 So, hey, great to see you all again here. So, I'll start with a review. We first described the two half BPS sectors of N equals two super conformal filters. That was the first lecture where we had the Higgs branch operators about which I did not say anything, but the belt is giving you a whole course about it. Then there are Coulomb branch operators, and these are the lectures I'm giving. So, there is a slight, I mean, I just wanted to make a small clear distinction between the various things that are known about these two sectors. So, in Lagrangian theories, this is a coupling constant independent. So, this is independent of the coupling constant. So, in the theories which have a Lagrangian, this can be determined at three level. But for theories that are non-Lagrangian, there are these techniques that belt is teaching you, and there's been a fantastic progress on these non-Lagrangian techniques. So, this is for non-Lagrangian techniques. Now, for Coulomb branch operators, there is already a lot of stuff to be said for Lagrangian theories. So, since there is a dependence on the Yang-Mills coupling and the Theta angle. So, the structure constants, the OP coefficients depend on those parameters in some normalization. So, there is already a lot to be said for Lagrangian theories but for non-Lagrangian theories, at least at the moment there hasn't been much analytical progress. It's mostly there are some bootstrap results, however, by essentially the Hamburg group, that I'll give more references to at the end. So, I'm more or less focusing on this, the perturbative or Lagrangian aspects of Coulomb branch operators and belt is focusing, I guess, mostly on the non-Lagrangian general properties of his branch operators. So, this is what the different strengths of developments are. So, we discussed those two sectors, and then we derived this relation between the four-sphere partition function and the Keller potential interior space, and the Keller potential interior space can also be related to the two-point function in flat space of marginal operators. Let me write this again. So, if the operator OI has dimension two, then it corresponds to an exactly marginal deformation of the theory, and the two-point function OI, OJ Degger captures the zoological metric as a function of the captains, and we have seen that, well, this is a flat space observable. So, this is measured in R4. So, we have seen that the four-sphere partition function captures the Keller potential, which is related to the metric by taking two derivatives. So, there is a direct relation between this particular extremal correlator and the four-sphere partition function. So, today, I'll study in detail the case of SU2 gauge group, and we'll study the perturbative properties of these extremal correlators, we'll study the chiral ring, and then I'll also show you how to solve the general extremal correlator, not just these particular extremal correlators. So, I'll do the most general case, and then we'll discuss connections to general aspects of perturbative series in quantum field theory, namely resurgence. So, this is the, are there any questions about the previous, whatever is unclear about the previous lectures, maybe I can start by addressing that? If there is anything, okay. So, I'll proceed. So, today, we do SU2 gauge theory, super conformal SU2 gauge theories. So, there are two interesting Lagrangian theories with just one SU2 gauge group. So, we'll discuss these two theories in parallel and compare them to each other. The first such theory is a SU2 gauge group with an adjoint hyper-multiple, adjoint hyper, massless adjoint hyper-multiple. This theory has another name. Does anybody know the name of that theory? So, again, this is a massless adjoint hyper. Very good. So, this is the n equals four, maximally supersymmetric Young-Mills theory. That's our first example. So, this has an exactly marginal parameter, which is G Young-Mills and theta for the SU2 gauge group, with theta being too pi periodic. Then, there is another theory, which does not have n equals four supersymmetry. So, it's a genuine n equals two. So, you take SU2 gauge theory and you add four hypers in the fundamental representation. This is again parameterized by the Young-Mills coupling and the theta angle. So, this is again the same story. But, this does not have n equals four supersymmetry. So, it's a genuine n equals two supersymmetric theory. So, let's start by describing the chiral ring in those cases. So, Coulomb-Brentch operators, chiral ring or Coulomb-Brentch operators. So, it's easy to convince yourself, this you can take as a small exercise if you want to get a little bit deeper into the subject, and familiarize yourself with this Lagrangian. Familiarize yourself with this Lagrangian, and the transformation rules under supersymmetry. You can show that the following is a chiral ring operator, Coulomb-Brentch operator, where phi is the scalar in the vector multiplet. Scalar in the vector multiplet. So, phi is in the adjoint representation of SU2. So, phi is in the adjoint of SU2. So, since it's in the adjoint of SU2, you cannot take a trace of just