 Thanks very much for the invitation and for the introduction. It's a pleasure to be back at IHGS. And yeah, so I will indeed talk about intercomability and planar ADS-CFT, and in particular on Youngian symmetry. So let me give you sort of an introduction where this talk is situated. So this is about the ADS-CFT duality, which you probably know as a proposerality between string and gauge, or more generally gravity and QFT models. And what makes this quite interesting, at least to me, is that some integral structures have been observed in the planar limit of these models. And well, intercomability is a nice word, but it's of practical use, at least in two ways. It provides a toolkit for efficient computations that you could perhaps otherwise not do. You could also say it has been applied for these systems to confirm, well, not ADS-CFT as a whole, but at least some central predictions of ADS-CFT. We now have much more confidence due to using these integral structures. And another way of talking about intercomability is to say that intercomability is a hidden symmetry that's somehow in these models. And this symmetry is, well, at least in this case, is known as a Youngian symmetry. So somehow the goals of this enterprise are to improve the understanding of some aspects of QFT, and in particular also gravity models in terms of, well, algebraic and analytical methods that come along with intercomability that help you to obtain some or get better understanding of non-perturbative results. And well, what's also interesting from the math point of view is that you get symmetries which are realized in a somewhat unconventional way. So for this talk, well, the first 20 minutes or so will be a review of the ADS-CFT correspondence, including intercomability. And then I will also give an overview of some central achievements and where this whole thing goes. And at the end, I hope I have some more time to talk about Youngian symmetry, which is more the current work that I'm working on. So yeah, let me start simple. And maybe this also helps to relate this talk to the conference. So well, ADS-CFT, you can view as well with the goal that you want to understand how strings propagate or how they work. If the target space is not just a flat spacetime, but a curved spacetime. And in particular, once you have a curved spacetime, all the equations of string theory will be highly nonlinear. In particular, the spectrum of string states will not be easy to understand. And if you don't understand that, then it's perhaps even harder to address questions like a scattering of string states. You don't even know how to get started if you don't understand the states too well. So in this sense, the ADS-CFT duality was a major achievement because it relates a string or gravitational theory on some ADS-curved target space to a conformal field theory on the boundary of the spacetime. And well, here in this talk, I will mostly address the prototype duality between two B strings on ADS-5 and N equals 4 supersymmetric Young-Mills theory, which is one particular four-dimensional conformal field theory. And well, there are good reasons to look at this example, mainly because it's highly symmetric. It's highly accessible. But even though you have these nice features, these are nonlinear models. And it's not easy to make reasonable computations that allow you to compare these two models, because essentially, this is a strong weak duality. But anyway, let me just tell you what these models are. So on the one hand, there are strings on ADS-5 cross S5, which you can view as, well, the embedding of a two-dimensional world sheet into this 10-dimensional curved target space. From a physics point of view, this is a two-dimensional nonlinear sigma model. So it's a, well, you can, or the way I would want to view it is it's a two-dimensional quantum field theory of a particular setup. And it has two coupling constants. There's the world sheet coupling lambda and the string coupling G string. Or at least these are the dimensionless coupling constants that you can form. It's a weakly coupled model when this parameter lambda is large. Of course, also you view one over lambda or one over square root lambda as the coupling constant. But here, I'll just consider lambda large as the weakly coupled regime. And this model has a certain amount of symmetry. And these symmetries come from the isometries of this target space, which is, well, it's actually a supergroup consisting of a conformal group and SU4 combined in a suitable supersymmetric way. On the other hand, there is any quiz for superimposed theory. That's more or less a conventional, well, you can view it as a conventional four-dimensional quantum gauge theory model. So you have a gauge field. You have four types of fermions, six types of scalars. For the gauge group, I'll typically take SUn or Un. It doesn't really matter so much. But there are some particular features about these fields. Namely, all of them are massless. And all of them are in the adjoint. So everything is just an n by n matrix, all these fields. So here we are. So here we arrive at the matrices. And, well, the couplings in this model, these are the standard couplings that you would also find in the standard model. So you have non-abelian gauge couplings. You have a Yukawa-type couplings and five-to-the-fourth couplings. But these couplings are arranged in a very peculiar way to have n equals 4 supersymmetry. And at the end of the day, there are only two coupling constants, like the M-Mills coupling constant and also the V-Mills constant. And also the topological angle, which doesn't really play a role here. But the nice thing is you do have a large amount of symmetry, which is not just the spacetime symmetries or the internal symmetries. But they all combine with supersymmetry and conformal symmetry into this super conformal symmetry, which is the same Li-algebra or supergroup that we had for the string model. Supersymmetry is a nice feature. It does help. Particularly here, it protects some quantities. And in particular, the beta function of this model is exactly 0. That's why you call this a finite model. But it doesn't mean it's anything trivial. It's a very non-trivial model. But it just happens to have a coupling constant, which does not run. And therefore, also this super conformal symmetry is an exact symmetry of the quantum theory as well. It's a model which is weakly coupled for G Young-Mills being small. And you can use