 Thanks. OK, thanks. Yeah, it would be, I think, very productive if you ask questions rather than just listen to what I say. It would be more interactive and more interesting for all parties involved. So my plan is to first describe some motivation and history of various chiral ring questions and how they play the role in the early days of the ADS-CFT correspondence and various other things. So I'll do a little bit of that, and then we'll plunge into the details. So I'll start with some history involving the n equals 4, maximally supersymmetric young mill theory in four dimensions. So this subject started its life around 1998. That's more or less when the first paper of the subject appeared in the context of maximally supersymmetric young mill theory. And I'll try to describe what people have been interested in back in the day, how that played a role in the ADS-CFT correspondence, and then we'll discuss modern developments, which are from 2010 and onwards, where people have managed to generalize many of the statements to n equals 2. And as you'll see, n equals 2 appears to be much more interesting. There are various perturbative aspects, instantons, resurgence. I'll mention a little bit at the end, some connections to resurgence, and many other questions of interest. So let me start. So in n equals 4, there is a family of n equals 4. We'll discuss some aspects first at zero-capting. So what is n equals 4 maximally supersymmetric young mill theory? This is basically an SUN gauge field, and six scalars in the adjoint representation, and then some while fermions that would not play a huge role in this motivational speech. This fermions and scalars are in the adjoint representation. So a family of operators that people have considered early on in the subject are the following. So let me define them. So these are single trace operators made out of k adjoint scalars. So I1, Ik, where each of the indices I runs over 1 and 6 from 1 to 6. And C is some tensor. So the label of I is just like an index of the tensor. There could be several tensors, and these are the actual indices. And it turned out that it's useful to, for reasons, this is just some history. So I'm not going to be extremely pedagogical or systematic. It's just like trying to remind you of some things that were important 20 years ago. So it turned out that these tensors have to be symmetric and traceless for what I say to be true. There isn't for this restriction that these tensors are symmetric and traceless. In that way, they would be irreducible representations of, irreducible representations of SO6. So this model has an SO6 symmetry acting on these indices. And in this way, they're in some symmetric, traceless representation of SO6. And these operators are half BPS. So they are annihilated by a half of the supercharges of this theory. So these operators played a very important role back in the day. Now I'll tell you what's special about these operators. So these operators have many interesting properties. I'll just try to remind you of a few of them. Sorry. My tag's bad. So the properties that would suffice, I'll normalize these tensors so that when we sum over all the indices, this is just one. So this is just the choice of normalization of these tensors. So more generally, I'll put two different indices here, I1 and I2. That will be just the delta function of I1 and I2. So that's how we'll normalize these operators. And now the action is going to be normalized as usual. So it's two times g and mil squared. And then there is a trace of f squared plus delta. So that's how the action is normalized as usual. And the propagator in the free field theory. So now we're discussing just free field theory at zero cupping. So the propagator of phi i to phi j from x to y is normalized as a consequence just by g young mil squared over 2 pi squared. And then there is a delta function in ij. There are some color indices here a and b. So these are like the elements of the matrix. 1 over the distance squared. So that's my normalization for the propagator. And using this information, you can now compute correlation functions of these operators at zero cupping, just like three-week constructions. It's a straightforward exercise. So we'll do a few examples. And then we'll talk about the surprising properties of this. Yes? Maybe trivial question. But by zero cupping, you mean the lowest border in asymptotic small. Yeah, I just mean, well, by zero coupling, you could mean the borne approximation more precisely that there are no loops. So you only use weak constructions. So oftentimes, there will be some dependence on the coupling, but it just arises from the normalization of the propagators. So you don't allow for any loops. So it's like the leading non-trivial order. So if you just use those formulas, you can compute the two-point function of arbitrary such two operators, just using weak constructions. So my convention is going to be, so there will be k of those, and then j of the j1 to jk. So you have to have the same amount, the same number of fields in the two-point function. Otherwise, it would vanish for obvious reasons. This is at x, and this is at y. So you have to have the same number of fields. Otherwise, there are no weak constructions to contemplate. And you can just take it as a small exercise to just do the weak constructions using the normalizations that I've defined. And you can just show, if I have not made a mistake, that this is g to the power 2k young mills. Then there is 2pi to the power 2k. And then there is a delta function of i1j1, delta function of i2j2, and then ikjk plus cyclic permutations. And this is the answer in the planar limit. So sorry, times 1 over the distance. I forgot about that, so to the power 2k. And this is the answer in the planar limit. So only if you take planar diagrams, you can just take it as a small exercise to check. Does everybody know what is the large n limit? What is the tuft expansion? Does anybody not know what is 1 over n expansion? OK. So this is the answer for this two-point function in the large n expansion. You can, of course, write an exact answer for any n. I suggest that this is what we will need for the discussion here. And now, using the fact that these coefficients C were chosen to be orthonormal, and using this formula, we can write the two-point function for these OIs. So this is all zero-capting. I haven't used the half-bps nature of these operators yet, so we can write OI, OI prime. Let me just not mess up the normalization. So this is at x. This is at y. So there will be lambda to the k, where lambda is the fifth coupling, as usual. So we have lambda to the power k times k over 2 pi 2k. And then x minus y to the power 2k. And then delta I1, I2. Oh, sorry, I and I prime. So this is the two-point function of these operators. And there's nothing interesting about it. It's just like the free-fil theory answer. And more interest. Then people went on, and they computed the three-point function. And that's going to be the main subject of our discussion. So we have I1, I2, I3. So the three-point function is where the physics lies. The two-point function can be viewed just as a normalization of the operators. It gives the dimension. Right. So the two-point function tells you the three-level dimension. We'll discuss the quantum corrections today. And the three-point function has a little bit of a miracle to it that I'm going to try to explain. So first of all, we have to discuss when does this three-point function vanish or not vanish. OK? So the picture in a three-level is quite clear. We have an operator O, which has K lines coming out of it. Let's say K1 lines from the operator OI1. And then there is another operator with K2 lines coming out and another operator with K3 lines coming out. We're only interested in the planar limit. OK? We're only interested in the planar limit. So we have to draw a diagram that doesn't have self-intersections. So how can such a diagram look like? So the idea is that you can connect some lines here, then some lines here, and then the remaining lines connect here. So that's how these diagrams are going to typically look like, OK, so that there are no self-intersections. So there are several observations you can