phi that would just be vanishing because it's an SU2 matrix, so it's traceless. But you can take the trace of phi squared. So, I'll denote this operator by phi 2. I'll denote this operator by phi 2, and I'll put also minus 4 pi i for future convenience. So, this is one chiral ring operator, and you can argue that, well, this is true in both examples. So, this is the chiral ring operator both in this theory and in this theory. You can argue that the chiral ring is spent, the chiral ring, as I mentioned, is finitely and freely generated. So, the chiral ring is spent or generated, but well, is given by powers of this guy. So, I'll define this operator's ON to conform with the previous notation as phi 2 to the power n. So, this is the whole chiral ring of this theory. The dimension of phi, the dimension of ON is going to be 2n, right? So, in particular, the O1, in particular O1 is the operator that generates the dependence on the cap, well, this is the operator that corresponds to changing the capping constants. So, this is the exactly marginal operator. So, this is the exactly marginal operator, and its coefficient is tau. If we add O to the action in this fashion, then the coefficient is tau, where tau is 4 pi squared over G angle squared with an i plus theta over 2 pi. So, O1 has a special role in this game. It's the coupling constant, essentially. Now, so, the chiral ring in this, so the chiral ring in this model is especially simple. The algebra of the chiral ring is in this convention and in this normalization. The algebra of the chiral ring is simply this, and it's independent of the coupling constant. The coupling constant dependence is in the two-point functions. You could have normalized these operators canonically, and then the dependence on the coupling constant would have been here in the algebra of the operators, but you can also just normalize these coefficients to be one, which is convenient, and then the coupling constant dependence is in the two-point function, which is a function of tau and tau bar. So, determining the chiral ring in this S2 gauge series is equivalent to finding these functions G to N. If you know all these two-point functions, then you can solve for any extremal correlator. Indeed, let's do it. So, let's suppose we have an extremal correlator, ON1 at x1, ON2 at x2, all the way to, let's say, ONK at xK, and then we have O dagger of N1 plus N2 all the way to NK. So, this is what we defined to be an extremal correlator, if you recall. So, given this product rule, and given these two-point functions, it's obvious what to do. This is just G, you just collapse these guys, and you get G2 N1 ta-ta-ta plus NK, which is a function of tau and tau bar. So, if we knew these two-point functions, if somebody just handed us these two-point functions in this normalization, we would be done. So, we would have all the extremal correlators. And I could even put this guy at y, if you want, let's be even more fancy. Let's put this guy at y, then we will have this factor that we discussed many times, y minus xi to the power two delta i, and delta i is two NK. So, it's two, four NK. And this is K, but four NI. This is I, and I goes from one to K. So, it can even be more fancy. So, we could compute all the extremal correlators in this series if we just knew the Gs. And that's our goal for today, to find these Gs non-perturbatively, hopefully, and then see what we learn from that. Are there any questions? This is the setup for today. So, we want to compute Gs in both SU2 gauge theory with four hypers and SU2 gauge theory with a non-joint hyper. And then we'll do some resurgence analysis and make connection with TT star geometry and other equations. Are there any questions about the setup? Anything that's unclear before I start? Okay, very good. So, the central result, oh, I should say that in this convention that I've defined here, G0 is given by one, obviously. So, it's just one. G0 is just one, and all zero is just one. Okay, so, these are the conventions that I'm using. So, yesterday we gave a prescription or I gave you the outline of a proof for how to compute G2. I didn't yet tell you how to compute the general GNs. But yesterday, we learned how to compute G2 as a function of tau and tau bar. So, this is the zomological metric in these two theories. And we discussed the fact that this G2 is the zomological metric on the space of these theories. So, the key formula was the G2 as a function of tau and tau bar is 16 times one over the four-sphere partition function squared times the determinant of the four-sphere partition function. The derivative with respect to tau of the four-sphere partition function, the derivative with respect to tau bar and the second derivative of the four-sphere partition function with respect to tau and tau bar. What is this? Oh, there is no six. Let me just get the normalization straight. I would hate to get it wrong now and then it will. Yeah, I think there is a 16. Another formula that's equivalent that was the homework exercise was to express this as a derivative of the logarithm of z. That was an equivalent formula. Okay, so now we can