Feynman graphs to do any perturbative expansion. But bear in mind that this is not in the. I mean, it's a well-structured problem, but it's very hard to obtain any sort of data, say, beyond the Wandoop level. So it's even computing a diagram like this will not be too easy. But there's one interesting limit that these two models have. This is the planar limit. In the gauge theory, it's the large NC limit, where the rank of the matrices becomes very large, while the coupling constant becomes small, such that you keep the top coupling finite. And this is the coupling lamina that we have in both models. Then for the gauge theory, we have only planar Feynman graphs that survive in this limit. So whenever you have propagators that cross, they will suppress the diagram. And that leads to drastic combinatorial simplifications. But you also get a two-dimensional structure from that by these graphs. You can draw them on the plane, and then this plane will be a two-dimensional plane. Which you can relate more or less directly to the two-dimensional world sheet of the string. So maybe it's easiest to see this when you turn the number of loops to be high, and then sort of you see really something like a surface that appears. And then in string theory, you get the same limit by sending the string coupling constant to zero. And that means strings will never split or join. And the world sheet will always have a trivial topology, maybe typically as a cylinder or maybe just a disk. But you will never encounter some splitting or joining of strings or these pair of pens, surfaces, or surfaces of higher genus. Those would typically be suppressed. Maybe now is a good time to just give a summary of how you relate the two structures, namely the world sheet and the gauge theory. So for the world sheet, well, maybe if I draw it as a cylinder, we have these two directions, which are sigma as the space and tau. And this string lives in some 10D space, which is AdS5 plus S5. And on the other hand here, we sort of have a 4D gauge theory. And the question is a bit, how can you relate these two pictures? Well, and I think the key to do this, at least in the framework that I will be using, is to consider the fields. These fields are N by N matrices. And what you can do with matrices is you multiply them. And so I'll just depict it like that. All of these are, I mean, since all fields are matrices, it's easy. You can just multiply them to matrix polynomials. And maybe you can even close them. And you should view this direction as the spatial direction of the string somehow. So whenever you have a polynomial of fields where you multiply the matrices as a matrix polynomial, then this more or less corresponds to a string, which sort of goes in that direction. It's more something like a discretized string world sheet to some extent. And this will be the framework that I'll employ in the talk. And it's interesting in particular because there will be symmetries which act in a non-local fashion on this string world sheet. And then if you translate this, then this works in a somehow non-local fashion on polynomials of the fields, which are polynomials of matrices. And I think it's always good to keep this in mind in particular when you're used to talking working with matrices. Anyway, so if you wanted to do any computation in either of these theories, or in particular in the gauge theory, you would use Feynman graphs. But as I said, this is never going to be an easy problem, in particular if you are at higher loops or if you have Feynman diagrams with many external legs. And the reason for that is simply that, well, all of these legs come along with some spacetime data, for example, momenta flowing in. And if you have many momenta, you can combine them in very many ways. And that just means the functional dependence on these momenta can be rather difficult, even if you want to respect the symmetries. But the nice thing is that this gauge theory is integrable in the planar limit. And then also the ADS-CFT dual string theory is. And that simplifies some calculations that you want to do in particular. The first thing that that was understood well is the spectrum of local operators, which we now largely understand how to compute. And the nice thing is that one can now compute observables not just perturbatively at small lambda, but even at finite lambda or at large lambda and compare them to the string theory picture. For example, one example I can give here is a simple integral equation that we've derived for the cusp dimension, which I'll just mention that in a few minutes again. But let me tell you about this duality to strings. Namely, this is somehow this picture that you have here when you compose some gauge invariant local operator in the gauge theory. So you take some fields. These are n by n field, n by n matrices. You take some derivatives of them if you want. And you take a product, and then you get a gauge invariant object, for example. And depending on how you align these directions, maybe one good thing that I lost my choke. Yeah, one thing that I should have mentioned is that, of course, the 10 dimensions you also get in this picture from looking at mostly at the indices of these fields. So if you have a scalar field, there are six different types, or this one it ranges from 0 to 3. So if you combine these two, you sort of get 10 directions. And this is somehow the way you encode the 10 dimensional target space, as usual also in matrix models. I thought it was useful to say. So somehow by aligning the indices of these derivatives or the directions of the scalar fields, you would get sort of elementary string states where these directions would tell you in what direction the string excitation modes work. So that's just the rough picture, if you wish. But there is a nice observable quantity that you can associate to these local operators. This is the scaling dimension in the CFT. And in the CFT, the scaling dimension tells you how, say, a two-point function scales with the distance between the operators, and namely this exponent is the scaling dimension. And that in the ADS-CFT dictionary is due to the energies of these elementary string excitations. So much for the very basic ADS-CFT. And what I wanted to show you a bit in more detail is how you get the spectrum, I mean, these numbers for either these string states or the local operators from integrability. And for that, there is the beta-unsets technique. And I don't really