make. So the first observation is that if one of those Ks is bigger than the sum of the two other Ks, there cannot be any diagrams. So we need that K, let's say 1, is smaller than K2 plus K3. K2 is smaller than K1 plus K3. And K3 is smaller than K1 plus K2. If any of these conditions is violated, then there cannot be any diagrams at three-level. And then it's useful to define, if any of these conditions, yeah, well, yes, if any of these conditions is violated, there are no diagrams. So no diagrams if not satisfied. Thank you. So no diagrams otherwise. Otherwise. Now, so it's useful to define alpha, which is, let's say, alpha 3 is going to be defined as K1 plus K2 minus K3 over 2, and so on. We'll define these combinations, which we'll call alpha 3, alpha 2, alpha 1. So you can just check as a homework exercise that at three-level, if I did not make a mistake, you find this answer. I'll define sigma in a second. So we have K1, K2, K3. And then there is a certain contraction of the tensor structures. This is just group theory. This is an SO6 invariant that you form out of the Cs by contracting the indices of the C tensors in an analogous fashion, where alpha 1 some indices from here contract with some indices here, and then the other ones with those, and the remaining ones come back here. So it's like a planar contraction of the SO6 indices. And then this is divided by x minus y, 2 alpha 3, x minus, let's say z to alpha 2, and then y minus z to alpha 1. So OK, so let's make a few comments about it. First of all, the space-time dependence of this two-point function and three-point functions is not surprising. This is the general dependence on the coordinates of any three-point function of primary operators in any conformal field theory. I was told that you know what primary operators are in a conformal field theory. So this dependence on x, y, and z is not surprising. And now there are these three factors. So what has been observed in 98 is the following. And that's where it becomes a little bit more mysterious. So far, we just discussed a free field theory. So there was nothing mysterious about it. So let me just tell you some observations from around 1998. 2000, about these three-point functions and two-point functions. Oh, I forgot to define sigma. Sigma is the sum of the k's. So sigma is k1 plus k2 plus k3. Thank you for reminding me. OK, so now there were some observations that were really miraculous at the time. So observation number one, which is, in some sense, a trivial observation, is that since these are half BPS operators, their dimension is not pre-normalized since it's related to some r-charge. So the dimension of OI is just k, where k is the number of free fields we used in the construction of that operator. So this is just because it's a half BPS. And it's a coupling independent, half BPS. And this is coupling constant independent. So in particular, the power here and the powers here are not going to depend on the coupling constant because they follow from just a conformal invariance. The more interesting observation, which in fact was not proved in the original papers, was that you could take this expression, which you got from a free field diagram, and compare it to ADS CFT. ADS CFT is infinite lambda. So the ADS dual of n equals 4 allows you to do calculations at infinite coupling rather than at zero coupling. And people have just observed numerically, so to speak, or empirically, more precisely, that it's exactly the same. So it seems that the time, therefore, the three-point functions are independent of the coupling constant. So you could just compute them at zero coupling and infer some properties of the infinitely strongly coupled theory, which has an ADS dual. And then people have done some very impressive checks of the ADS CFT correspondence where the same pre-factors can be obtained in the ADS dual from three-point couplings in the bulk. So there are these written diagrams, which correspond to some fields in the bulk that correspond to these primary operators. And the three-point couplings in supergravity exactly reproduce these pre-factors, even though they were computed in completely different regimes. And, a priori, it was not justified. So that's comment number one. And then there is another property of these operators, which is very nice. These are not extremely. Now I'll define what is extremely. Are there any questions about that? So in the original papers, nobody gave a proof that this is supposed to be independent of the coupling, but the proofs using harmonic superspace and various rather advanced techniques appeared later. So we now know that this is true, that this particular three-point functions can be computed exactly at three level. In fact, not just in the planar limit. This is true for any n. And that's why these comparisons were retrospectively justified. So the ADS stuff was basically saying that loop corrections are always around? Yes. So on the filter side, it means that the loop corrections in the filter, like these kind of diagrams, would have to vanish. Miraculously, they all have to vanish. And on the ADS side, it means that loop corrections in the bulk have to vanish. So these corrections have to vanish. So both have to vanish exactly for this class of observables. OK. Yeah? Are there simpler proofs? No. So the point of this lecture is that this is about maximally supersymmetric Young-Mills theory. I haven't yet defined what extremal correlators are. These are non-extremal in general. That's how the terminology goes. I'll define now a subset of observables which are called extremal. I'll show you that they obey even slightly more restricted properties. And then we'll discuss analogues of those constructions with less supersymmetry. But there is no simple proof of this fact. I mean, even now, if you were to come up with a proof, you would not have an easy time. It's extremely non-trivial, in my opinion. Notice that there are half BPS, but they are not necessarily half BPS with respect to the same Q. So this preserves some other half of supersymmetry, but they are not necessarily the same half. You see what I'm saying? So it's not an obvious statement. Also, if you're familiar with n equals 1 supersymmetry, you know that when you have a half BPS operators, the correlation functions are distance independent. Well, here, they are definitely distance dependent. So the non-normalization properties, like in some sense of the pre-factor, not of the distance dependent. There is a non-trivial distance dependence. It's just that the pre-factor doesn't run. So it's a non-trivial fact that even nowadays, there is no easy proof as far as I know. Now, you could ask, what about endpoint functions? Those are two and three-point functions, but what about endpoint functions? So as you know from conformal field theory, the position dependence of three-point functions is completely fixed by conformal invariance. So the fact that we knew the position dependence here exactly is not surprising. Endpoint functions, however, have non-trivial cross ratios in general. So the question is not about just the normalization of the four-point function, but also do we actually know the position dependence? So here, this is where extremal comes in. This is where the notion of extremality comes in. So it's useful to label the operators instead of by this indices i1 to in. I'll just label them by delta 1 to delta n. It's just more convenient when we, that will be more convenient when we discuss n equals 2. So delta is the dimension. Delta i is just Ki. So it's the number of fields we used to define that single trace operator. So that labeling is just more convenient. In fact, so what is extremal? Extremal correlator is a special case of this family of correlators where there is like a delta which is the sum of the other deltas. So it's exactly the case where one of those inequalities is saturated. So for a three-point function, that would mean that let's say delta 3 is delta 2 plus delta 1. That would mean that the diagrams are in fact a little bit simpler. They would look like this. So there won't be any constructions at real level between two operators, but there will be constructions going from some mother operator to the