proceed. We can at least start from G2, see what are the properties of G2 and then we'll generalize to all the other extremal corridors. So, let's do it. So, let's say compute. Let's start computing something now. After all this general introduction. So, I'll start from, let's, I'll give you both cases in parallel. We'll do both cases in parallel. So, for SU2, with an adjoint hyper, also known as n equals four supereagnals theory, the four-sphere partition function is a very simple. It's the integral of some real variable a from minus infinity to plus infinity of e to the minus four pi imaginary part of tau. So, the imaginary part of tau, as you can see from here, is essentially one over the angle scapuling. So, four pi imaginary type a squared and then there is two a squared for normal is that that's the SU2 with adjoint hyper and for SU2 with the four flavors, four fundamental hypers, a four fundamental hypers, the formula is significantly more complicated as many of you know, but it's still explicit. So, it's still an integral for over the a from minus infinity to plus infinity and then there is the same e to the minus four pi imaginary tau a squared and then there is the same normalization factor two a squared, but then there is a bunch of a few special functions that I think that somebody must have talked about. So, I'm not going to review them here. So, there is a bunch of special, well, sorry, this is to the four and then there is the instant on factor that's also known explicitly, which you need to raise to, which you need to take the absolute value of and then raise to the second power. So, the h's are some special functions that are explicitly known, but I'm sure there was somebody who talked about the S4 partition function. Okay, so you should know that. I'm not going to review that, I'm just going to use that to solve the three more correlators. Okay, so, given these formulas, we can now just compute, right? Given this and given that, we can just compute the two point functions. For in the, so in the first case, let's do the first case first. The first case, SU2 plus an adjoint hyper, we find, well, this matrix integral, well, this integral is trivial to do and what we find is the G2 is a, I have some funny normalization, but it's actually a kind of canonical. So, it's independent of the theta angle. That's a surprise number one, perhaps. And it's just quadratic in the angles coupling and that's it. So, the coupling constant dependence in n equals four is trivial and that's it. So, this is exact. This will soon be, I'll soon connect this result to what we saw about extremal correlators in n equals four in the first lecture. This is essentially saying that only the three level diagrams contribute to the zomologic of metric. So, it has the same content as what we discussed before. In the second example, G2 is much more interesting. So, the three level contribution is the same and you might not be surprised by that because it's the same three level diagram for the adjoint scalar. It's the same operator. So, at three level you get the same answer. But then there is a series of corrections which are perturbative in nature. So, let me just write down the few next corrections and they contain the zeta function with odd arguments. And then there is one over the imaginary part of tau to the power four. So, this is like G and mills to the power eight, I think, in my convention. Okay. And then there is a 1,500, I'm not gonna write all the terms just if the first few ones, four pi cubed. Then there is one over the imaginary part of tau to the power five and so on. So, these are perturbative corrections. So, you could reproduce these corrections from diagrams. So, the first diagram looks like this. It's essentially a three level diagram. But then there are diagrams that look like this and so on. There are many diagrams. So, you can reproduce these results from an explicit perturbative computation. So, these are perturbative corrections but there are also instanton corrections and amazingly we actually know them. So, for example, the first instanton correction depends on the theta angle. So, while in n equals four, there is no dependence on the theta angle cause there are no instantons for this observable. This does receive instanton correction. So, there is some dependence on the theta angle and there is your typical instanton factor that you like. And then there is some perturbative series like six over imaginary tau squared, three over pi, one over imaginary tau cubed and so on. So, the structure here is basically that there is the perturbative series and then there are instanton factors but each instanton factor is also addressed by a perturbative series. So, and that's, yep, the second. So, the question is if I can do the perturbative part exactly. I could probably write it as a sum of very infinitely many poles. Is that considered exact? Well, if it's combined with a good function, I'm asking. I don't know, we did it like monkeys, just expanding it, yeah. But maybe there is a nice way to resummit. So, let me now tell you what it means. All of these things, what this means. This is a