 So now I want to continue this set of lectures and finish them by computing all these observables in SU2 gauge theory with four fundamental hypers and then with us, well, and at the same time in SU2 gauge theory with an adjoint hyper. And we'll see the connection to integrability, Todde equations, the T-star geometry. That's the topic of the last lecture. I will not present the subject for other gauge groups. I'll just mention a few of the open questions and ideas and results in that field. So since I want to get to the interesting things quicker, what I'll do is the path that I chose in the presentation. So here I gave you a derivation. I mean I gave you sort of an overview of a derivation through the trace anomaly for this basic relation. One of the tests that you should run on this relation is exactly what we discussed at Lens in the previous session, which is that this is invariant under Kepler transformations. So this is invariant under these holomorphic functions that the partition functions are ambiguous by. So the path that I am choosing to giving you the most general solution for the extremal correlators is not by deduction. I'm not going to give you a proof. There are some proofs in the literature. I'm just going to give you a formula, and then you can try to check by yourself that it's actually invariant under holomorphic times anti-holomorphic ambiguities. So the idea is the following. So the idea is to generalize this formula. So this will be our starting point, and I'm just going to write an expression that generalizes that formula, and it will give the answer to these observables. So the first step is that you construct a huge matrix, m and m, which consists of essentially derivatives of the partition function. So we have the partition function in the denominator, and then we have n derivatives with respect to tau, and m derivatives with respect to tau bar. So you construct such a huge matrix, m. It's an infinite matrix in principle. So there is one here, and then there is d tau of z over z. Then there is d tau bar of z over z, and then there is d tau bar of z over z, and so on. So you can observe, it's a silly little observation, that g2 was essentially the determinant over the 2 by 2 block of the upper left corner of this matrix. So the determinant of that is exactly g2. Is that okay? Oh, factor of 16, you mean? No. Wait, just a second. What did I do? There is a small discrepancy, indeed. So what is the discrepancy? Can you tell me again? What? Maybe it will be useful if I write this as z over z. Then when you take the determinant, clearly z squared pulls out, and it's the same, no? So I claim it's the same, okay? So the formula is the following, right? The claim, I'm not going to derive this claim, I'm going to just tell you what are the consequences and the structures that arise from this. So g2n has a silly normalization factor in front here. This is just generalizing this 16 that we had in that. And then what you do is take the determinant of the n by n upper left block and divide by the determinant of the n minus 1 upper left block. And that's it. This is the idea. Well, this is the formula. It's very nice. Very nice and very neat. Okay? So this reduces to the formula for g2 that we had before. Now an exercise for you, this is a really nice exercise. It shows that this is quite a miraculous formula. So as we discussed many times, the partition function is not actually unobservable. You can shift it by a holomorphic plus an anti-holomorphic function. This is a purely holomorphic function and this is a purely anti-holomorphic function. So you can check that this is invariant under this transformation, which is extremely non-trivial exercise but technical. It's a technical non-trivial exercise. So g2n is invariant. This is a very non-trivial property of this construction. So that's of course necessary if what you want to construct is an actual observable. I think I messed something up a little bit with the indices n. So dn is the determinant of the 2n by 2n block, not n by n block. Okay? Yeah. dn is the determinant of the 2n by 2n block, block on the upper left corner. Okay. So I'm not going to prove this formula, I'm just giving this formula as a fact. Now we'll see that this leads to some very interesting properties for these two-point functions in flat space. So this is supposed to be a formula to compute two-point functions of extremal correlators in flat space. And as I remarked, if we knew g2n, we know all the extremal correlators. So with this formula in principle, you know everything. Amazingly, there is a recursion relation between the various g2n's. Okay? I don't know why there is such a recursion relation, but there is a recursion relation between the different g2n's. So in some sense, they're not entirely independent. And this recursion relation is what's known as 30-star geometry. A property of this recursion relation is that it's an integrable, you can think about it as an integrable system, which we're going to see soon. So inside every n equals 2, super-conformal filter, this is general in fact. These are general facts, not just for SU2. So inside every n equals 2, super-conformal filter, the Coulomb-Wrench operators obey the recursion relations of some 30-star geometry, and they're always an integrable system. In some cases, this integrable system can be identified with a classical integrable system that is in textbooks. And in some cases, we don't know what this integrable system is. So a small exercise, this is easier than that exercise, is that you can check the d tau bar of the logarithm of this dn obeys the following equation, dn plus 1, dn minus 1, dn squared minus n plus 1, d1. So these are these determinants, and they obey this differential equation, relating