daughters. But there won't be any constructions between the two light operators. So this is the definition of extremal, and that definition will carry over to less of a symmetry. Let's say I'll just change the notation a little bit. So we'll just add another operator with some big delta and put it at y. So delta is the sum of delta i. So it's a special case of this general family of correlation functions, and it's an interesting notion because that notion carries over to less of a symmetry, naturally. So if the operator is extremal, we can make a definite statement about the position dependence of general n plus one point functions. Let me make that statement. That also can be contrasted with the ADS-CFT correspondence. Sorry, let's see if there is anything here This definition is reserved for four dimensions, otherwise you would add the dimension of the field on that definition? No, no, no. That definition, that one scaling dimension is the sum of all the rest is uniform in dimension. But the delta i could scale. Oh, that one is just the relation between the dimension of the operator and the number of fields. This is special to four dimensions, but the notion of extremality is this. This is the notion of extremality, and this is more general than just four dimensions. Is the point of this that you can somehow reduce it to products of two point functions? In some way, yes, yes, in some way. Yeah, this is just a historical discussion. Yeah, we'll see what are the, but in some way, yes, you can reduce it to lower point functions in some sense. So let's write the answer. So we have delta 1, delta n, and then we have o delta y. So the general position dependence of this n plus one point function is known. So there is some prefactor a, which depends on all the dimensions, and on the coupling on n, and on the Young-Mills coupling constant. We'll later argue that in fact it's independent of the Young-Mills coupling constant, but never mind that. So then there is a product of one over y minus x i to the power 2 delta i, and this product runs from one to n. In fact, this is independent of the coupling constant up to some trivial prefactor like we had here. Lambda to some power. So there is some lambda to some power, which is given basically by the sum of dimensions. Then there is some function of the deltas nn, which is analogous to this thing, and then there is some position dependence which is fixed. So the position dependence is completely fixed, and the coupling dependence is trivial. So the position dependence is fixed, and coupling dependence is trivial. So these are the extremal correlators. That's what's special about extremal, the descent point functions take this very simple form. If this condition is not obeyed, then this answer is not correct. So it's only for extremal correlators that this is the right answer for endpoint functions. This was tested against the ADS-CFT correspondence, and it led to very interesting non-trivial checks of the ADS-CFT correspondence. So if you want to read a little bit more about this subject, in the context of the ADS-CFT correspondence, and in the context of just filtering, the best source is the TASC lectures of, one of the best sources is the TASC lectures, the whole care, so you can read about the construction of this half-PPS operators and the extremal correlators, and you can see there is a whole discussion there of how this expression matches the ADS-CFT written diagrams. So there is no singularity in two of the X's touch? Exactly. So there is no singularity when two of the X's touch. So this was believed to be a special property of just the maximally supersymmetric Young-Mills theory. But now we know that let me just now what I want to do is to give you a broad picture of what is true for n equals 2. So for n equals 2, not everything that I've said is true, but some things are still true and they're very interesting. That's the main subject of the talk. So let me just tell you what is true with a little bit less supersymmetry. So this is just an n equals 2 overview. So the first thing that is true, which I'll argue soon, is that we can define an analogous sector of operators, which are called chiral ring operators. That's how I'm going to call them, and we can define extremal correlators. So you can define extremal correlators, you can define extremal correlators of chiral ring operators. This is some of chiral ring operators. It turns out to be still true that the endpoint functions of these operators obey this property. So the dependence is just this funny function, 1 over 1 y minus x to the power 2 delta i. So that still remains true. The thing that is different from n equals 4, which makes it more interesting, is that the pre-factor is now depends on the young mills coupling. So the pre-factor is a function of the dimensions of n and of the young mills coupling. Unlike in n equals 4, this is a non-trivial function now. So there are corrections in perturbation theory, and also due to instantons. They do not change the dependence in position, but they change the pre-factor, which is physical. Yeah. Ask about the physical intuition, because let's say I didn't have this operator delta, then when I had two operators nearby, the object exploded. Then I insert this operator somewhere in Andromeda. Yes. Suddenly it's impure. All right. Yes. How do you see this happening? So, I mean, you can take two of these operators, let's say at x1 and x2. You can bring them close to each other, and you have some heavy operator in Andromeda. But this heavy operator is hungry. It needs to soak up all the lines. So if the OP is when you take some of the lines and you contract them between each other, that's how you get singularities in filtering. But if you contract even one of the lines, then this operator would not be well-fed. So it's like saying that if you pick up the singular terms in the OP, what remains has zero overlap with O. So it's just that the form factor vanishes. So there is an OP between these two operators, but the only term that contributes in that OP is the regular piece. The singular pieces turn out not to contribute. They just have zero overlap exactly with that guy. But is this the statement only in margin? No, no. So this statement is true for any n. It turns out. So this statement is true for any n. And so the most interesting thing about the n equals 2 counterpart of this story is that the function actually depends on the Young-Mills coupling. And not only that, it turns out that we can compute it exactly. So it's not just that we know that there is such a sector and it depends on the coupling. The coupling dependence can be computed exactly. So this is exactly computable in any n equals 2 theory. That's the first interesting thing about it. But furthermore, you might remember that in gage theories that there is also a theta angle. And it's actually natural to trade a coupling constant for 1 over g Young-Mills plus i theta. So you have tau and tau bar. You might remember the work of Cyberg-Witton also uncovered some interesting dependence on the coupling constant. But it was purely holomorphic. So the interesting thing about this exactly computable observables is that they are non-holomorphic. So this goes beyond all the tools that were available in the 90s where people knew how to extract holomorphic quantities. So this is non-holomorphic. And we can still compute it. Even though it's non-holomorphic, we can still compute it using modern techniques. So this goes beyond this has, I mean, I'll talk about the physical meaning of these pre-factors. You'll see that they can be, in some special cases, used to measure the metric interior space and there are various other connections to mathematics. But basically, it goes beyond the holomorphic data in the Cyberg-Witton curve. And in the special case of n equals 4, you will see explicitly that this just drops out. So it's easy to see that in n equals 4, we recover the old results. OK, so are there any questions about this quick historical overview? So this totally cannot be done with the product that I'm publishing, the one that you wrote. I'll talk about it now. Are there any questions about the historical overview? They just try to put the subject into