very interesting, in some sense, dual, two dual interpretations. So, there are some resurgence aspects about it, which I'm gonna discuss soon, but first I wanna tell you what it means for geometry. So, the space of theories in both cases is spent by the young mills capping and the theta angle. And in both cases, there is an S-duality group that acts on these two parameters. So, in fact, the space of indistinguishable theories is in the fundamental domain. Well, I'll discuss that soon, just a second. The space of indistinguishable theories is in this fundamental domain, okay? So, both cases describe, in both cases, the exact marginal parameters lead to a conformal manifold that looks like the fundamental domain. So, since this G2 is interpreted as the zomological metric, so we basically, we should interpret these two formulas as the different metrics that we can put on the upper half plane, mod SL2Z, namely on the fundamental domain. So, in the first case, the metric, the zomological metric is gonna be given by simply this, times a DG young mills squared, which is like the line element. Or if you write it more, if you wanna write it a little bit more precisely, you get d tau, d tau bar, which is the upper half, well, the coordinates on the upper half plane, and then we get six over imaginary tau squared. So, this is the metric that we induce on the fundamental domain, domain from the physics of n equals four, super young mill theory. This is a familiar metrics, right? So, how is it called? What is this space? This is a constant curvature metric, which is normally put on the upper half plane, hyperbolic hyper half plane. So, this is the Poincare disk, Poincare disk. So, it's a constant curvature metric. So, we see that in n equals four, the metric on the space of theories does not receive any corrections beyond the three level diagram, and you get a constant curvature metric, negative constant curvature metric. In the second case, we see that the metric receives corrections, both perturbative and non-perturbative. So, in the second case, the metric receives corrections. It does start from this piece, which is the dominant piece near wick-up. So, wick-upping in this picture is very far up. So, very far up, the metric that is induced by SU2 with four hypers coincides with that metric. But there are, of course, these corrections, perturbative as well as non-perturbative corrections in the coordinates on the upper half plane. So, this is like another metric. It's not your canonical metric. It does not have constant curvature, but it does have something in common with that metric. So, first of all, it's probably true that the volume is finite. The volume of this disk with this metric is finite. This is an exercise. You can try to show that using this metric, the volume of the fundamental domain is finite. This property remains true also for the metric that's induced by SU2 with four hypers in the fundamental representation. And secondly, wick-upping is at infinite distance away logarithmically. So, wick-upping is logarithmically far away. Is logarithmically far away, infinitely far away, and the distance diverges logarithmically, which is obvious from here. If you take this line element and you just go straight up, it diverges logarithmically. But the volume remains finite. So, wick-upping is like an infinite throat, an infinite, very narrow throat. So, total volume is finite, but the distance is infinite. So, while this is a non-standard metric, and I mean, it has some properties which are similar to the n equals four metric. So, this is the zomological metric interpretation of these results. But then there is another way to interpret these results in the language of AGT. So, AGT allows for a different interpretation of these computations. So, in AGT, you can think about SU2 with four hypers as arising from compactifying some six-dimensional theory on a sphere with punctures. So, let's start from the second case. The second case is a sphere with four punctures. So, the interpretation of this formula in AGT is that it's in, you might think about it as the zomological metric on the space of theories, but you can also think about it as a metric on the space of spheres with four punctures. Because the coupling constant G and mills comma theta is associated to the location of these four punctures. So, this space is also one complex dimensional. The space of spheres with four punctures, modulo SL2R, is also one complex, modulo SL2C is also one complex dimensional. And this metric can be interpreted as a metric on this space. And in this case, you can think about it as a tec-muler metric. So, it's a non-standard metric on some space of Riemann surfaces. It's different from the Wild-Petersen metric. So, it's not the same as the classical Wild-Petersen metric on this space. So, you can think about this computation as furnishing some interesting new metric on the tec-muler space. And in the first case, the way you construct n equals four supri-anguinal theory from six dimensions is just a torus with some A marked point. And in this case, you get the standard metric on that space of tori. So, n equals