various components of these determinants. This is not terribly hard to show. So that can be rewritten in a slightly more suggestive form, this differential equation. So this follows just from the definitions that I've given you. There is nothing more to that. So to understand the meaning of this differential equation, it's useful to change variables a little bit. Can you just pick up a little bit? In four dimensions. Any n equals 2 is 15 in four dimensions. So integrable systems could be very complicated, it doesn't mean that it's straightforward. In some cases, it's not even easy to identify them. But you can just prove that there is a lack spare in these formal properties. So there is a change of variables that allows to uncover the meaning of this differential equation. So you rewrite dn in terms of an exponential of some qn, you just take the logarithm of dn essentially, minus the logarithm of the four-sphere partition function. And there is some factor of 16 that I'm carrying around for some conventions. So you do this change of variables. And then this differential equation in terms of the qn's looks a little bit more familiar. So the differential equation in terms of the qn's after you do the change of variables becomes d tau, d tau bar qn is e to the qn plus 1 minus qn minus e to the qn minus qn minus 1. And this equation is familiar from the literature on integrability. This is called the tau-da equation, I'll explain what it means now. It's called the tau-da equation. So here, this tau-da equation lives in a space which is totally different from maybe the other tau-da equations that you've seen. So there is one tau-da system in AGT that describes something completely different. Then there is a tau-da system that lives on the cyber-witten curve that also describes something completely different. This tau-da equation lives in the space of extremal corridors. So it's a recursion relation for extremal corridors in some sense. So you can think about this tau-da chain as saying that points, which are like masses, so the imaginary part of tau plays the role of time. So you can think about it as Newton's second law, if you ignore for a second the real part of tau, which I'll make a comment about it, just ignore for a second the real part of tau, then this looks like q dot dot equals something. So you can interpret it as a chain of masses with some springs. So this is the boundary. So this is like a half-infinite chain of masses and springs. And the force between two nearby springs and nearby masses is given by this. So this is like a half-infinite tau-da chain. So this is half-infinite. But the meaning of these sites is like extremal correlators. So this is the extremal correlator that gives the zomologic of metric. This is the extremal correlator for the next element in the chiral ring, like all three all three dagger, then all four dagger, and so on. So this is like an equation that allows you to evolve in time or evolve in tau the different extremal correlators. Now this kind of system of oscillators is exactly solvable, given the boundary conditions. So this is exactly solvable given the boundary conditions. The boundary conditions sit here. So somebody is just, somebody is oscillating this mass. And this will send some waves down this chain. And this is a solvable system. So this is like a solid tau-da chain. And given the forcing on the open end of the tau-da chain, you can determine everything. So it's a solvable, yeah, it's a classically solvable equation. And if you open the integrability literature, how do people solve such a tau-da chain? They write the solution in terms of ratios of determinants. How surprising? So yeah, so it's a, the solution is written as a ratio of determinants. Okay? So any time you see ratios of determinants, you should immediately say integrability. Many, many, many integrable systems can be solved by ratios of determinants. So now let me just make a comment about the meaning of the real part of tau. The real part of tau is absent in the classical tau-da system, just like there is time and it's like Newton's second law. So the real part of tau is a compact variable. So you can Fourier transform in the real part of tau, and that effectively just adds another index. So in fact, what we have is the lattice of tau-da chains. And the interpretation of this vertical direction is what? Can somebody say? So what is the physical interpretation of this direction? This is the instanton number. So there is like a tau-da chain for each instanton sector. And this equation means that they don't talk to each other. Because you can just, if you go to Fourier space, then it's sort of diagonal. So we have a lattice that is a slight generalization of the tau-da chain, where the additional coordinate has to do with the cosine theta, dependence of the partition function of the extremal correlators. Okay, so this is the complete solution of SU2, of the chiral ring of SU2 gauge theory with four hypers. These equations are true both in n equals four, and in SU2 with four hypers. Just like in n equals four, the initial condition is trivial. So n equals four supra-yang-nil theory is only different from SU2 with four hypers in this language by the initial conditions. So the initial conditions are just trivial, and the wave is trivial, and everything is three-level exact. In SU2 gauge theory, the initial condition is non-trivial, the tau-da wave is not trivial, and all these things have an interesting perturbative and non-perturbative contributions. So the equation is the same for the two theories, but the solution is in one case, the solution is just trivial, and in the other case, the solution is very non-trivial. Yeah? How can I perform Fourier here because it's stronger than