want to go into the details. But what it gives you is a set of algebraic, well, or just a set of equations that you can solve. And for each solution to these equations, you'll find one state in either of the two models. And for each state, you can read off the energy or the scaling dimension using a simple formula. But I think what's interesting here is that the coupling constant enters these two expressions. And it enters them in a way where you have sort of the exact dependence. So the lambda dependence here is a very well-described form. And so you can, in principle at least, solve these equations for small lambda, perturbatively, or for large lambda, or for any finite lambda. And even this expression for the energy will then just be an analytic expression lambda, and you can evaluate it. This is not exactly that there are some approximations in these for which tools are known, how to improve them. But let me not get into that now. But you can apply these equations to maybe the simplest types of local operators that exist, namely the twist two operators. So for twist two operators, you just take two matrices. You apply some derivatives because the space-time dependence you can also take into account. And in some sense, these correspond to some tiny bits of string which rotate around an axis in this alias 5 space. And these objects have been looked at for a long time in QFT, and their anomalous dimensions are well studied in QFT and QCD. The anomalous dimensions are responsible for scale violations in deep in elastic scattering. Some evolution equations have been set up for them and using them as the famous D-Glob and BFKL equations. And what maybe excites, what's nice here is you can look at the large acid behavior. That's sort of when you turn the string slightly longer or take the long string limit of that. And here it's sort of, you can also view it as sort of pulling these two fields slightly apart. But the nice thing is that the anomalous dimension then has a particular scaling behavior. And the leading term goes like log of the spin. And the coefficient in front is called the cusp dimension. And that's just the number which you can, in principle, compute. And that. In O of s, there is a coincidence limit. It's the same point x in 40. Say again, in O s, yeah. The argument is phi of x and phi of x. Yeah, right. It's phi of x and phi of x. But once you put in lots of derivatives, then sort of it's like a tail expansion and slightly pulling them apart. Which is perhaps more easier to understand if you do it the opposite way. But anyway, so an interesting quantity, observable quantity in this theory and also in string theory as the cusp dimension or the corresponding energy where you just pick the coefficient of this leading log behavior. And you can apply these equations to determine this quantity for any finite value of s. But if you take s to go to infinity, you'll end up with a certain integral equation of this sort. So there is a certain kernel, which I'll describe in a second. And the solution psi of x. And this is the kernel that you need. You can write it using the Bessel functions j0 and 1 in a particular combination. And then there is also some term where you have some convolution of these elementary kernels. And the nice thing is, once you have a solution to this equation with certain boundary conditions, of course, you need. Then you can easily read off the cusp anomalous dimension as this function evaluated at 0. You can, I mean, the nice thing then, as I explained earlier, is that the coupling constant lambda enters this expression here analytically. For example, here, it's just a pre-factor in the exponent. And it doesn't enter anywhere else. Also enters here, of course. And then you can expand it sort of at small lambda first. And you get all these terms, which you can also compute by other means in quantum field theory. For example, by gluon scattering amplitudes, all these terms appear. And they have been confirmed up to a very high loop order. In particular, here in gluon scattering amplitudes, you have a certain expectation, how things exponentiate, how the divergence is exponentiate, and the cusp dimension plays an important role in that. So that's interesting because it also draws some connection between integrable models and scattering amplitudes, which I'll cover later. You can also do an expansion at strong coupling. And then you get a leading term, which coincides with classical string theory and then also some quantum worksheet corrections that also match. That's nice. So you get agreement with both sides of this duality. And so let me show you in this graph what the result is. So you can even now compute the cusp dimension using this integral equation at finite lambda. And then you'll see sort of a smooth interpolation between the small lambda regime, where you have perturbative gauge theory, and the large lambda regime, where you have perturbative strings. And it sort of matches nicely with these perturbative extrapolations, at least as far as their radius of convergence goes. I mean, the gauge theory does have a finite radius of convergence, apparently. But the string theory is just an asymptotic series. Nevertheless, this interpolating function, which is in red, seems to match well with the perturbative expansion. So well, that's just one particular quantity here. Here I draw the coupling constant, the cusp dimension, which was computed first numerically by this team. And the nice thing is that this you can view as an exact result in a planar four dimensional gauge theory at finite coupling. So you have this function, which is determined by this particular integral equation. Well, I said the beta equations were not quite exact, and they are not quite exact, when you have a finite size meaning when the number of matrices in your trace here is finite. Well, in this case, it's also a finite number, but you have a large number of s, and that somehow compensates. But whenever you have a finite size number of matrices, then there need to be some corrections. And that's because the scattering picture that you need assumes an infinite world sheet. So you have sort of think of, in the string theory picture, you think of an infinitely extended world sheet. And there you can set up a scattering picture, and then you can compute the scattering matrix. But once you look at the actual string states, these are on a finite cylinder. And the finite cylinder, the scattering matrix