimensional CFTs, there are examples of theories with a continuous spectrum. In higher dimensions, it's a conjecture that any CFT has a discrete spectrum. So it's not a proven conjecture, but it's a conjecture. So we can make this assumption, but actually if you look closely at how the method works, this assumption is not even needed. There was some other question up there. Yes, so the question is how these operators are distributed. So that's also a very interesting question. So it turns out that the density of the separators as a function of scaling dimension grows in any CFT in the dimensions. It grows as e to some constant times dimension to the power d over d minus one over d. So in 2d, you get e to the central charge times square root d with some constant. In higher dimensions, instead of central charge, you get some other number, and the power changes from square root to d minus one over d. There are exponentially many operators in high dimension. So that's the first part of non-perturbative data. And the second part is OP coefficients. So most of you know, most of you heard about the OP. So OP means that we take two operators, OI of X and OJ of Y. And if these two points are close to each other, then we can replace this product of two operators by a sum over all operators K of some numbers Fijk times operator OK. Let's say inserted the point X plus Y over two at the midpoint times times what? Times some factor. So there's going to be some factor which depends on what is going to be this factor. Let's write it down explicitly. So X minus Y to the power delta I plus delta J minus delta K. But can you see it here? Is it's not blocking the view? Probably I should. Can you see this equation? All right. So here I wrote this equation for the case where the operators are scalars. Clearly if the operators are tensors, then there are going to be some Lorentz indices and then you have to modify this equation. So this is for scalars. So these numbers Fijk are called opaque coefficients and these numbers represent also a piece of non-perturbative data about CFT. So if you remember about the two dimensional CFT, they were these numbers also here. They were taken in particular, sigma times sigma was, so here there was sigma times sigma equals one plus F sigma sigma epsilon times epsilon and this number was a known number two dimensions. And so here, similarly here, there is some number here. So actually in order to make everything fixed, you know, you could change this number by changing the normalization of operators. But let us fix the normalization. So we fix normalization by requiring that the two point function OI of X, OJ of zero goes as one over, let me take the same index, over X to the power two delta I. So this number is fixed to one. So there's in general there's going to be some number here called n and I'm going to fix it to one. So I'm going quickly. Anything not clear about this equation? Yes, no, no, the hash is a positive number. Because, so the question is, you know, it looks like this density grows with numbers. So this density counts the number of operators. How many operators are there? So what this equation says is that there is exponentially large number of operators. Now, no, but what you are saying is distribution. But what you are saying also, you know, there is an interesting follow up to what you are saying is because suppose now that you want to consider the partition function of this theory. Then at some temperature, you put this theory at some finite temperature. Then, you know, there is a way in which this dimension of the previous can be interpreted as energy if you put the theory on some sphere. So if you compute the partition function, then there's going to be the Boltzmann factor. And it's important that the Boltzmann factor is e to the minus delta over temperature. So it's precisely exponential. Well, here we have a behavior which is slightly slower than exponential. So it means that, yes, there is exponentially many operators, but the growth is not so fast so that the partition function is actually finite. Any other question about the OP? So why is the OP important? Actually, these two things is basically everything. So in the good zero approximation, these two things are everything that you need to know about CFTs. If you know these two things about the CFT, then basically you know 99% about your conformal theory. So in particular, if you know this spectrum and you know these OP coefficients, then you can compute any correlation function of your conformal theory in flat space. Not just the two-point function, but also the three-point function, four-point function, and so on. This is something that I would like to discuss next unless there are any questions. Yes? No, we don't know all of the CFT data, but it turns out that because of this phenomenon of decoupling, which I mentioned, that you can get high precision numerical results about the low part of the CFT data, so about the low-lying operators, about their OP coefficients, without solving everything about the high-dimension CFT data. So this is very fortunate because since there are infinitely many numbers to determine, if all of them had to be determined at the same time, this would be a hopeless problem. So in two dimensions, you see we were lucky because the problem was finite dimensional because of this interesting property of the rational conformal theories, you just have finitely many objects, finitely many numbers, and then you just write down some system of linear equations and you solve period. So