linear? Can you say again? It's stronger than linear. How can I perform Fourier here? Well, the question is it's strongly non-linear, so how can you do Fourier, right? I was just saying that because the real tau is a circle, you can just trade the derivatives, you can diagonalize the derivatives in real tau. So there is one direction which is non-compact, that's the time direction, and then there is one direction where all the dependence is periodic, so you can just expand everything in cosines. And the exponents as well. Right. So it's not that they are completely decoupled, but the meaning of these different layers of the chain. That's a strongly coupled. You could say that they are coupled, yeah. But the system is still solvable by the same techniques of tau, the integrability. So this is a mild, this is lattice tau, this kind of lattice tau is just a very mild generalization of tau, of the classical chain, because it can essentially solve for the other direction with no cost. You just take the usual solution and you add another direction, and it's still a solution. So in some vague sense, it's decoupled, but I agree that technically it's not. Okay. So in fact, one interesting open question is that if you look at these two-point functions, O and dagger, and if you take N to be very large, let's say N much bigger than one, then the techniques of effective filtering should work, because it's like a heavy operator. This can be analyzed using the large R-charge technique of Heller-Manett-All. So you can analyze these special cases by the large R-charge technique, and it would be interesting to try to understand what this means in that sense. So this is the behavior very far down the chain. So the ideas of effective filtering of Heller-Manett-All imply that very far down the chain here, there is some universal behavior that is true in any QFT, even those ones that we're discussing here, which obey the Todda equation. So in general quantum filters, we don't have such a nice equation for correlation functions, for large-charge operators, they're intructable, but here it's supposed to be controlled by effective filtering. So here there is some effective filtering regime, which would be nice to interpret using the Todda chain language. Okay. Now let me, I want to make some comments about other gauge groups, and are there any questions about SU2 gauge theories? Any more questions about SU2? And then I'll make a few comments about SU3 gauge theories or SUN gauge theories. Okay. So now I'll make some comments about SUN gauge theories. I'll try to say what we know about this aspect of the problem for higher gauge groups. For SUN, let's consider SUN plus 2n-hypers in the fundamental representation, just to bring you up to speed with the literature about what's known about it. So let's start from describing the chiral ring. So we still have this scalar phi, which is in the vector-multi-plate, the scalar in the vector-multi-plate, a complex scalar in the vector-multi-plate, and it is still true that all the chiral ring generators can be constructed by polynomials of phi. So we have trace phi squared, but unlike SU3 gauge theory, we can construct now independent traces all the way to n-1. So for SU2 gauge theory, the special thing is that this is the only guy, because all the other ones are given by polynomials in that guy. Because if you have a 2 by 2 matrix and you take the trace of the matrix cube or the matrix to the 4, it's expressible in terms of lower powers. But for SUn, there are n-1 generators. So for SUn, there will be n-1 generators. I think this goes all the way to n, actually, yeah, to capital N. So there are n-1 generators, and any operator is constructed out of those, is constructed out of those letters. So those are like the letters in the alphabet, and any chiral ring operators constructed out of these letters. Yeah, these are the casemiers of SU capital N. So in principle, the job of computing extremal correlators is to, we can just think about computing all the two-point functions of any word that is made out of these letters. So for example, you could say, I would like to compute the two-point function of trace phi squared, trace phi cubed, trace phi bar squared, trace phi bar cube. That will be a reasonable question to ask as a function of the capping. Now, how many capping constants are there here? Again, just one, one complex capping constant. There is only one gauge drop, and therefore there is only one exactly marginal parameter, which is a Gm ms and theta. As before, the distinguished operator that corresponds to changing the capping constant is this, since this has dimension two. So it's exactly marginal. So there is some distinguished operator here. Okay. So what do we know about this theory? So there is a similar formula with determinants, even though it has more indices. So such a formula still exists for all the extremal correlators. I'm not going to tell you in detail, I'm just giving you a quick vague introduction to the subject. So the generalization of the ratio of the determinants formula, exists and you can prove that it leads to an integrable system, and it's some kind of tether chain, but it's not really a tether chain, because now there are many extremal correlators, you can construct many words, not just powers of trace five squared. So it's not natural to put them on a single chain now. In fact, there are infinitely many chains, in some sense, all of which are half infinite, and there are complicated interactions, but it's still an integrable system of some sort. As far as I know, it may not have a name. So it's a very complicated system of interacting chains, and it's still integrable formally, as I said, but we don't know what the name of that system is. So an interesting