for the world sheet is not quite exact. And then you can introduce some Lischer terms, and these correspond to virtual particles, which sort of wrap around or walk around this cylinder, which is now finite. There is a whole technique associated to that, which goes by the name of the thermodynamic beta ansatz, and also there's a double wick rotation. The idea behind that is basically that if you have a cylinder, then space has a finite extent, but at least time is infinite. And instead of considering what you usually do evolution in time, you could also do an evolution in space. So instead of having this time evolution, you do evolution in space, and then your whole time axis is still an infinite line, and on this infinite line, you can set up a nice scattering picture. And the next thing that's nice about it is that this alternative picture can be obtained by a double wick rotation in two dimensions, because if you do a wick rotation on time, you get two spatial directions. But if you then also do a wick rotation on the spatial direction, you get another time direction. And then so you get from a 1,1 theory, you get to another 1,1 theory, which happens to be more or less the same. But it allows you to treat your scattering picture with an infinite line, which is what you need. From this, you obtain similar types of beta equations, but now this becomes an infinite set of coupled integral equations, which is a bit hard to deal with. And then people have used a big arsenal of techniques related to integral models. These are known by the name of TY system, Herota equations, Baxter equations, quantum curves, or finite line linear integral equations. So these are like the integral equations that I just mentioned, but just somehow there's a way to reduce them to finitely many. And doing that, you can apply this to particular states. For example, the Konishi state, which is sort of the same state that we had here, but not at the upper end of the tower, not at the large spin limit, but sort of the smallest one. And that receives some of these, well, finite size corrections. Those are very important. And then you can compute with these techniques the anomalous dimension, for example, perturbatively up to many orders in the loop here. I just displayed the first four orders, but many more are known by now. And again, that gives you a nice interpolation between the weak coupling regime here and a particular finite string solution at large lambda. I can also show you a bit what was achieved using this map of parameter space. So here's the Toft coupling lambda. And this would be the string coupling GS or the inverse rank of the matrices. And if you look at where the classical regimes of these theories are, well, the classical gauge theory is where lambda is small, but where the rank of the gauge group doesn't really matter. So you can do gauge theory for SU3. You can do it for SU20. You can do it for large n. That doesn't make much of a difference. So this is the classical regime of gauge theory. For string theory, you sort of sit in this corner where lambda is large and also the string coupling is small. So because you would always have to do an expansion in the string coupling. When you want to go to the quantum gauge theory, you typically do this by Feynman diagrams and loops. And in that way, you go from this classical regime somewhere to the perturbative gauge theory towards the quantum gauge theory. In string theory, you can do two different types of expansions. Either you look at sort of adding interactions to strings where strings can split and join. And that amounts to adding handles to the string world sheet. And here, well, when you have a curved target space, then the world sheet theory is a quantum field theory with a parameter. And you can do expansions. These are curvature corrections to the classical strings which sit here. And in this way, you can also approach the quantum string regime, which sits anywhere in the middle of this chart. The planar limit somehow connects along this line where the rank is large or gs is 0. So you're on this line. And the nice thing is that this line connects the two regimes. And this is the regime where you get integrability. Using the tools of integrability, one can now compute functions, well, interpolating functions going all the way from here to there, at least for some observables. And that's what has been achieved. Let me mention some achievements using integrability. I already mentioned there is some scattering picture on the two-dimensional world sheet. So the string theory here is a two-dimensional field theory on which you can set up a scattering problem. And well, that's what you would typically do also in strings in flat space. You look at the excitations of the string world sheet around the trivial solution. And these are geometric things because you get eight bosonic excitations which correspond to the eight transverse directions of the string and maybe also eight fermionic excitations because it's a super string. And in this particular scattering picture here, you have a residual symmetry coming from the isometries that are left over. And these are two copies of this Lie super algebra SU2 slash 2. And you have a scattering matrix which respects the symmetry. And the nice thing is that using just the symmetries, actually, you can determine what this scattering matrix for two particles is as a function of the momenta of the world sheet momenta of these particles and as a function of the world sheet coupling lamina. So you have an expression which works at finite coupling string. And the nice thing is that, well, this two-particle scattering matrix is of a particular form, which is compatible with integrability. It allows factorized multi-particle scattering because it satisfies the Eier-Bekstra equation. And so then at least with the assumption that this is a factorized scattering problem, you can derive all n-particle scattering processes only through the two-particle scattering process. Well, concerning symmetry, this scattering matrix gives rise to some unusual non-local type of symmetry, which is known as the Jungian algebra. And the Jungian algebra here is more or less the Jungian of this algebra, SU2 slash 2, or maybe U2 slash 2, or maybe some deformation of it, which is not completely under good control in terms of mathematics. But because it has some central extensions and derivations which are not quite covered by the usual framework, so