four theory gives the standard Wild-Petersen metric. So, there are several parallel ways to think about these computations. Now, I want to make some contact. Are there any questions about this? Yep. N equals four is constructed from six dimensions by compactifying on a torus. So, this also has the one complex dimensional modulate space. And you can think about this as parameterizing g n mils and theta in N equals four. So, this metric can be interpreted as the metric on the space of marked tori. And in this case, this metric can also be interpreted as the metric on the space of spheres with four punctures. Or in the zomological language, you interpret them both as the metric on the fundamental domain. So, there are several different interpretations. Yeah. What is not written? No, this comment is not written anywhere. But you can think about it as a new metric on the tecumular space. It's like a quantization of the standard Wild-Petersen metric. It's a quantum version of the standard metric. Shouldn't it be the metric on the complex structure of the double cover of this thing? Yeah, you're probably right. You're probably right. The cover which is a torus and complex star at the space of fixed complex star. You're probably right that I might need to take some covers of the tecumular space. You're probably right. Yeah, the tecumular space of a double cover of the surface. You're probably right that I need to do that. Thanks for the comment. What is the markupon on the first one? It's just like the way that n equals 4 arises in the AGT correspondence. I'm not going to review, of course, this thing. I'm just saying that this is a possible interpretation of these computations, which is complementary to the interpretation as a zomal logic of metric. OK. Are there any more questions about these aspects? Now, I'm going to describe some connections to QCD. Analogy is connections. Connection to resurgence and QCD. Say that again. Oh, that's a great question. So one of the things that you would like this metric to obey. So the question is whether I can see that it's consistent with s duality. So one of the things that this metric has to obey is that it would be a good function on the fundamental domain. Or it could be invariant under s o to z. So you remember that yesterday we discussed the fact that the partition function on the force sphere is not well-defined because of a counter term. So suppose you try to take this formula and you ask, is it invariant under s o to z? It turns out that it's not, but it's only non-invariant up to a holomorphic function. Can you answer it if you can? No. I don't know how to see it directly. But it's supposed to be true. It's supposed to be true. So for example, this you can do directly. In this case, you can see that the force field partition function is not invariant under s o to z. But it's invariant up to a holomorphic function. So that nicely ties with what we discussed yesterday, that in fact, there is a counter term. And when we compute g2, this goes away. So by the time that you compute the metric on theory space, it should indeed be invariant under s o to z. But I don't know how to see it explicitly here. I've never seen anybody showing that. Is it a function, or is it something? Is it a formula for what do you expect? The force field partition function transforms like a bundle of L times L star, a section of L times L star. So this is not invariant. But by the time that I do this procedure with the determinant, it's good. So you can show that this prescription with the determinant is a scalar function. It's not a section of L times L star. This is why this is a good formula. But I'm asking from point of model of properties, do you expect this transform as a model of functional problem? Let me say my mask is so written, wrote in 95. Yes, I know what you mean. It transforms a model of functions. Yes. Yes, yes, yes. Yeah, I think that, well, here if you do an s transformation, you pick up some holomorphic function. And you can compute it explicitly here. So I can actually do it right. I can do it in real time. So I can do it in real time. So this formula, let me just make an s side to address Maxime's question about whether it's a modular function or not. So the logarithm of the force field partition function in the first case, we can do it in real time. It's going to be a logarithm of tau minus tau bar. So if we do an s transformation, let's apply an s transformation. So we get log of 1 over tau minus 1 over tau bar. But this is the same as log of tau minus tau bar up to a holomorphic plus anti-holomorphic function. You see? I mean, I'm missing signs, but I don't care now. So once you do an s transformation, the partition function will pick up a holomorphic plus an anti-holomorphic piece. So this would go away when I take the determinants. So what I'm doing is OK, but the force field partition function itself is only invariant up to a counter term. And the same is true for Witten's 95 paper for Donaldson Witten Twist. But is it a generic feature for all the details of these things where it depends on the theory or not? There is an interesting question