open question is to identify that integrable system. Another topic, which is not completely settled, perhaps it's more important, is that these determinants were made out of derivatives of the partition function. In SU2. So in SU2, all we needed to know were derivatives of the partition functions. So SU2 was an especially easy case, because the determinants consisted of derivatives of partition functions that were already written explicitly by a piston. So in principle, you had all the information to compute anything you want. However, here the determinants contain some elements that are not in the piston's paper. So I'm gonna explain what those elements are. This is an open problem to try to understand how to include them properly. So you can always deform the Lagrangian of an n equals two supersymmetric filtering by the integral of any Coulomb branch operator, let's say ON. So even if the dimension of ON and put some coupling constant, let's say lambda. So even if the dimension of ON is bigger than two, it's an okay deformation. It's an irrelevant deformation in general. It's an irrelevant deformation in general, but the power series in lambda should exist. So maybe the theory doesn't exist as a UV complete theory, but the perturbative expansion in lambda, so the perturbative series in lambda should exist. So when we construct these determinants in the general SUN case, because not all the words are made out of just trade phi squared, you also need to know the derivatives of the S4 partition function with respect to various parameters of irrelevant operators that we could have added to the action if we liked. So we need various derivatives of this sort. How do we include such higher operators into the force-phere partition function? Well, the perturbative part is easy. So the force-phere partition function typically takes the form of an integral over the carton, and then there is some kind of perturbative dependence on A here, and then there is the necrosis partition function squared, called the instant part squared. So including lambda here is trivial because that amounts to doing some one-loop computation in localization, and that's easy. However, we don't know how to include lambda here. Nobody performed localization in the presence of such heavy operators, as far as I know. Too much of part-loop is a sake. You're putting a classical part right on this. Yes, excuse me, I meant here, you need to add lambda A to the power n. But one-loop is the same. Yes, you're completely right. I meant here. But why just curiosity, why you wouldn't think that, I mean, instant part is exactly the same? It's not. It's not. It's not. So what do we, so suppose you just want to study this extremal correlators to this. So already the perturbative part is interesting. As I said, there are many aspects of resurgence that can be studied just from the perturbative series. So for the perturbative series, we have all the information that's needed. So these determinants and tada chains can be completely understood to all orders in perturbation theory. So these integrable systems can be understood to all orders in G Young Mills. What you need for the determinants are various derivatives of the partition function with respect to lambda at lambda equals zero. So those we have complete control over to all orders in perturbation theory because we can get them from integrals of this part. So we do understand this integrable system to all orders in perturbation theory. So it's not like we don't know anything. There's a huge amount of information here and there were very nice works by Papadodimas and Baggio and Yarkos checking the predictions of this, also making some predictions for this prescription from explicit Feynman diagrams. So these integrable systems are understood to all orders, but since I don't think that this is actually known explicitly, like it's not known how to localize. I think it's not explicitly known how to correct the non-perturbative part of the partition function in the presence of higher casemures. So this is- I do know there was a paper by Necrosse from March of 19. Right, right. So Necrosse- You also talk about the toda, but is it something different toda or- It's completely different. So yeah, I'll tell you what is known. So what we want is to understand how to deform the omega background partition function by higher casemures. So if you want to understand the complete non-perturbative content of this extremal correlators in arbitrary SUN gauge theory, we have to do that. We need to understand how to include higher casemures in omega background. Now, for UN gauge theories, it's relatively easy. It seems like a somewhat technical difficulty. The issue is that it's SUN. So removing the UN seems particularly nasty. I think for UN, there are papers by Necrosse and Marchakov and Okonkov and I think the results are somehow or also Fujito and Morales. I think the results are sort of implicitly known. But for SUN, you know, it's very hard to get rid of the UN. And it might be a technical difficulty, but it's actually an interesting concrete problem because if you want to understand completely this extremal correlators also including the non-perturbative terms, this has to be done. So the state of the art is that in any Lagrangian theory, this integrable systems and all the extremal correlators are understood to all orders in perturbation theory. And in special cases such as SU2, also the non-perturbative content is completely understood. But in general, since we're lacking the modification of the instant on part of the partition function in the presence of higher casemures for SUN gauge theories, unfortunately we don't know. So we don't have the instant on contributions to these integrable systems yet. So