 So let us continue. I introduced this concept of OPE. Let me write it a bit in a different, more condensed form. So OX1 of O2 of Y. So this is equal to a sum of over K, F1 to K. And then here I said there's going to be an infinite series and so I can write this series as I can introduce here some notation for this infinite series. So it's some operator. Let me denote it P, X minus Y and derivatives which acts on the operator OK inserted at Y. So it's an infinite series in DY. And as I said, the coefficients of this series can be all at least in principle determined by matching with a three point function. So we will draw some pictures and we will denote this OPE by this sort of picture that we take the two operators O1 and O2 and we fuse them. So sometimes one uses this terminology fusing fusion instead of OPE and we replace it by a sum over operators K. And so this line here which denotes not just operator OK but also all of its descendants, all of its derivatives. And so we have this equation that this with F1 to K, this formula it has to be equal to really the correlation function where I insert the two operators 1, 2 and K. I can compute this three point correlation function by using the OPE. So one comment I think I let me stress it once again is that if you're familiar with two dimensional CFT, of course you're familiar with the OPE also in the two dimensional context. But there is going to be one change of emphasis in our story. So in two dimensional CFT, one often pays primary attention only to the most singular terms in the OPE. So if you remember for example, in 2D one discusses the OPE of the stress tensor operator. So let me remind you. So one writes equations like dz t0 and here, so here there are several terms. There's the first term C over two divided by z to the fourth and then there are terms proportional to t divided by z squared and another term proportional to dt divided by z, you know with some coefficients which I don't remember. And then there are infinitely many other terms in the OPE and you hardly ever see those terms in two dimensional CFT literature because the way for example, the way this OPE is used often in textbooks is to derive the voracero algebra. There is some counter argument if you remember and for this counter argument it's important the singular terms of the OPE are important but these subliminal terms are not important. So that's why they are not mentioned. So for us all terms in this OPE are going to be important and the reason is that if you only keep the leading most singular terms in the OPE, then you can describe, so this is my cell phone, you can describe the correlation function only in the limit where these two points are close to each other but for us it's going to be important to describe the correlation function also at finite separation. So this is something that I already mentioned. So if you include all of these terms in the OPE, then you can determine the correlation function not only in the limit when the points one and two are infinitesimally close to each other but also at finite separation. So as I already mentioned, the radius of convergence of this series if provided that you use this OPE to compute, so let me say it once again. So suppose that you have endpoint correlation function O1 of X1, O2 of X2 and here you have some other fields sum of OY, XI, so you have X1 here, X2 here and then there are other points here where something is inserted and you want to compute this correlation function using the OPE. So you say, okay, this correlation function is equal to the sum F1 to K, this operator PX1 minus X2 DPX2 acting on the operator okay and here you have N minus one point correlation function. You have okay of X2 and all these other operators in the same ones. So when is this expansion going to converge? It will converge, so I'm not going to explain it but let me state the result. It will converge provided that X1 minus X2 is smaller than the maximum of the distances X2 minus X, no minimum, minimum of the distances X2 minus all these other points XI. Provided that a point X1 is closer to X2 than all the other points, then this is going to converge and this is going to be crucial that the OPE provides the description not only in the infinitesimal limit but also at finite separation. Any questions about that? And so, okay, now in the language of pictures I can already tell you what is going to be the main idea of this conformal bootstrap. The main idea is going to be to take a four point correlation function. So we are going to take a four point correlation function and we are going to, so, a one or two or three or four and we are going to say that we can compute this four point correlation function in two possible ways. We can take the OPE of points a one or two and a three or four. So we are going to compute this correlation function as in the language of the diagrams that I reproduced is going to be this diagram. So there's going to be sum over k of these things. Now let me make sure that everyone understands what this diagram means. It means that I take the points a one or two, I replace them by an infinite sum of these differential operators acting on operator okay. So let me write, this is important so I'm going to write it in detail. So this is equal to sum of p x one two this is the difference d x two okay at the point x two and then analogously o three four so here I forgot f one two k and analogously I replace I replace three and four as well. So f three four k p x three x three four d x four and here there's operator okay of x four. In principle you have a sum over k here and you have a sum over k prime here but now we have to compute the two point function okay of x two okay of x four. And as we said the two point function is diagonal. So only two