in high dimensions, this is not gonna work, so the method actually is different that we are going to discuss. So there are some details that I omitted, so let me try to fill them in. So first of all, this distinction into primaries and descendants. So these operators, which I mentioned there, these are going to be the primary operators. So if you have a primary operator, it means, so primary means that if you take an operator at the point X and you apply a conformal transformation, then after the conformal transformation, I presume you know what are the conformal transformations. Should I discuss conformal transformations in D dimensions, or is it more or less, you know, you have inversion, if you have inversion out of inversion and translation, you can build special conformal transformations and that's basically all there is. The only non-trivial one. So if you apply conformal transformation, F of X, then you get an operator at the point F of X times some factor, which is, you know, any conformal transformation is a combination locally. It's a combination of rotation and dilatation. And this factor takes into account that since there is a dilatation involved, it means that this operator O has to be rescaled because there is a scaling dimension. So I'm not going to write down this factor. It's in my notes, you can look it up. But the crucial thing here is that it transforms homogeneously. So you start from O of X and you get after the conformal transformation O at a different point, F of X. This is the defining property of a primary operator. It's an operator which transforms homogeneously under conformal transformations. So this is for primary. So descendants, so in 2D, there are tons of descendants because there are all of these generators LN of the Veraserra algebra. And if you act with any of these generators with negative generators, then you get a descendant. So in higher dimensions, it's simpler. So we only have global conformal group. It means that the only raising operator that we have is P mu. So descendants in D larger than two, these are just derivatives. So it's just D mu, D, D, D, O. So these are, you can contract some of these indices, but these are all descendants. So the descendants are the derivatives in high dimensions. It's very simple. And if you know how the operator itself transforms under the conformal transformation and this rule is fixed completely by conformal group in terms of, if you know dimension and spin of the operator, then you know this rule. You differentiate, you determine the rule how the derivative is going to transform. And the derivative is not going to transform homogeneously because if you act with the derivative, then the derivative can fall on the operator itself or it can fall on this factor. And if the derivative falls on the factor, you will find that the transformation rule for the derivative involves not just the derivative itself but also the primary operator. So it's not going to be homogenous. So transform not homogenous. But the important thing is that these descendants, well, what is the role of the theory? So on the one hand, we feel that we don't really need to care about these descendants because it's the primaries which contain, if you know something about the primaries, you know about the descendants. On the other hand, do they play any role? Can we completely forget about the descendants or should we keep them in mind? Well, we cannot completely forget about them. We could at least not at this stage. And the reason is the following. So cannot forget about descendants because when I wrote that OPE, I didn't write it completely. So let me write it again in a better form. So let's OI of X or J of Y. You see, I wrote in the right-hand side, the operator OK, but I inserted it at the middle point. So why did I insert it at the middle point? I didn't have to insert it at the middle point. I could have inserted it at X or I could insert it at Y or I could insert it anywhere between X and Y. So if I make this choice, this arbitrary choice, all these arbitrary choices can be related to each other. If I take this operator, say, so for example, I could insert it here at the operator at the point Y. But then the difference between OK of Y and OK of X plus Y over two can be expressed by just Taylor expanding as a sum in derivatives of operator OK. What this means is that if an operator OK occurs in an OPE, then clearly its derivatives will also occur in the OPE. So we cannot set the derivative terms to zero. So the more correct way to write the OPE would be, OK of Y plus derivative terms. So plus, let's set Y to zero. OK of zero plus, here they're going to be terms of the form say X mu, some number times X mu D mu OK of zero plus some other number, X mu X mu, X mu X mu D mu D mu OK plus dot, dot, dot. So there is an infinite sum of terms. There is an overall coefficient the same as I wrote there, X delta I plus delta J minus delta K. But this is going to be the full form of the OPE. And you see, so that's why we cannot forget about the descendants because they occur in the OPE. And now comes the crucial point that yes, the descendants occur in the OPE, but the coefficients with these coefficients with which they occur, they are all fixed by conformal symmetry. So there is an overall number, F I G K, which we don't know at this point. Some free parameter of the theory. But the relative coefficients of the descendants in the OPE we do know. Or rather, what do these coefficients depend on? They depend on, they only depend on the dimensions. They