it's still confusing you up with the casemures, what the position from what was up just before. Right. So Maxim's question is why do the casemures affect the instant on partition function, right? So you're completely right that the supercharger remains the same. You're still going to localize on small instantons, but they back react. So remember that the instant on partition function dependent on G and mills. So if you add these higher casemures, they will depend on lambda too. Lambda and G and mills are very similar from- Young rule centers are all the six total counts of parameters. Right. But in each instanton sector, there will be a non-trivial lambda dependence. So it's incorrect to assume that there is no lambda dependence here. Mathematically, what it will be is typically called like subwordable characteristic class of formulas. What are you supposed to do now? I don't know. But for UN, you can, for example, read there is a paper by Fujito Morales, who present rather concrete formulas for UN gauge groups. The formulas are not just modifications of the one dough part. So there is a modification at any instant on them. I don't actually know what it would amount to do that, but this is something that's missing from the story. So that's the state of the art. The extremal correlators are understood to all orders in perturbation theory. In some cases, non-perturbatively, but not in all cases. However, people have been able to come up with general proofs that the integral system of extremal correlators, sorry, the system of extremal correlators is always integrable. It's always a ratio of some determinants, and this has to do with steady-star geometry. So this is known for any n equals to super conformal filter. And more or less, that's it. So, yeah. Is the integrable to fix the modified necrosure? I would fail to see how. I wouldn't know how to do that. You can try to use Borrel business. Borrel probably has some chance, at least of saying something. Yeah, I don't know. So even for SC3, this integrable system has not been completely identified because the instantons are not exactly known yet. And of course, for non-lagrangian theories, there's been almost no progress yet, except for the bootstrap results of the Hamburg group. So, of course, one also wants to venture eventually to non-lagrangian theories, but that would be a nice starting point to complete this program. Okay. Any other questions? Very good. Yes, yes, indeed. So, very good. So the comment is that we might, that this is known in some limit, which is like the small epsilon limit through the cyber-wit and pre-potential. And indeed, you're right. But for this, since the sphere has like radius one in some conventions, it's not enough. So you have some partial information here. We do have some partial information about it, but not complete information. Perhaps these tools of effective filtering that I mentioned of large charge could be useful because they should really apply independently of the underlying filters. So they might be able to teach us something. Another comment is that in the AGT correspondence, this hierarchism is mapped to WN algebra generators. So that's another possible way to continue. So WN symmetry maps to hierarchism years. That's how the AGT correspondence works. So in particular, the necrosis of partition functions in the presence of these coefficients lambda should correspond to some partition functions in 2D with some chemical potentials for the WN symmetry. So it's not guaranteed that anybody would know how to compute it, but that's another formal. Can you say one more time what you said? Adding one of those operators to the Lagrangian corresponds in AGT to adding some chemical potential for the WN charge. So you have some Riemann surface, and you pick a cycle, sorry, a non-contractable cycle. You pick a non-contractable cycle, and you wrap some WN, well, you just put a chemical potential for the WN charge. So now what you are studying is some partition functions of Toda-like theories or Toda CFTs in the presence of some twists for WN. So in principle, it should be possible to compute those objects from that point of view, but that may be. What is turning on these extra coupling constants? Well, this coupling constant here maps to the chemical potential. So you have some kind of parameter here, which is the chemical potential of the WN charge. So that's how the correspondence works. This is one possible way to approach this subject. I don't know if this is technically the easiest. Any other questions? What about KDV hierarchy, which appear in large, large wavelengths limit of Toda? Yeah, the question is about what about the KDV hierarchy, which appears in the large wavelengths limit. So maybe I'm wrong, but maybe you can remind me. But that has to do with long-time behavior, right? Big wavelength. So, well, I don't know the answer, but let's see. Well, here, the driving force of the chain is kind of fixed. So the driving force is fixed. You're talking about the infinite chain, right, from both directions. Here, the driving force is fixed, and your job is to find the solution on the chain. So I don't know if there is such a limit. But there is a long-time limit, which is interesting. That's the weak coupling expansion limit here. I think that there are two interesting limits of this Toda chains. One is the long-time limit, which is the weak coupling expansion limit. And the other limit is what I said, large distance limit from the boundary. You might expect that maybe if you're very far from the boundary, it's approximated by an infinite Toda chain. And here, the tools of effective field theory, a charge should apply. So that would be also interesting to understand. Any final questions? OK. So I'm done. Thanks.