some slightly broader context. It will be clear why we are discussing that. OK, so now I want to discuss the question that already was asked a few times about in what sense we can reduce these two products. So if the theory is confining, then it's not a parameter. So this is kind of void of content. But here, I should have said that this is about n equals 2 super conformal field theories, where the coupling is an actual parameter. So there is a vast class of such theories, in particular this class s theory and Cyberg-Witton theories. And they all have some coupling constants, which are exactly marginal. Any other questions? OK, so let's talk a little bit more about what it means for the endpoint function to take that particular form. So we won't care about the normalization now. I just want to study this particular dependence on the coordinates just to try to understand what it means, following aliases and things, questions. So let me remind you of something something rather basic in conformal filtering, so just to try to understand what it means. So let's suppose you have two primary operators, O1 and O2, X and Y. And suppose they're primary operators, and suppose they have some dimension delta. So it's known that this takes the form 1 over X minus Y to the power 2 delta, right? Now suppose you have a three point function, O1, O2, O3 of some operators of dimension delta 1, delta 2, delta 3. And here I'm not restricting the dimensions to obey this sum rule as for extremal correlators. Sorry, this is Z, Y, X. So this is going to take the form, I'm dropping the normalization, which is not important now. This is going to take the form X minus Y to some power, then X minus Z to some power, and Y minus Z to some power. And now I have to figure out the powers, unfortunately. So this is delta 1 plus delta 2 minus delta 3. This is delta 1 plus delta 3 minus delta 2. This is delta 2 plus delta 3 minus delta 1. Is that correct? Seems correct. OK, so again, yeah, if I put X minus Y squared, then maybe I need to divide by 2. But in this notation, probably not. OK, so following this kind of, so this is generally true for any two primary operators, there is a very interesting observation that is known from two dimensions. But it's also very useful in higher dimensions. So using the conformal group, we can formally put any operator. So any operator can formally put at infinity. So any operator can be formally put at infinity using a conformal transformation. But you have to somehow do it carefully. You don't want to destroy these formulas. What does it mean if Y goes to infinity? It just goes to 0, and that's it. So when you put an operator at infinity using the conformal transformation, you have to do it in a careful manner. And the prescription is the following. So an operator at infinity is defined to be the following its Y to the power 2 delta of that operator times the operator at Y. And now we send Y to infinity in a limiting process. So when we talk about operators at infinity, that's what we mean. That's the limiting procedure that we're doing. And now it turns out that we can just take these formulas now and put one of the points at infinity and write down the answers. So for this formula, this will just become, let's say, 0 infinity. Following this prescription, this is just 1, right? So we didn't lose any information. It's just like 1, the pre-factor. And also this turns out to just be 1. So let's look at the powers of Z. So Z appears here with power delta 1 plus delta 3 minus delta 2. And here it appears with power delta 2 plus delta 3 minus delta 1. So in total, the power of Z in the denominator is 2 times delta 3. So if we multiply it by 0 times delta 3, these things go away. And what remains is just so what? OK, there isn't enough space here. I'll just use that blackboard. So the 3-point function 0 1, 0 2, let's say x, y infinity, this 3-point function, where this is of dimension delta 1, this is of dimension delta 2, this is of dimension delta 3, this just becomes x minus y to the power delta 1 plus delta 2 minus delta 3. And the statement more generally is that all the correlation functions in conformal filters have behaved nicely in this limit. So all correlation functions behave nicely. So all correlation functions behave nicely. So using these variables, it's actually very useful to rewrite this answer in these variables. It's very useful to put the coordinate y at infinity and see what comes out. So using this procedure, this statement that this is true in conformal filter is entirely equivalent. Oh, I should have said that this process of putting a point at infinity is reversible. Because the conformal group allows to put any point at infinity. In principle, if somebody tells you the correlation function where one point is at infinity, you can go back and put all the points at final distance. So this is a reversible process. It's not like you do it once and that's it. You cannot go back. This process is irreversible. So an alternative way to write this correlation function, which adds a little bit more light on the meaning of this correlation function, is to write, is to put the last point at infinity. So we have O delta infinity. So this is one, completely independent of the distance. So that's what it means. This correlation function is secretly independent of the distance. That might bring a bell for those of you who started supersymmetry in flat space with n equals 1, because there are some correlation functions that are independent of the distance. But this is much more subtle. This is only independent of the distance, because one point is at infinity. It does depend on the distance if all the points are generic, but it's independent of the distance when one point is at infinity. So it's a slightly different statement than those that you have encountered in classic books about supersymmetry, but it's true. So what does it mean? It means that since it's independent of the distance, we can just OPE these guys, which you can just OPE these guys until everything dies out. Only the regular pieces can survive. So let's write it down concretely. So the consequence of this is now clear. So consequently, the operators at the x i at the points at x i can be brought near each other. So you can just pile them all together in a small section of space, O1, O2, On, and just OPE them. So you OPE them all, and you get some new operator O quiddle, which sits at like 0. So these correlation functions are really, in some sense, correlation numbers, but not correlation functions, because the functional dependence is kind of a myth. There isn't really functional dependence here on the coordinates in some sense. So what you want to compute are just these numbers. And these numbers come from the relations of multiplying these operators and extracting the regular pieces. So only the regular pieces matter. So it's not like you can straightforwardly compute these correlation functions. You still need to know how to multiply these operators. And it might depend on the coupling constant, as I said. So in principle, this will be a function of g and mils and stuff. It's not trivial, but it's simpler than what it could have been. OK, so now n equals 2. Are there any more questions about the setup? OK, so now n equals 2. So first, we have to identify the operators in n equals 2. Conformal filters that may enjoy these interesting properties. So I'll just identify that sector on this blackboard. And then we'll try to analyze it. So we're talking about n equals 2 in four dimensions. And we're talking about super conformal filtering. There hasn't been much work about it. There hasn't been a lot of work on what happens if the theory is not conformal. And then the questions that you might want to ask are a little bit different. People have not quite pursued that. But for conformal filters, there's been quite a bit of work. So we have supercharges, which are q alpha i and q bar alpha i, alpha dot i. These are the ordinary supercharges. And then there are the super conformal supercharges of n equals 2. And as usual, i runs from 1 to 2. This is dn equals 2. And alpha is a Lorentz index, which likewise runs over the indices 1 and 2. The theory has conformal symmetry, which