operators with equal dimensions and spins can have nonzero two point correlation function. So in practice okay in practice it means that operator okay will only have nonzero correlation function with itself. So you can you can diagonalize the correlation function. So that's why we have a double sum over k k prime which collapses to just a single sum over k. And so we reduced the four point correlation function to a sum of two point correlation functions acted upon by these differential operators and OP coefficients multiplying it. So this is a yeah this okay are primary operators. Well if this happens then you should diagonalize should do a linear transformation to make sure. So first of all you know generically if you have an interacting conformal field theory then you are not going to have two operators of the same dimension. All dimensions are going to be different. It's only in very special theories like in free theories you will have operators which are not identical but the dimensions are identical. So in that case you have to do a diagonalization. Two remarks first of all this equation solves for us the problem of computing four point correlation function. So I told you that if you know all dimensions and all spins and all OP coefficients then you know everything. Well here's an example so if I know these OP coefficients and the dimensions of the operators which occur in the OP then I can compute also the four point function by doing this infinite sum. And this sum is going to converge. So that's the first point. The second point is that this expansion also gives us a constraint. Also gives us a constraint on these numbers on Fs. The point is that I can now do, I can compute the same four point correlation function in the opposite order. I can do the OP of points one, three and two, four. Here there's going to be some of our other operators k prime. And it's the same correlation function. So the two expansions should agree that they agree is not going to be automatic. Meaning that if you take just some random, if you take just some random numbers delta k for the separators that you exchange and some random numbers for the OP coefficients, you compute the four point function using this way or in that way you're not going to get the same function. It's only for very special choices of OP coefficients and the dimensions that you are going to get this agreement between different channels. So in, you know when, so these diagrams they look a little bit as fine diagrams although they are not fine and diagrams. And so just like in fine and diagrams we speak about S channel and T channel. Also here we talk about direct channel and the crossed channel for the OP. So there is this terminology. And all channels should agree. So this is the conformal bootstrap. The conformal bootstrap. So conformal bootstrap says that first of all, all channels for all four point functions should agree and this gives you a constraint on allowed deltas and fs. So this is clear. What is less clear is whether how strong this constraint is. And so the second point is that it's basically all there is. I'm going to discuss this. Yeah. No, it's possible to find positions where both channels are going to give convergent functions. I'm going to show this in a second. So let me focus on this point. So in some zero approximation in some zero approximation, basically if you have a set of deltas and fs which satisfies this constraint, then you basically have a CFT. You have a consistent theory or you have 99% of a consistent theory. So there's some subtlety here. So there may be some additional constraints. There may be some additional constraints that sometimes you need to impose but basically already in the 70s. So when people first were playing with this idea, they in particular Polikov, Polikov when he was thinking about this idea in the 70s, he said that, well, if you find the system of deltas and fs which satisfy this constraint, well, this is your theory. There is no other constraint. There's no other obvious constraint that you should impose for consistency. So basically that's it. If you should accept this as a valid conformal field theory. And so if you take this point of view, then the problem of classifying conformal field theories in any number of dimensions is basically reduced to the problem of classifying solutions to this equation. Any questions about that? I'm sorry? In two dimensions. So I'm sorry, I didn't understand the question. Yeah, so that's precisely why I said it's 99%. So in two dimensions, it's known that there are extra constraints which are known under the name of modular invariance. And basically what this amounts to is that you can sometimes take a conformal field theory and you take a sub-sector of conformal field theory which is completely closed under operator product expansion for some symmetry reasons. So from the point of view of conformal bootstrap, everything is fine. Conformal bootstrap equation was satisfied for the original CFT and it's satisfied also for this sub-sector. One example is the stress tensor operator in any two-dimensional conformal field theory has a closed OPE with itself. So if you just truncate the conformal field theory to the stress tensor in two dimensions, then conformal bootstrap equation is satisfied. But that's not a full CFT, that's not a full CFT, that's a truncation over CFT. And the way you go, the way you see it is by looking at the modular invariance constraints. I'm not gonna explain what it is, but for those who know modular invariance. So modular invariance tells you how you should put together various sectors of the theory which are closed under OPE to get the full consistent theory. But basically, if you already solved conformal bootstrap and you found all consistent sectors, then how to