investigations of this algebra are still going on. Another, well, so this is about the scattering matrix, which I just wanted to mention because it plays a central role in anything, any calculation that you want to do on this string world sheet. And you can even apply it to the computation of correlation functions. Now, correlation functions, what I mean is correlation functions in the gauge theory. So you take several local operators. Each local operator would be maybe a trace of a product of matrices at different spacetime points. And so these are the essential observables you can compute in any CFT, be it two-dimensional or high-dimensional. And then for such objects, you can also do a genus expansion. And this would sort of be a genus one contribution to a four-point function. The trouble is if you want to apply integrability, then you realize that this works best if the string world sheet has the topology of an annulus or even of a disk. These are the cases which you can address well with integrability. And this is a case which you cannot really address well. But what you should do is you should do what you usually do in string theory if you want to look at higher genus corrections, namely, you patch your world sheet together from pairs of pans. And a pair of pans is not among these, but you can even glue a pair of pans from two, basically, two hexagons. So you take two hexagons, glue these at these indicated sides together to form a pair of pans. And that's what's at the heart of what's known as the hexagon approach using integrability. So then once you have a pair of pans, you can glue this together to any surface that you want. And then in principle, you can compute correlation functions. This is still, I mean, any genus expansion if you've ever tried it, this is a difficult thing to do. But at least, I mean, the promise at least in this approach is that you can understand these patches here at finite coupling strength, at finite lambda. So that at least gives you some hope of computing these correlation functions at finite coupling strength. And then in principle, you can do a genus expansion, or you hope that someday one can do it. It might just be difficult in practice. But by gluing together these things, you can get more and more complicated remand surfaces that give you higher genus corrections. And that would be in this picture where you engage usually start with the classical gauge theory and then do a loop expansion to go towards the middle. This may open another way of approaching the middle of the diagram by starting at this line and then sort of doing the genus expansion and going from here, where you can get exact results going perturbatively towards the middle of the diagram. Another context where integrability plays a role is planar scattering engage theory. So for a long time, collared planar scattering amplitudes have been investigated. Well, because it's the natural object you would look at anyway at tree level and then also quantum corrections, loop corrections can be computed. Much progress has been made over the last 15 years, mainly using on-shell methods, but then also some geometric methods and methods using integrability. One particular aspect of scattering amplitudes is that they have infrared divergences and there's a certain infrared factorization scheme for such a scattering amplitude. And roughly, well, schematically, you can say scattering amplitude is given by the tree level times loop corrections, which you can write in sort of some exponential form. And this exponent still has some divergences, but these divergences are just given in terms of the one loop divergence times the cusp dimension. And that's where, again, the cusp dimension plays a role. And once you have removed the divergences, some finite remainder function remains. Well, so that means a scattering amplitude is expressed through this tree level scattering factor, the one loop factor, which is iadavergent, the cusp dimension, and some finite remainder function. And a long time ago, an intriguing observation has been made for scattering amplitudes with at least with four and five legs, namely that this remainder function turned out to be zero, or maybe not necessarily zero, but just a constant, so no interesting function. And this has been computed and confirmed, even at four loops using uniterity methods. And if you believe it, then that's actually an exact result for scattering amplitudes at finite lambda. So you do get, you have the option in particular, I mean, if you believe this conjecture, then you realize that you know a scattering amplitude in a four-dimensional gauge theory at finite coupling, which is quite interesting. So of course, if, well, you should ask yourself, why is this so simple? And maybe is there a way to generalize this even beyond five legs? Now, beyond five legs, at six legs or more, this finite remainder function is going to be non-trivial. But that doesn't mean that you cannot determine it. Maybe there is a way to obtain it for six particles at finite coupling. And that that would be great. Now, let me see the, yeah, I guess I'll explain why this is so in a minute. So the ADS-CFT correspondence provides a string analog for planar scattering amplitudes. So if you have a planar scattering amplitude in the gauge theory, you can relate this to the area of a minimal surface in ADS-5. And here, yeah, this minimal surface is determined by a certain null polygon which ends on the boundary of ADS-5. And the nice thing is that by using this duality, you can also use another correspondence in this ADS-CFT dictionary, namely saying that a minimal surface in ADS-5 ending on some contour on the boundary corresponds to a Wilson loop observable in the gauge theory. And that's nice because now you have a correspondence or a duality between an object, between a scattering amplitude, and a certain Wilson loop in the gauge theory. And that's something you can compute directly in the gauge theory without any ADS-CFT considerations. And it's even something which you can compare perturbatively. So there is a way to compare perturbatively objects of this kind to objects of this kind. And the ADS-CFT term would be that these two objects are related by what in string theory is known as T duality. And that's a nice thing because it also gives you some relationship to integrability and to enhance symmetries. And so let's look at the symmetries back of n equals 4 super n mils. So as I told