is s of 5 comma 1 in Euclidean signature. And in addition, it has an r symmetry. There is an r symmetry in such super conformal filters, which is su2r times u1r. So local operators are labeled by some highest weight representation of this algebra. You must have seen this algebra before. I'm not going to review it. So what are the primaries? The super conformal primaries, these are the highest weight states of this algebra. And they are annihilated by s and by s bar. Because s and s bar should be interpreted as raising operators. So the highest weight states or the super conformal primaries are annihilated by these guys. And the quantum numbers of these primaries are, so the quantum numbers of such primaries are first the dimension of the operator. Then there is the spin. The spin in four dimensions is labeled, the spin of a local operator is labeled by two su2 quantum numbers, j left and j right. Because the Lorentz symmetry in Euclidean space is s of 4, which is su2 times su2. And so it's labeled by two su2 quantum numbers. Like all the operators are labeled in this way. And then there are the quantum numbers under the su2 times u1r symmetry. So the spin under the su2r symmetry is s. And the u1r charge is going to be labeled by capital R. So this is the su2 quantum number. And this is the u1 quantum number. So all the highest weight states are labeled by these five quantum numbers. All the local operators are labeled by these guys. So of course, in principle, we're interested in all the operators of the conformal filter. And this is what the bootstrap program is about. But then there is a huge amount of work on various special cases of these highest weight operators that obey various additional conditions. So there is a huge amount of work on special subsets or special subsets, special classes of such operators. So the super conformal index, the SURE operators, the chiral algebra, the chiral ring, all of those things are just names for various subclasses of these operators, which obey additional properties. So here I want to remind you what is known about the half-BPS operators of n equals 2 super conformal filters. So I'll just give you a quick reminder of what is known about half-BPS operators. So I'll actually start with those that I'm going to be eventually interested in. So by half-BPS, what we mean is that, OK, so highest weight operators are already annihilated by all the s's. So by half-BPS, I mean that they're also annihilated by a half of these q's. So in total, we have eight q's here and eight s's. So half-BPS means that they are annihilated by a half of the eight q's, meaning four q's. So there are two natural ways to do it. One is that it's annihilated by all the q bars. So this is an additional restriction we can impose on the operator the highest weight operator O. And with this additional restriction, it will obey further conditions on its quantum numbers. And in fact, such operators are called chiral operators, or chiral ring operators. Sometimes chiral ring operators. Sometimes people call them Coulomb branch operators. No, it looks misspelled, right? It's fine, yeah. Coulomb branch operators. So that's another name. So these are all names for this class of operators, chiral operators, chiral ring operators, Coulomb branch operators. These are all the names representing the same thing. And in fact, those operators are going to be the main focus of these lectures. So what are the properties of these operators? You don't get chiral Higgs branch operators? No, I'll talk about Higgs branch operators soon. They're not of this type. So I'll talk about all the half BPS operators of both types. So first I'll talk about these operators. And then I'll talk about what you call the Higgs branch operators. OK, so from the n equals 2 super conformal algebra, the n equals 2 super conformal algebra, very, very schematically, is that if you take q and s and you commute them, let's say q bar and s bar, let's say that you anticommute q bar and s bar, you get a bunch of epsilon tensors, delta plus r with some coefficients. This is very, very schematically. Plus Lorentz generators and plus the scaling dimension. Well, sorry, I meant the r symmetry. The scaling dimension is already here. So very, very schematically, that's the n equals 2 algebra. If you do the commutator of q and s, you get the scaling dimension, some r symmetry, and this is the s u 2 r symmetry. This is the u 1 r symmetry. So that's the very schematic structure of the n equals 2 super conformal algebra. So you could just look it up. Let's say this is written very carefully in the paper of Minwala et al on the super conformal index. So they have all the conventions and all the factors. So you could reproduce what I'm telling you by looking at their appendix. So if you just impose that O is the highest weight state that's annihilated by s bar, but also by q bar, also annihilated by q bar, then this imposes various restrictions on the possible u 1 r symmetry, quantum numbers, the dimensions, the Lorentz quantum numbers, and the s u 2 quantum numbers. So these conditions impose various restrictions. And what it turns out is that it imposes the following condition, that j r has to vanish. So this is a general property of any half BPS operator, which is a Coulomb branch operator or a chiral ring operator, j r has to vanish, s has to vanish. Remember that s was the s u 2 r quantum number. And in addition, delta is r over 2 in absolute value. Or r is the u 1 quantum number. So the dimension is fixed in terms of the r symmetry quantum number. So u 1 r quantum number of that operator fixes its scaling dimension exactly. So this cannot receive quantum corrections. It follows from kinematics from the super conformal algebra. And also, one of the spins inside SO4 has to vanish. Now, there is an interesting conjecture here that restrictions on the j field. Exactly. So people have been looking at that for ages, for 10 years now. And it's striking that there is no restriction on the left-moving spin of that operator. This is especially puzzling given the fact that we have tons of Lagrangian constructions of such super conformal filters using cyberguiden theory and Quiver gauge theories. There are tons of Lagrangian constructions. So we have tons of Lagrangian constructions of such n equals to super conformal filters. And in all of them, all the Coulomb-Brentch operators satisfy, in addition, this restriction in all of them. In all the known Lagrangian theories, this is true. So a natural conjecture is perhaps this is always true, but nobody has been able to prove it. This is an open question. So there is an open question of whether there exists any n equals to super conformal filter out there with a Coulomb-Brentch operator that is spinning. That's an open question. For these lectures, I'll assume that they don't exist. I'll just make this assumption. I do not know. I'm not an expert on the non-Lagrangian constructions, but I don't want to quote Med Buick and incorrectly, but Med Buick and told me that there are no examples. In fact, he has a paper saying that there are no examples, I think. I believe this is a reasonable conjecture to make that there are no such operators, as far as I know. Now, in Lagrangian theories, it's very easy to prove, by the way. In Lagrangian theories, it's really like one hour of thinking, and you can prove it. It's not a hard question. Assuming that you know how to write down all the operators and stuff like that. So these are some basic facts about Coulomb-Brentch operators. Let's continue discussing Coulomb-Brentch operators. So the interesting thing, which you will resonate with what we said before, is this equation. I have not been very careful about what this absolute value means. So for the half BPS operators that obey this equation, but you could also envision half BPS operators obeying the other equation, where you replace q bar by q, those will be called anti-kiral ring operators, or anti-Coulomb-Brentch operators. So those are anti-kiral operators. Now, you could try to ask what happens if you impose that O is both kiral and anti-kiral? That would mean that it must be the unit operator. So it has to be either a dead or a none, but it cannot be both unless