put this together is in a sense a simpler problem. That's why I'm saying it's already 99% of the thing. So in higher dimensions, the number of allowed truncations is presumably even less, because for example, the stress tensor operator in higher dimensions does not form a closed sector by itself, unlike in two dimensions. So probably the only truncations you're allowed in the higher dimensions are the ones which are justified in terms of some global symmetry. So once again, if you do conformal bootstrap, you might be able to solve not the full CFT, but just the subsector, and then you should ask, well, can you extend it to an even bigger theory? But again, I consider it as a 1%. So the modular invariance and unfortunately in high dimensions, so in two dimensions, there is modular invariance constraint which very easily allows you to complete this 1%, but with high dimensions, the modular invariance is not as nice because the modular invariance constraint cannot be expressed in high dimensions in terms of the CFT data in flat space. There was some other question? Yes. Yes, that's an important question. It's enough to consider only four point functions provided that you consider all possible four point functions because there are various ways to see it. There's various ways to see it. So one way to see it, perhaps the nicest way is to think of this conformal bootstrap constraint as a sort of condition for OPE associativity. So if you take the OPE 1 and the 2, and then you take the OPE over 3, then it's the same as doing things in the opposite order. And this fourth operator O4 serves for you to project the result of doing the repetitive OPE on some particular channel. And if you view it this way, then it's clear that by considering all possible four point functions, you already enforce OPE associativity, and then by going, and then all high point functions are going to be automatic. But again, yes, so, but, so this condition, which one? This condition. Okay, so what's gonna happen is that this correlation function, this correlation function are going to be smooth functions apart from some coincident points. So if you find an open region where they agree, I mean, by not at least they basically have to agree everywhere. So I'm not sure it answers your question, so yes. So basically what's gonna happen is that for any correlation function, any endpoint correlation function, there's going to be some finite region of possible point configurations where you can impose this conformal bootstrap constraint in a way that both sides converge. And what I'm saying is that morally that's enough. Enough that there should be at least some finite region where both expansions agree, where both expansions converge and agree. Then the rest follows by electricity. I'm sorry, I didn't understand. You can consider one or four, yeah. Sometimes yes, yeah, it depends. Yeah, you should consider all possible permutations. Okay, so philosophically, this is very satisfactory because yeah, so we have equations which are mathematically well-defined equations. So these are equations for convergent quantities. You don't have to worry about renormalization or anything, all these things that we usually worry about when we do quantum field theory. So in principle, if somebody gives you deltas and delts, you can put all this in the computer, you can sum the series, you can plot the function. So if you are mathematically minded, then this is very satisfactory. So the only non-satisfactory thing is that how do you actually solve these equations? So this was the advance of the recent years is that finally we have, we discovered the method which allows us to solve these equations, at least numerically. So I'm going to discuss this method and I'd like to mention straight from the start is that you don't yet understand analytically while this method works. But given that numerically it works so nicely, there's basically a little doubt that sooner or later this understanding is going to be achieved. So not only for the first time in many, many years we have a method which works to solve this problem, but also we have a clear way to go. We should just understand better why this method works and then we are certain to learn more about this problem. So that's a very nice time in the history of this problem. So we are kind of on the crucial junction. So the next thing that I have to discuss is goes under the name of conformal blocks. So what is the complication with this problem? Especially if you are a two-dimensional person then what are the complicated things? So in two dimensions there is this nice separation between Z and Z bars and you can basically forget about Z bars. You can think only in terms of Zs. So this is holomorphic anti-holomorphic factorization. So in high dimensions this is not possible. So you have to work with all Xs. And there's a lot of tensor algebra involved. So there are lots of Xs out there. And so it may look a bit intimidating to have to deal with all these tensors. And so the first thing you have to do is to simplify this tensor algebra as much as possible. So that's what I'm going to explain now how we deal with this. You have to develop some technical tools in order to make the problem manageable. So this goes under the name of conformal blocks. So let me consider a four-point correlation function and now I'm going to do some simplifying assumptions. So before until now I was writing the equations as if my operators were scalars but in fact this was just to save some notation. But now I'm really going to focus to the case when the correlation function I'm considering is a correlation function of four scalar operators. It's really a scalar operator. So I'm going to focus on that case and moreover I'm going to consider the case where these four