you, it's a super conformal theory. It has this PSU 2 comma 2 slash 4 symmetry. And in this context, I like to draw all the generators of this group in this disk. So you have momentum generators. You have super symmetry generators, some rotation and scale generators, and the special conformal generators. And the point is that once you have a symmetry, then the observables typically respect the symmetry. So scattering amplitudes should be, in some sense, conformally invariant. And also Wilson loops should be conformally invariant. However, the conformal symmetry that you have here acts in a different way on amplitudes and Wilson loops. And so you have some symmetries which sort of act in the ordinary way on amplitudes, but these are not the same symmetries as in the other picture. And so you sort of get two different sets of conformal symmetries which, by this duality, are related by T duality. And so then you see, well, you have two different sets of symmetries, but the interesting point is that these are not completely distinct. They overlap on a certain subset of the generators. So some of these generators act the same on both pictures, and some generators act differently. And when you are in this situation, you can do commutators of these symmetries. You can sort of mix the two symmetries and generate new symmetries. And when you go on, you find more and more new symmetries, and you get a whole tower of new symmetry generators. And that is what is known as the Youngian algebra. So in this sense, this duality between scattering amplitudes and Wilson loops or the T duality makes, at the end of the day, gives you a whole tower or infinite-dimensional tower of new symmetries. And then, as I said, this also explains why some of these scattering amplitudes are so simple. And that's basically because you have no conformal cross ratios for four and five external particles. And that's not ordinary conformal cross ratios, but these are dual conformal cross ratios. And they act like the ordinary conformal cross ratios, but they have a different form. And no such dual cross ratios exist for four and five external particles, because in some sense, these are null directions. And so you need more than five to come up with something which is non-trivial. And that just means if you have no conformal cross ratios, this remainder function can actually depend on nothing and must be a constant. So that explains the triviality, whereas starting from six particles, you do have conformal cross ratios, and then this remainder function can now depend on them. OK, so yeah, as I mentioned, these scattering amplitudes are related to Wilson loops. One problem, if you want to address symmetries for observables, is if these observables produce divergences. And unfortunately, these non-polygonal Wilson loops, just as the scattering amplitudes, they produce some divergences. But there's a better object you can look at, namely just ordinary smooth Wilson loops and so-called malasena Wilson loops, which also coupled to the scalars in a particular way. And the nice thing about them is that they avoid having divergences in n equals 4 superegmails. And you can then act with symmetries on these Wilson loops. So a Wilson loop is a contour in spacetime. And if you act with conformal symmetry, then basically you just transform this contour in spacetime in the way conformal symmetry dictates to you. And that's how a conformal variation acts on a Wilson loop. But there is a different way of viewing this. Namely, if you look at an infinitesimal transformation, you take this contour and shift it slightly. But this shift can also be represented by an insertion into the Wilson loop, where you just shift the contour in that direction locally. And that means if you have this Wilson loop operator, which is this path-ordered exponential, you insert sort of the transformation acting on a single gauge field and integrate that over the point to have this infinitesimal transformation realized. That's just how you represent symmetry or infinitesimal symmetries. And the point is then if this is a symmetry, then the expectation value of this combination would be 0, because an infinitesimal transformation should not change the expectation value, right? What is the cycle here? What is the first integral? This is just your contour or? I'm placing you written first one. This one. Another second one. This one. This one. What is the integral? Ah, OK. Well, you have your Wilson loop. It's a closed contour. You integrate once around it with this. And then this just means you insert this in. I mean, because this is under the path ordering, a Wilson loop is a path-ordered exponential, right? So it's p trace px. And then here it's an integral over the gauge field around. I mean, this is the parallel transport of the gauge field. And here what you do is basically you insert this under the path ordering into the appropriate spot. It's just a fancy way. You could also write this as going from your start to this point, then inserting JA, and then going from there to the end. Yeah, I don't know how to explain this much better right now. But so why I wanted to show this to you is because the Youngian, these additional generators that I just talked about, have a similar form acting on Wilson loops, but they insert two of these insertions into the Wilson loop at different points. And each of them is determined by the conformal transformation. And then you somehow contract these by structure constants. And this gives you a new sort of object. And if this is a symmetry, then the expectation value of this combination should be 0. So that gives you new identities for your theory. As I said here, Youngian invariance of the Wilson loop then means that the expectation value of, well, the conformal variation is 0, but also of these additional symmetries is 0. Now, if you expand Wilson loops at weak coupling, you do this by basically inserting propagators between two points on the Wilson loop at the first order. So the first particular correction is you insert a propagator between two points and integrate overall pairs of points. And then for the conformal variation, what you get here is basically you do the deformation at one of these two points, and then once you integrate over this, you get 0, irrespectively of which contour you take. That's the symmetry statement. And for the level 1 symmetry, the invariant statement means you vary both ends of this propagator by conformal