of whether the holomorphic function is dependent on the theory or not, and I have no idea. Because I think that we're making claims that it depends on topology of, I mean, of pretty general data. No, that's a different question. There is a question of whether it depends on the theory. And there is a question of whether it depends on the manifold. So here it's a fixed manifold, so the second question doesn't arise. So you're asking if it depends on the theory, and I have no idea. I don't know if anybody knows the answer to that. OK, so are there any questions about this? Now I'm moving to making some connection with Resurgence theory and QCD. So it is typically expected in theories with a small coupling constant that the expansion would morally look like what we got, that there will be a perturbative series. And on top of the perturbative series, there will be an instanton corrections. An instanton correction would be dressed up with a perturbative series of its own. So for any quantity, let's say Q, you expect some expansion in lambda, and then some instantons, which are dressed by some expansion in lambda, and so on. So this is the general structure that's expected in any theory. And it coincides with the structure that we find in this case. So now I'll tell you about all the conjectures about the Resurgence theory. And we can test them here. So there are conjectures about properties of a perturbative series from the 70s and 80s. So first, and also from before, there is the Dyson argument. Dyson wrote a very nice paper in the 50s, I think, where he argued that the coefficients of any reasonable perturbative series, so let's call this A1. This is A2, and so on. So Dyson argued a half a century ago or more that in absolute value, this has to be more or less n. The arguments of Dyson have to do with some instability if you reverse the sign of the coupling. But you can also understand this factorial growth of the coefficients from the fact that the number of Feynman diagrams proliferates very fast. So there are of the order of n factorial Feynman diagrams at order n in perturbation theory. So to show you that these supersymmetric theories So the amazing thing about these computations is that on the one hand, we can do them exactly. On the other hand, they obey all the standard rules from perturbative quantum filtering. So from the point of view of just perturbative quantum filtering, this is obeyed. So I'll show you something even more stringent soon. So the number of Feynman diagrams does grow factorially in this computation. So it does obey the standard rules of perturbative series in quantum filtering. So you can check that this is true asymptotically for this perturbative series. The second conjecture is a more recent. It's a Brodsky, Karl Liner, and several other people were involved in this thing. And they have a more sophisticated criterion that is supposed to be true always. But I think this is the only case where you can actually check it in four dimensions. So that's nice. So they are saying the following things. Suppose you knew the first n loops. So you were very diligent and studious, and you computed the first n loops of some perturbative expansion. Let's say that n is even. For what I'm going to say, it's going to be important. There is some modification when n is odd. So then the procedure is the following. There is something that's called the Padei approximation, where what you do is you construct a rational function in the coupling constant that has a numerator of degree on n over 2. That's why I want n to be even. So you construct the rational functions with some coefficient c i of degree on n over 2. And you construct some rational function in the denominator, not a polynomial function in the denominator of degree n over 2. And you solve for the coefficient c i and d i by requiring that they reproduce exactly the n loops that you computed analytically. But once you have this expression, you can now make predictions for the coefficient of the n plus 1 term. So this makes a prediction for an plus 1. Is this clear? This is called the Padei approximation. The c i and the d i are determined by matching the first n terms that you've already computed. What's the difference between c and d? The c i's are the coefficients of the polynomial in the numerator. And the d i's are the coefficients of the polynomial in the denominator. So is it the ratio or the approximate? The ratio is a rational function, which is called the Padei approximation to the perturbative series. And it's called the symmetric Padei approximation. It's called symmetric because the degree. So in the literature, this would be called n over 2, n over 2 Padei approximation, because the degree of the polynomial in the numerator in the denominator is the same. For some experimental reasons, empirical reasons, it seems that the best Padei approximations are of this sort, where the degree of the new. It's voodoo, but it seems to work the best. So once you construct this rational function as an approximation to the perturbative series, you can ask, how is my result for a n plus 1 different from the actual result? So