operators are identical. So this is going to be the simplest case. So I'm considering correlation function phi x1, phi x2, phi x3, phi x4, the same operator. So to extend what I'm going to say to non-identical scalars is possible and has been done to extend it to the case of non-identical scalars is also possible but much more complicated and has been done only partially and has not yet been implemented numerically in the numerical analysis. So this is kind of an open problem. So this correlation function of four scalar operators it can be expressed in the following form. There is some correlation, some function g uv. And in the denominator I have x1, 2, x2, x3, x4, to the power 2 delta phi and x3, 4 to the power 2 delta phi. So xij is xi minus xj. And these use and these are the cross ratios. So u is equal to x12 squared, x34 squared divided by x13 squared, x24 squared. And v is equal to the same expression as u provided that you interchange points one and three. So this is no. This is exactly the same as in two dimensions. So the four point correlation function can be written up to this factor can be written as a function of cross ratios. So these cross ratios u and v, when you apply conformal transformation, they are left invariant. And so this correlation function written in this form transforms coherently under the conformal transformations. So this is just conformal kinematics. Any questions about that? But now I already told you that I already told you that there is a different way to write this correlation function namely by using the OPE. I could compute the same correlation function by using the OPE and doing it like this. This means that I can compute, so this function GUV, which from the point of view of conformal kinematics is arbitrary. If I use the OPE, I can actually compute this function GUV, right? So this function GUV is going to be computable as a sum over all k's, the same k's as there. There's going to be F1 to k, well, one, two here is the same. So phi phi k squared because I'm applying the OPE two times. Times some other function, let me call it G delta k Lk of UV. So where does this equation come from? So the OPE coefficients, F phi phi k, they suggest these things, F1 to k and F3 for k. Now here, I was talking about these differential operators acting on the two point function or k or k. So what I'm saying now is that if you were to make this computation, if you were to do this computation, then at the end of all the algebra, which is considerable algebra, you are supposed to get that everything should collapse. So this one term and this sum, either infinite, infinite many terms, but each term and this sum should collapse to some function of cross ratios again. You notice before you even do the computation that in the end it should collapse. And this function is called conformal block. Is this clear? So this function is a conformal block and this function, it depends only on the dimension and spin of the exchanged operator k. So it's completely fixed by conformal symmetry. So if you know this function, then, and if you know the OPEC coefficients, then you compute this function gv. So let's cause, yeah, I'm, yeah, you're right. I should be a bit more specific about choices at the time. So I chose all my fields to be real to her mission. And if I do this choice, then the OPEC coefficients are going to be real. So there is no, in a unitary theory. So I'm choosing here that, let's suppose that my theory is unitary and all operators, I choose the real basis of operators, her mission basis of operator. These OPEC coefficients are real and this needs to be shown, but one can show this. So think about the easing model or about the fight to the fourth theory. This is anyway, it's good to keep it in mind, you know. All fields in that theory, I mean, it's a real theory. So all operators in the theory are made out of phi and its derivatives. Now, what I'm saying is that if you go to the critical point, you go to long distances and you measure three point functions and these are the OPEC coefficients Fijk. Since everything is real, it's clear that this OPEC is also going to be real, at least in this case. But there is a large class of theories where this is also going to be true. So now we are facing the following problem. We need to compute, we need to get some handle on these conformal blocks. So if we know the conformal blocks, then we can impose this condition, this crossing symmetry condition. Because this crossing symmetry, it means that we should interchange the values of U and V. The crossing symmetry will say that this formula has to be equal to the same formula when I interchange points one and three. Because if I interchange your points one and three, then on the left-hand side, nothing changes. So the right-hand side also has to be invariant. And so this is equal to G V U divided by X 1, 3, no, X 2, 3, sorry, the power two delta phi, X 3, 4, X 1, 4 to the power two delta phi. And so what this means is that I get the equation that G U V is equal to U over V to the power delta phi G V U. Okay, I'm being a bit quick here. So in general case, so this constraint, which I call crossing symmetry constraint, it requires you to do the UP in one channel, do the UP in the other channel and then compare the two expansions. But here we are in a better situation. In this particular case that I'm considering, we are in a better situation because we are dealing with a four point function of four identical scalars. So all channels are the same because meaning that the operators that I'm going to get when I expand in the cross channel, the same operators that I'm getting when I expanding in the direct channel. So the crossing symmetry constraint just is expressed as a certain constraint on the function G U V, which is the