transformation, and then you contract them with the structure constants of the conformal group. And again, that after integration recklessly gives you a 0, and that means you have some additional symmetry in your theory somehow. And that means you have Youngian invariance for Wilson loops at the first vertebrative order. Now, one thing I'm certainly interested in now is computing higher vertebrative variables. Computing higher vertebrative orders and see whether at higher vertebrative orders where you have more propagators or maybe some vertices inside here, whether these still respect the symmetry, and this is in particular a question for anomalies. So let me summarize here the status of ADS-CFT integrability. So as I've shown you, implications of integrability have been understood quite well for several observables. They have been applied to compute finite coupling corrections. Interestingly, some very well-defined math concepts of quantum groups appear at leading week order, and some vertebrative corrections in terms of math are under control. And it's also understood, well, what is the integral structure of classical string theory on ADS-5 process 5? But altogether, it's not really known or was not really known what is integrability for this model at all. In particular, how can you define it? And you would have to define it before you want to prove it, and then how can you prove this integrability? And as I said, well, the math concepts at leading week order are under good control, but what happens at higher loops than at finite coupling? So many results have been obtained, and they are trustworthy. But of course, we would like to know for sure that all these techniques work, and for what reason do they work? And so let me then talk about this Youngian symmetry of this theory a bit before stopping. So if you think about it, what you should be aiming for is to show that the action of your model of N equals 4 super young mills, or maybe correspondingly of the string theory, that this has a certain extended symmetry, namely a Youngian symmetry. And I'll just call J hat some particular Youngian symmetry generator. But the question was, for a long time, how can you apply this symmetry generator to the action in order to verify the statement? In particular, if you look at the action of a gauge theory, it's hardly clear what you mean by the planar contribution or non-planar contributions to the action. How can you distinguish them? And that's in particular relevant because you expect integrability to be there only in the planar limit. So whatever statement you make must only be valid in the planar limit. And also, which representation of the symmetry should you take? Maybe some free representation or non-linear or quantum. How does that work? But at least there have been some hints that some statement like this should hold. And one reason is that you can look at Wilson loops and the Wilson loop, as I said, does display Youngian symmetry. And when you do a Wilson loop OPE, that is, if you take the Wilson loop to be smaller and smaller and do an expansion around small radius, then in some order you find the Lagrangian of the theory. And so if you think, well, the Wilson loop is invariant, then perhaps the Lagrangian should also show some signs of integrability of this invariance. And there are some essential features of the action that, in principle, make this possible. The action is single trace. So in the large n way of thinking about things, it has a disk topology, which, as I mentioned earlier, is something which is nice for integrability. The action is conformal. And that's also a requirement in this particular setting. And also, maybe something that is very useful is that the action is not renormalized. The coupling constant is not renormalized. The beta function is exactly 0. And that maybe helps in establishing that there are no anomalies even for this symmetry. But we didn't know what this statement actually means. So we, first of all, applied this symmetry generator to the equations of motion and wanted to see whether whatever it produces, if you have a symmetry, whatever it produces, it should again be equations of motion. In particular, if you want a consistent formulation of a symmetry, this statement really needs to be true in the quantum theory. And the Dirac equation is easiest. So let me schematically show you the Dirac equation. It's d slash psi, where this is a covariant derivative. Plus maybe this Yukawa term. So if you write it out, it's these three terms. And now you can act with some bilocal level one generator on these fields. And you get certain numbers of contributions. In particular, those here, those are very non-trivial. And these are the non-local corrections. And but their form is precisely known. And what I mean, then we looked at this equation and we found some way of putting a consistent, well, I mean, these expressions were not known. But we could invent something so that this equation holds. And not only this equation holds, but all the other equations of motion did display the symmetry also for all the other generators. So we could establish that the equations of motion are young and invariant with a consistent set of, well, representations. Now, if we have an exact symmetry in a quantum field theory, that usually gives you some Ward or Wartah-Kahashi identities. But what you need is that the action is invariant. And the invariance of the equation of motion would not quite be sufficient for that. And in addition, you would want for the quantum theory that there are no quantum anomalies. So in principle, quantum theory could invalidate some classical theory. Quantum theory could invalidate some classical symmetries. So we wanted to show the invariance of the action, but there were some difficulties, namely, cyclicity and non-linearity. Don't have a good chance to explain what the problem is here, but using the invariance of the equation of motion, which we had already established, we could construct this j hat on S in such a way that it's exactly 0 for n equals 4 supernmills, and it's actually not 0 for other models. This way, the symmetry acts as some very unusual features. Somehow the coefficients depend on the number of fields they act on. You have some strange overlapping bilocal terms, and also the gauge invariant of this construction is not really manifest. But the nice thing is, it does give you 0 after you cancel many terms. You get 0 in classical n equals 4 