it's the unit operator. So the meaning of this formula is that deltas are over 2 for kiral ring operators, or Coulomb-Brentch operators, and it's minus r over 2 for anti-kiral. And that leads to some interesting consequences for the OP of two kiral operators. So this should be possible to expand in terms of the other operators in the theory, let's say x, with various, OK. So let me just write it down more carefully. So it should be possible to expand that product in terms of k, i, j, x, y. It should be possible to write down such an expansion, but this would not be kiral, not necessarily kiral. These are just like all the operators in the theory at first sight. OK. So the fact that these two operators are kiral does lead to some restrictions on this OP, which I would like to mention now. So the total, let's say this is o1, this is i, and this is j. Sorry, this is oi, this is oj. Let's say that the r charge of this guy is ri, and the r charge of that guy is rj. So the r charge of the right-hand side must be ri plus rj. So the r charge of the right-hand side must be ri plus rj. So the only operators that could appear on the right-hand side are operators whose r charge is ri plus rj, and that they have no s2r charge. So there must be neutral under the s2r charge. And because there is a general inequality, so you can prove a general inequality for any operator in the theory, this is always true. This is a unitarity bound. And this unitarity bound is saturated by chiral and anti-chiral operators. So you can prove such an inequality. It means that there cannot be singular terms here. So the only operators that can appear on the right-hand side have dimension at least bigger than the sum of the dimensions of the left-hand side. So it's like a triangle inequality of some sort. And therefore, there cannot be singular terms. So no singular terms. And that may resonate with what we discussed a little bit before about this endpoint functions that they have no singularities. So in the OP of two chiral operators, there cannot be singular terms. And in fact, the term that is non-singular, so the term that is independent of x and y, is a chiral ring operator, since its dimension must be exactly ri plus rj. So the non-singular, there's order 1, the order 1 term, meaning the term that doesn't depend on x minus y. So the order 1 term is a chiral ring operator, is another chiral ring operator, is another chiral ring operator, so we can write, loosely speaking, without any x dependence now, plus non-chiral ring. That would be enough for our purposes. And these are chiral ring operators. And there is no x dependence here, absolutely no x dependence. So this could be xy, but this is independent of x minus y. So you can put this at x, this at y, this at x. So the chiral ring operators form a ring. That's where the name comes from. You multiply two, you get the third one with some coefficients. And these coefficients may depend on the coupling constants. And as you will see, the computation of extremal correlation functions is equivalent to computing these OP coefficients as a function of the coupling constant. Why is that? You remember that we argued that the endpoint functions can be computed by just OPing all the operators. So if you knew all these coefficients, you could just reduce any end product of operators to a sum of our single operators, and then you would be done. The two-point function would be the only thing that remains. So these are coupling constant dependent. So I haven't proved that yet, but we will prove that those don't make a contribution. So if you picked up any contribution in the OP from the non-kiral ring operators, it would be exactly orthogonal to the extremal correlators. So you'll just get 0. So these coefficients are coupling constant dependence, and they are the main task to computing these coefficients as an exact function of the coupling constants. And as I said, it turns out to be a non-holomorphic function of the coupling constants, unlike the holomorphic data in the cyberwitten curve. So this cannot be extracted from the cyberwitten curve in any way. So this is one comment about these operators, about the Kiral ring operators. Another thing about the Kiral ring operators, which is what made me interested in the subject in the first place, is that there is a special case where the R charge is 4. So if the R charge is 4, these operators are quite interesting. If the R charge of these operators is 4, then the dimension is 2. And therefore, we can add them to the action. So this is the superspace integral. So to speak, the n equals 2 superpotential. And we can just add these operators. Oh. So this is a supersymmetric deformation. And it's exactly marginal, because the dimension of this guy is 2, and the dimension of this guy is 2. So therefore, this is the deformation of the Lagrangian. So therefore, the Lagrangian is deformed by a dimension 4 operator. And this is why it's exactly. I haven't proven that it's exactly marginal. From this, it follows only that it's perhaps marginal, but it turns out to be exactly marginal. That I'm not going to prove. So in particular, if we can compute correlation functions of OO dagger, for instance, that would be the zomologic of metric. Because these are exactly marginal deformation of the theory, and the two-point functions measure distances between tiers. I'll explain this in much more detail in the upcoming lectures. So this measures distances in theory space. So this is like a special case of an extremal correlator that measures distances in theory space. And this is a very canonical observable in any family of conformal filters. And the fact that we can compute it exactly is very, very nice. So that's an additional interesting thing about extremal correlators in n equals 2 and the chiral ring operators. OK, I want to now mention one last conjecture about chiral ring operators, and then I'll talk about Higgs branch operators, which is the other family of a half BPS operators, which has some own interesting properties, but it's far less interesting in Lagrangian tiers. It turns out. So I want to mention one last property. Here, before I move on to the other half BPS operators of n equals 2, the last property is also a conjecture. Well, it's a conjecture that may be false, but it's like a false conjecture in some. I'll explain what I mean. So in all the Lagrangian theories, in all Lagrangian SCFTs, in all the Lagrangian SCFTs, this is true about the spin. Remember, I told you about this open question about the spin, but there is another property that is true in all the Lagrangian SCFTs, which is that the ring is freely generated. So there is really a ring structure here that you multiply operators. There is no singularity in the OP, and you get some new operator, which is in the chiral ring. So the chiral ring is closed under multiplication. So it's a real ring in the mathematical sense. So it turns out that in all the known Lagrangian constructions, the ring is freely generated. Freely generated. What is a freely generated ring? It would be important for what we discussed on. What is a freely generated ring? A freely generated finitely, finitely and freely generated ring. Sorry, wait, is it finitely? No, I'm suspecting that this might not be always true. A freely generated ring is a ring that has some elements, let's say, a i, a k, let's say, a l, such that any other ring element is given in a unique fashion by products of a 1, a 2, a l with some powers. And there is no double counting, so there is no redundancy. Any word that you write using these letters is like a new element in the ring. And products of words obey the intuitive multiplication law, where you add the powers as appropriate, and that's it. So a freely generated ring is essentially a ring where there are no relations. A ring that could have relation is like a 1, a 2, is the same as a 3, a 4. That would be a ring with a relation. But a freely generated ring is something where every word counts differently. So every word is different from any other word. So all different. So these are all different. So any operator can be generated by some term like that, sums of terms like that, and every such term is unique and independent of the