one that I derived. This clear? And so you already see the pattern. So I impose that G U V is expandable in terms of conformal blocks with these coefficients, which are unknown. That's the first step. Second step is that I demand that the result of the expansion satisfies this equation written here. This is for this particular four point function, the crossing symmetry constraint. And in order to impose it, we have to know something about the conformal blocks. We have to understand these conformal blocks. Let me draw one more picture here. So yeah, here I was writing this as a function of U and V, but sometimes it's convenient to... So in this form, this equation is really applicable for any points x1, x2, x3, and four. But you can also view it slightly differently. And this is usually the approach we also take in two dimensions. You can say, well, I'm going to use freedom under conformal transformations to move these points around. And I'm going to fix them to some particular positions. So I can, usually the way people do it is that they fix the point x1 to zero. They send the point x4 to infinity. Then x3, maybe you said to one particular point, you can call it one. And then you're left with just one point x2. So this is what we usually do in two dimensions. But this can also be done in any number of dimensions. So in any number of dimensions, I can put the point x1 at zero. I can take the point x3 at one. So x4 goes to infinity. Now I'm left with the point x2. And the point x2, I still have rotation variance around this axis. So I can use this rotation variance in order to put the point x2 in the plane of the blackboard. And so this means, since I have, my blackboard is two dimensional, it means that any four, so the four point function in any number of dimensions has the freedom of two real variables, right? And this is exactly what I have here. So I have three, I have two real variables, U and V, which are in this frame. They are related to the coordinates of the point x2 in this plane. And actually, you know, this plane is two dimensional. So I'm allowed to pick whatever coordinates I want in this plane. In two dimensions, we pick the complex coordinate z. And in two dimensions, it's particularly nice because everything is holomorphic and anti-holomorphic are factorized. But I can also pick this coordinate z in any number of dimensions, also in d dimensions. So let me do this. I'm going to use this coordinate z and the complex conjugate coordinate z bar, also in d dimensions. Of course, things are no longer going to be factorized in z and z bar. So as you will see, there are going to be some equations which are going to involve both z and z bar, unlike in 2G, but I can do this. And so if I write it like this, then my four point correlation function becomes a function of z and z bar. If you look at U and V, then it's going to become z, z, z bar because x12 is, oh, maybe I should write just z squared, absolute value of z squared because x12 is z squared and here I sent x4 to infinity. So all the other factors go to one. x13 is one and V becomes one minus z squared in this particular, for this particular choice of coordinates. So then I have then G z, z, z, z bar is equal to the sum fk squared Gk of z, z bar. And then this equation, this equation star, I can write as, I can write as, let me write it like this, one minus z, one minus z bar to power delta phi. G of z, z bar has to be equal to z, z bar to the power delta phi E of one minus z, one minus z bar. Basically the crossing transformation is the transformation which interchanges x1 and x3 and so what was z becomes one minus z. So the crossing transformation changes z to one minus z and z bar to one minus z bar. And here I can, now I can answer, now I can answer the question. So there was a question asked, whether I can find a frame in which both OPE expansions are going to be convergent. And here I clearly see that there is such a frame. So because if I, if this is my choice of points, x1, x2, x3, x4, then the OPE expansion for x1, x2 is going to converge provided that the point x2 is closer to x1 than any other point. So it's going to converge in this, in this circle, in the circle of radius one around x1. And in the other channel, x2, x3, it's going to converge in this circle, in the circle of radius one around the point x3. And so you see that there is this whole region, there's this whole region in which both OPE expansions are going to converge. And so we can impose the crossing symmetry constraint by demanding that at least in this region, the two expansions should agree. Actually, I don't understand the question, sorry. What's the U-channel? Ah, the third channel you mean? Well, the third channel is not going to converge for this, for this choice of points. You have to choose, you have to put points differently in order to make the third channel converge. No, no, no, you cannot have a region where both the channels converge simultaneously, I think. Yeah, you should compare one to two, then in some other region two to three, when you need to do it. Here we don't need to do it because since we are dealing with a four point function for identical scalars, actually the moment you compare this channel one, two and two, three are basically done. We don't need to do anything else. So let me recap. I told you that at least in this particular case, the conformal booster of equation takes form of this equation, which has to be satisfied by four point correlation function. And in order to impose this equation, we have to expand the four point correlation function in conformal blocks. So I already defined for you the conformal blocks. And now my next task is going to be tomorrow to tell you how you actually compute this conformal blocks in practice. I think I'll stop here.