depend only on deltas of the fields. So if you know the dimension delta K and delta I, delta J, delta K, then you know all of these coefficients. At least in principle. This F I G K you mean? Well, let me discuss how you fix them. There are various ways to fix these numbers. So one way to fix these numbers is to demand indeed that this OPE be invariant under conformal transformation. So you act on this OPE with some, you commute this OPE with conformal generator. And then you demand that everything works out as it should. Then, as you yourself said, because the conformal generators, they transform primaries to descendants and so on, you will see that in order to close the transformation property, you will fix these numbers one by one, including the first one. This can be done, but there is a simpler way, there is a simpler way to do this computation, which is to start from the three point function. So this is probably the last thing I will say today. So most of you said that they know more or less the constraints imposed by conformal symmetry on correlation functions. So in particular, the fact that correlation functions, there is a famous fact that if you take a two point correlation function or i of x or j of y, then this is nonzero only if delta i is equal to delta j can be nonzero. And then okay, the behavior is x to the two delta y. So this is really already consequence of conformal symmetry because if you had just scaling variance, then you could write here delta i plus delta j and it will be scaling variant. But conformal symmetry tells you no, only if the two primary operators have exactly the same dimension, their two point function can be nonzero. And then there is an analogous equation for the three point function, or y of x or j of y or k of z. Let me put it, let me put three operators, or one x one or two x two, here x minus y, sorry. O three x three, this is equal to some constant and this constant is going to be exactly the same as the OP coefficient f one to three. And here I will have x one minus x two to the power h one to three, x one minus x three to the power h one three two, x two minus x three to the power h two three one and h i j k is equal to delta i plus delta j minus delta k. So this formula, which many of you I presume have seen follows completely from invariance under conformal transformations. Now if you take this formula and you expand it in the limit, x one going to x two and require that this formula be reproduced by the OPE. So you see OPE says that you can, OPE says that you can get exactly the same formula by taking the OPE O one or two, focusing in the right-hand side on the term involving the operator O three and then match term by term in the expansion in powers of x one minus x two. And since this formula is exactly known, it means that all the coefficients in the expansion here are also going to be exactly known. This was somewhat fast. Is this clear the logic? You mean if all these operators have spin, what happens with these coefficients? Well, yeah, then still there is an analogous story. So let's start. But you see, when we take this OPE and you insert it in the three-point function, you get infinitely many terms of the form O three O three, D mu O three O three and then D mu D mu O three O three and so on. So this term you know is just normalization. It's one over x to the power two delta three. These two-point functions you just find by differentiating the first one. And so you get a series in which you know all the terms apart from these coefficients that you have to fix. But you match this first series on the series that you obtain by expanding this three-point function and this allows you to fix all the coefficients. Okay, so I think that's enough for the first lecture. Unless you have any questions. I use this one more. Could you repeat sir Ed? Well, so the question, so if you shift the insertion point here, like from zero to the midpoint and so on, then of course all these terms change. So all these terms they depend on which point you insert. But the leading Fajk does not change. So all these different prescriptions, they only change the sub leading terms. Yeah, so no, you cannot. So the argument that I gave that you know they can insert text plus one over two, you can insert the text and you can insert it, why? This argument tells you only that this derivative, this descent of terms, they should be present in the OPE. But in principle, this does not tell you with which precisely coefficients. It just tells you that there is this ambiguity so you probably cannot extrude those terms. But in order to find the coefficients you have to do this argument. Yes, the sum is convergent. Well, in this particular case, you just, I mean, since you know the function, right? You know the function, there is x one, x two and x three. So the question was, what is the radius of convergence of this expansion? And in this case, you just see explicitly that the radius of convergence is x one minus x two should be smaller than x one minus x three. So the points, the two points that you are doing the OPE, they should be closer to each other than any other point. So here you see it by expansion and then you can give a general argument that this is always going to be the case, even for higher point functions. No, for high point functions you have to give a separate argument because for high point functions you don't know what the two point functions are, right? Actually, I think we should continue probably since there has to be a break, let's, I, yeah. Yes, yes. Okay.