others. So that's what freely generated means. And OK, so since it's true for all the Lagrangian CFTs, the chiral ring is freely generated, people were compelled to make this conjecture more generally, but then there are recent claims that there are some non Lagrangian theories that violate it. But these are not really true violations of this conjecture. If you look in the details, then the ring there is not freely generated for like stupid reasons, because you projected out something. So it's not a genuine violation of this conjecture. So what do you mean by that? So there are these constructions where you? We have a paper on the super-conformal angle. Right. And you see that it's not freely generated. Right. But when the ring is not freely generated, it could be that it's genuinely not freely generated, or that perhaps you could obtain the theory from gauging some discrete symmetry in a theory where it is freely generated. So as far as I know, all the constructions where it's not freely generated are obtained by just gauging some freely generated ring. Right? Now, Argyris has also claimed that he has the construction of some models where it's not like that. But these constructions are still, the verdict is still out. It's not yet out. Yeah. The verdict is not yet out. So it could be that this conjecture is still true. In any case, it's true in all the Lagrangian theories. OK, so this is what I wanted to say about this sector of half-BPS operators. And now I just want for the completeness of your, so to speak, education. What is the rate for the conjecture that the conformal magnitude of the conmentality of the theory is a coupling conjecture? This is not a conjecture. This is a trivial. I mean, what we can say is the following. So here, we describe some exactly marginal deformations. OK, you can ask, could there be also other exactly marginal deformations? And the answer is no. This is not a conjecture. So these are all the exactly marginal deformations must be integrals over chiral ring operators. Is that what you meant? I mean coupling constant. Gauge coupling, gauge constant. Well, I don't think that this. How do we have the boundary there? OK, you could say that in all the known examples, these exactly marginal deformations can be viewed as gauge coupling constants of some gauge groups. So yeah, we could add that. So all the exactly marginal operators must be of this form. This is easy to show. And yeah, and also we can say that in all the Lagrangian examples, these exactly marginal operators can be viewed as the Young-Mills coupling constants and theta angles for some gauge groups. For some simply connected, sorry, simple gauge groups. Same simple gauge groups and u1 factors and so on. OK, so these exactly marginal parameters are always just Young-Mills coupling constants and theta angles. OK, now I want to finish this last 10 minutes. I want to tell you about the other half-BPS operators so that you have a broader vision, a broader view of the literature, because there is a huge amount of literature on the chiral ring, the Coulomb branch operator, and then there is an ever-increasing literature on the other half-BPS sector of the theory, which is the Higgs branch. So the Higgs branch operators are defined by a different condition. So our operators were annihilated by 4 out of 8. Remember, so we have these 8 guys. And the chiral ring is defined by all those operators that are annihilated by those 4 out of 8. But the Higgs branch is defined by taking 2 from here and 2 from here. OK, so it's a different subalgebra. So the Higgs branch operators are defined by the condition that they are annihilated by q alpha 1. So I'll call these operators H and by q bar 1 alpha dot. So this is another half-BPS sector of the theory. And this also has many names. Sometimes people call it the Higgs branch, which is analogous to that Coulomb branch. So this is like the Higgs branch sector. And sometimes I think more is mostly, well, sometimes people call it the Shuri sector, but this is not entirely precise terminology. So let's just leave it at that. Higgs branch sector. So you can follow a similar logic and find the quantum numbers of these operators. And here, what one finds is that j left equals j right equals r equals 0. So unlike the operators that we've described before, which had non-vanishing u and r, r charge, these operators have vanishing u and r charge, but they have non-zero dimension and non-zero SU2, r symmetry spin. And the relation is that delta is equal to 2s. S is the casimir of the SU2 representation. So the dimension of these operators is 2 times the casimir of the SU2 representation. So this is another sector of operators. And they also have interesting correlation functions. But what turns out is that the OP of these operators is also interesting. So there is some kind of analogous structure to this. But here, there is coupling constant dependence. So these products, these ring coefficients have perturbative as well as non-perturbative corrections. But in the Higgs branch, it turns out that the coefficients c, i, j, k for the Higgs branch, those turn out to be three-level exact. So there are no quantum corrections. And at least for Lagrangian theories, this is therefore very boring. But for non-Lagrangian theories, these are still interesting observables. So this is interesting, especially in non-Lagrangian theories. But in Lagrangian theories, there isn't much to discuss. This can be computed with coupling, and that's it. So there is nothing to discuss. So this is the other sector about which there is a lot of work, but mostly about non-Lagrangian theories. While the chiral ring, the Coulomb branch operators are interesting even in perturbation theory, non-trivial even in perturbation theory. So this is a. It's interesting that this Higgs branch is, I think, very interesting. Say again? Say again? I don't know. How to say it's trivial. How to say it's trivial? How do you see it there? Oh, how do you see that this is the independent of the coupling constant? That's a very good question. So the answer is actually intuitively the following. So the coupling constant dependence comes from the exactly marginal deformations. These are our coupling constants. So we have a space of theories, which are parameterized by the Young Mills couplings and the theta angles. And changing the Young Mills coupling is like adding a deformation. But as I told you, all the exactly marginal operators, they are Coulomb branch operators. So vaguely speaking, it is true that the Coulomb branch operators and the Higgs branch operators don't talk to each other. So the data of the Higgs branch operators does not care about the deformations in the Coulomb branch sector. And it just so happens that the Young Mills coupling constant is a Coulomb branch parameter, not the Higgs branch parameter. So that's why, roughly speaking, the Higgs branch operators do not care about the Young Mills coupling. It's just that in any costaure, the Young Mills coupling is a Coulomb branch parameter. It's the coefficient of a Coulomb branch deformation, not the Higgs branch deformation. OK, so this is, roughly speaking, the intuitive reason. If you know a little bit about two dimensions, you can make it much more precise, because there are chirals and twisted chirals. And it's kind of analogous to this statement. Now, physically, these two half-BPS sectors have another important interpretation. That will be the last thing I want to say today. So typically, when we study supersymmetric field theories, especially those n-equals-tool theories that we have in mind, typically we have a super conformal field theory, which is some interacting super conformal field theory. And this theory has some coupling constants, Young Mills and theta. But in addition, it has a vacuum manifold. So there are non-trivial vacuums, like the generate supers selection sectors. So we must not be confused about the space of theories, which are parameterized by the Young Mills couplings and the theta angles, and the space of vacuums, which are just different super selection sectors in infinite volume. So for example, in n-equals-4, we have a huge modular space of vacuums, which is like R6 mod gamma. So these are