this was important in the 80s because people wanted to compute the beta function four or five, six loops of QCD because they wanted to make ever more precise comparisons with deep-in-elastic scattering. So people who are lazy don't want to compute the six loop. So they want, OK, we have the first four loops. Let's just guess the six loop. So that's where it comes from. So the conjecture is that in any quantum field theory, the coefficient of the n plus 1's term that you get from the Padei approximation divided by the coefficient of the n plus 1's term that is actually correct, minus 1, approaches exponentially fast to the right answer, where C is some positive coefficient and sigma is some positive coefficient. And this is asymptotically true. This is supposed to be asymptotically true, OK? So the idea is that you are making only an exponentially small error for high orders in perturbation theory if you use Padei, asymmetric Padei approximation. So using this, I think that using these results, you can test these ideas in four dimensions now. So you can plot, for example, here would be n and here would be the precision. So the vertical axis is this combination. So it's, let's say, 10 to the minus 10, 10 to the minus 20, 10 to the minus 30, 10 to the minus 40. And this is n. So we just plotted on the vertical axis this combination in absolute value and on the horizontal axis, the order in perturbation theory. And we went to more or less order 100. We did not prove that this is true analytically, but we went to more or less order 100, which you can do using this explicit expressions. And it falls on a beautiful straight line with sigma, which is 0.7. And amazingly, in QCD, they found a similar value. So amazingly in QCD, how much is log 2? Log 2 is 0.71, right? 0.72. So in QCD, people believe that sigma is around log 2. So it's a very similar value. Yeah. So people in QCD estimated that sigma might be somewhere around log 2. So why do sigma depend on the observable or the total amount? Well, we did a lot of tests of this sort. And it seems to be always around 0.7. We have around 10 different tests. So it seems to slightly depend. Maybe some theories it's 0.71, 0.72, but it's always around 0.7. I don't know why. We tested many extremal correlators, and it was always like that. So you can do many other checks of this sort, but we haven't exhausted this, of course. And then there is another question about perturbative series that is, of course, very commonplace nowadays, which is given that there is such a factorial growth of the coefficients, what can we say about the convergence of the perturbative series? So we have the perturbative series in the 0 instant on sector, then we have a perturbative series in the 1 instant on sector. So the question is about what about Borel's summability? So as you know, for a power series that diverge in such a fashion, there is a formal procedure, which is called the Borel transform, to render those perturbative expansions more manageable. And the idea is that perhaps these expansions are Borel summable. So here, we actually were able to prove that the answer is yes. For all the observables we considered, even in the non-trivial instanton sectors, like even the perturbative expansions in the instanton sectors, this series turned out to be Borel summable. But they're not summable by ordinary series expansion. So they are Borel summable. And there are some poles on the negative Borel axis, which have not been studied yet. So nobody knows what they mean. So there are some poles on the negative Borel axis that would be nice to understand what they mean. So they- Do you know the position of that one? I don't remember them. I see. So you don't see that they don't belong to them. So in QCD, there is an everlasting debate about poles on the positive Borel axis that would make the power series not even Borel summable. But then, so Borel summability is disturbed by it. If you have poles on the positive Borel axis, then you cannot even Borel sum the series. But this is called renormalance. Renormalance, and they may or may not exist. It's really the jury's out. As Eliezer told me to say, the jury's out. The verdict is not out. The jury's out about those in QCD. It's like they have to do probably with an infrared, the non-safe observables. That's my belief. Anyway, it may not be a substantiated belief. But here in the supersymmetric theories, we find poles only on the negative Borel axis. So in fact, it's Borel summable. But the meaning of those is not clear yet. They have to do these are in the QCD literature. You interpret these poles on the negative axis as they're having to do with instantons. So it's like saying that let's say that you have just the perturbative series. So it's not Borel summable. The perturbative series, sorry. Suppose you have just the perturbative series. It's not summable. So the idea is that that means that you're missing some instantons. So the poles on this negative axis have to do with instantons. And therefore, it should be true that the poles on the negative axis have something to do with the instanton corrections. So there might be some recursion relation where we can get the instanton corrections in the necros of partition function from the zero instanton sector. Because those guys should be able to teach us about the instanton corrections. So there should be some kind of resurgence equation. Do you expect that to be instanton, not like a complex? It could be also a complex saddle. I simply don't know. But ordinary interpretation of poles on the negative Borel axis is that they have to do with instantons that you need to add. And since we already know what the instantons are, it might be that that means that there is some recursion relation on the necros of partition function. OK, so surprisingly, many of these bold conjectures about QCD turn out to be analytically correct for no good reason, as far as I can say. What was the original motivation for that conjecture? The original motivation is the laziness of people. They did not want to go to sixth order. They wanted to guess the sixth order from a rational function. It was wishful thinking based on, I mean, there is lots of observables in QCD that are computed to four loops. And it seemed that if you know the first three loops, then the first loop is kind of close to the approximation. So then they wrote this conjecture. Where was it? Yeah, that conjecture. OK, at what point should we take a break? Like for just five minutes. OK, let's say if there are more questions, I can take them now. And then we can take like a 10 minutes long break. And we'll finish the extremal correlators after the break. Or I can just spend some time giving references. Are there any questions now? Or maybe I'll spend like five minutes giving references now. Because, OK, I got a few emails and a few people asked me about references. So I prepared a list of references for everything that I'm saying. So I'll just spend five minutes giving you all the references in an orderly fashion. And then we'll take a break. So I'll divide the references by topics. So derivations of z equals e to the minus k of this business. This you can find in two papers from two different points of view. So 0, 8, 5, 1, 1. This is from the supersymmetric point of view, a point of view which is more suitable for supersymmetric localization. And this is from the point of view that I explained, the point of view of trace anomaly. So this is about z to the e to the minus k. The extremal correlators, which is the subject of the next hour. Extremal correlators, which is the subject of the next hour you can find in this paper. Now, people asked me, also, over email, about global properties of trace anomalies. So people are like, what can we say about the space of, let's say, the conformal manifold from these properties is that it's a section of L times L star and these kind of things. So global properties were recently studied in two very nice papers. When I say very nice, it means it's not my own papers. 0, 7, 3, 6, 6. So there were two papers recently addressing some global issues. This is on the physics side and on the mathematics side, there was some very nice work by Donaghi Morrison, whose number I forgot to copy. So these are like the physics on the physics side. This is on the math side. Now, this also in this paper, there was quite a bit of work about the resurgence properties of these perturbative expansions. You can find some of one of the first papers is this and you can look at the follow-ups if you want to see Morrison developments. This is one of the first papers on the subject. Then these extremal correlators, they have like a limit where the dimension is very large. This is when this little n goes to infinity. These extremal correlators describe correlation functions of very, very heavy operators. So this limit of very heavy operators has to do with the large charge expansion of Hellermann at L. So that is some story in effective filtering. But since here there are exact results, you can make some useful comparisons between those two approaches. And you can read about that stuff in some recent papers that I'm quoting here. So these are like the most important papers to read at the moment. Then there is a topic of the next lecture, which is Titi-star geometry, Toda chains, Toda chains, and integrability. These topics arise when we study these extremal correlators of heavy operators. And that has been first initiated in 2009. And well, there was a lot of work since I'm just quoting some of the papers that are relevant. There are more and more, I'm just quoting a few, and there are more and more papers about this. And also everything I'm saying has an analog in two dimensions. Also in two dimensions, you get interesting dependence on various coupling constants on the complex structure moduli of Calabi-Au manifolds. That's the subject in and of itself. And this has been recently, there were two papers that came out almost at the same time about the two-dimensional analog of all this story. And I presume there is much more to do there. And in fact, this paper is by you, right? So he's a second one. Oh, that's right. Yes, yes, yes, yes, yes. You're right. This is yours. Yeah. OK. We can take a break now.