a vacua, not the formations of the theory. In any case, what I wanted to say is that typically, in n-equals-tool theories, if you look at the space of vacuums, there are vacuums on which there is the U1-R symmetry, is spontaneously broken. And then there is an orthogonal space, so to speak. Orthogonal space, where SU2-R is spontaneously broken. So this vacua have a Goulson bosons. So the Higgs branch operators, their expectation values in the new super selection sectors parameterized this space, and the Coulomb branch operators parameterized this space. So typically, the vacua manifold of n-equals-2 theories look like Higgs branches and Coulomb branches, and they're essentially orthogonal. Essentially orthogonal. They're also mixed branches, but they're essentially orthogonal. And so the vacuum expectation values of Coulomb branch operators and Higgs branch operators parameterized those spaces. So you can think about Coulomb branch operators and Higgs branch operators. So these are Coulomb branch and Higgs branch as order parameters for the spontaneous breaking of SU2 and U1. So this is a U1 order parameter, and this is SU2 order parameter. So the physical meaning of these operators is that they parameterized these spaces. And then you can ask, what is then the physical meaning of these coefficients? Well, these coefficients, roughly speaking, captured the geometric structure of these spaces, but their coupling constant dependence is more subtle. In some cases, we know exactly what that coupling constant dependence means. We'll discuss some of those cases. But in general, I don't know geometrically what it means. But there are some cases in which we'll discuss this question. So these are the half BPS sectors. And as I already hinted, we'll be interested in those in the Coulomb branch operators. And we'll show that they obey very similar properties to the extremal correlators in n equals 4. And we will compute those coefficients exactly using localization and then study some of their resurgence properties. Are there any questions? Can you go from one branch to the other? Yes, yes, there are various massive deformations of the theories that allow you to, like you have a super conformal field theory, and all these buckwares degenerate. But you could add a mass. And then in some situations, you can think about it as if the mass just forces you to go on one of those branches. Yeah, and you can also start from the Higgs branch and some terms to the action, go to some points on the Coulomb branch. Like the famous story of cyber-guit, so there is cyber-guit in theory. All the action, so to speak, all the interesting stuff happens on the Coulomb branch. The Higgs branch doesn't participate in that story. It's not very normalized. There are no instant tones. And it's all essentially because the Coulomb branch, the Yang-Mills coupling is the Coulomb branch parameter. So the Higgs branch doesn't participate in most of those examples. So all the action is happening on the Coulomb branch diaries. There is a monopole point, a dion point, and there is no conformal filter in the first construction. But then there is the conformal version. So the Coulomb branch is quite complicated, generally. There are lots of singularities, like Argyris-Douglas singularities. What do you mean by super-selection rules? Super-selection rules? Yes, in the back wall. I'm just talking about, like, say, OK, I'll give you an example. Maybe an example would clarify what I mean. I just mean, like, vacuo in infinite volume. But like, in quantum mechanics, if you have such a potential, then how many ground states have you got? One, right? Like the superposition of psi plus and psi minus. Where, like, there is a Gaussian here and it's a Gaussian here. But in quantum filtering, when we draw such a potential, like phi squared plus phi to the 4, how many ground states have you got in quantum filtering? One or two? Two. So not one. Two, OK? So I don't know. Maybe I should give a separate lecture about what is the large volume limit. But more seriously, the instanton action has a factor of volume in it. So as you remember, you can compare these two states. In quantum mechanics, you can compare these two states. And the energy difference is an exponential in minus the instanton action. Now in quantum filtering, it's not quantum mechanics. We don't have just one degree of freedom sitting at one point. We have infinitely many degrees of freedom filling out an infinite volume. And so you can think about it as if there is, like, a quantum filtering in a lattice approximation. So when we write such a potential in quantum filtering, that means that each lattice degree of freedom has such a potential. And so to tunnel from this to that, you don't need just one of the quantum mechanical degrees of freedom to tunnel. You need all the quantum mechanical degrees of freedom to tunnel. So the instanton action is e to the minus the volume, like the total number of lattice sites, times the action of the quantum mechanical instanton tunneling, which is finite. So that kills the exponent in the thermodynamic limit, in the large volume limit. So in quantum filtering, actually, when we draw such a potential, we have two grounds, say it's not one. And this is what's called superselection sectors. That's what people say. When people say superselection sector, that's what they mean. It's an artifact of infinite volume. So this is very typical in supersymmetric filter is that there are many vacua, many superselection sectors. So the potential, the scalar potential in quantum filter is typically, with supersymmetry, is typically flat. So in quantum mechanics, when you have a flat potential, you still have just one ground state. What quantum mechanics with the flat potential has this kind of wave functions, and there is only one ground state, which is the constant wave function. But in quantum filter, if you have a flat potential, you have infinitely many ground states. For the same reason, the volume suppression is not in it. If you want to know more, then there is an exception for two dimensions. In two dimensions, if you have an exactly flat potential, there is some logarithm. And in fact, it's more like quantum mechanics. But above two dimensions, if you have a flat potential, then you have genuine superslection sectors. It's related to why there's no Goldstone bosons. Right. There are no Goldstone bosons in two dimensions, because this picture is too crude. There isn't being that if the potential is exactly flat, in addition to the volume suppression, there is also you have to take into account correlations. So here, I ignored completely correlations, as you've seen. I just assumed that each lattice site has some probability of tundling, and then I just multiplied by the volume. This is a correct approximation if there are no strong correlation functions. But if there are strong correlation functions, like in massless theories in two dimensions, then this volume factor can get erased. And that's what happens in two dimensions. But above two dimensions, there is no such subtlety. Yes? Get curious, how is it possible to compute the OP for the Higgs branch operators in the future? The Higgs branch operators can be written like, if you have a capping constant gmms, then you can just write down all the Higgs branch operators by just finding all the classical operators made out of the classical fields that obey this condition, that q bar annihilates them and q annihilates them. You just write them all down. And that's it. You're done. There is no renormalization there. You can compute it at zero capping. He's asking about non-lagrangian. Ah, you're asking about non-lagrangian? No, for non-lagrangian theories, it's an art. I mean, because there is nowhere to start. So you have to use much more sophisticated techniques. Also, the chiral ring operators, we don't know how to compute them in non-lagrangian theories in general. We don't know how to compute the CIJK in non-lagrangian theories in general. Because there is no capping constant to expand in. Yeah, so there are some special cases in which people have found in genuine methods to do it using very more sophisticated techniques. Any more questions? OK, thanks so much.