supernmills, and you don't get 0 in other gauge theories that may be n equals 2 supernmills. And that's basically our eta straight on the field. Well, I don't show it here, but it's a few lines of expression, and it's non-linear in the fields, and you need some framework to write this down. But yes, I mean, it acts in a very well-defined way. You don't get it where n equals 2, you said. Well, so for example, pure n equals 2 supernmills is not expected to be integrable, and the symmetry does not hold. Even the confrower of f is equal to 2nc. Well, there are some quantum conformal n equals 2 models, and there you do indeed get a chance of having integrability for the reason that n equals 4, you can often somehow obi-fold or reduce to an n equals 2 model, and then sometimes you can preserve integrability. But these are indeed the cases where integrability is most likely preserved. But what I meant here is if you just have pure n equals 2 supernmills without additional matter, then it fails. I should really get to an end. Just let me say what else we've done. We've looked at correlation functions of several gauge fields, and then the Youngian symmetry implies some Ward-Takashi identities, which you can sort of sketch like this. And we've verified this, the problem in conceptual problem is that once you compute correlation functions, you need to fix the gauge theory. And once you fix the gauge, this fixing of gauge may be in conflict with the symmetries. So we had to look at BST gauge fixing. Well, at the end of the day, the structures are much more complicated, but you can still write Ward-type identities and show also that the gauge fix action is invariant. And one thing that remains to be settled is whether you can have anomalies, whether the symmetry has anomalies and whether it can have anomalies. I mean, tools for investigating anomalies are well known, but it's not so clear, because here you have a non-local type of symmetry. It's not really non-local in spacetime. It's rather non-local in color space, and sort of refers to what I have here. So you have a symmetry which probably just acts locally in spacetime, but non-locally on such polynomials and sort of correspondingly in the string theory, where she did act non-locally. However, I mean, I don't believe that there will be anomalies, because we know that integrability appears to work well at finite lambda. So if somehow integrability would be lost, then somehow these results should be wrong, but they seem to be OK. So one way we want to address this is consider the Wilson loop at higher orders, and then we get some additional contributions which potentially can see the anomaly, but we'll have to regularize that very carefully, and that actually produces many, many terms of this kind and see whether this all sums up to 0 for generic contours. OK, yeah. So my conclusions, or just what I did in this talk is I reviewed ADS-CFT integrability, shown you the motivation, and some constituent models. I've told you what is the planar spectrum and how you can obtain it at finite coupling, and also further progress in terms of correlators, scattering amplitudes, and Wilson loops. And in the latter part of the talk, I've discussed Youngian symmetry of planar and equals 4 supian mills. And in a nutshell, we've shown that the equations of motion and the action of the model are invariant. So in that sense, classical planar and equals 4 supian mills is an intergovernable model. We've set up Ward-Takashi identities, which arise due to this extended symmetry. We've shown that the symmetry is compatible with gauge fixing, but it actually does introduce some new interesting BRST-type structures, which I guess are not under very good control mathematically. And we expect that there will be no quantum anomalies, but that still needs to be settled. Thanks very much. Questions, comments, thank you. Maybe on this last topic, you mentioned the Ward identities. It's catchy. Sure, sorry. For example, for a three-point function, does it change the number of traces? Meaning the Youngian operator just keeps the number of traces three. No, sorry. This might be confusing, but here I want to look at correlators of individual fields. The thing you do in your quantum field theory, one textbook, correlators of three or four fields. And since these are colored objects, you'll need to fix the gauge to make sense of this. The space is not the invariant operator which you wrote before. No, no. I mean, these are in some sense at the heart of them. You can always patch them together. But then you have coincident points and that introduces some other problems. Would the number of traces change if you looked at it? No, no, sorry. There is a reason why we do this because we like the disk topology. Whereas if you had three, four traces, then you would have to connect this by a higher, not really a higher genus surface, but a surface with holes. And that leads you to some situation which integrability does not really address. So we can only do disk topology. And that's why we need these types of correlation functions. Martin? I'm not an expert at all. But since you talk about integrability here, so is that the usual sigma model integrability with values in the symmetric homogeneous? I mean, the string theory picture is exactly that. So it corresponds to these old works by Paul Meijer. And so the Jungians, you have a classical R matrix with a spectral parameter. So as I said, this is all under rather good control in the string theory and the classical string theory. But the picture in the ADS-CFT dual gauge theory is not so clear. And that, I mean, it actually more relates to quantum integrability, whereas this is classical integrability. I also sort of, when you said the non-procure. How about the question? I'm wondering, so how much of these results go through if you switch on a v field so that the n equal to 4 becomes non-committal if n equal to 4? Which is in a sense a matrix model? Do you know anything? I don't know. If you're lucky, everything. Well, OK. OK, there is a nice thing. What you can do on the string theory, you can deform the string theory background. And you can deform it in such a way that you get sort of the two form field. And then, if you then transform it to the gauge theory, you would expect it to be a non-commutative kind of gauge theory. It's just that this type of, I mean, you need to do the right transformation. And then you would have to have a nice, well, non-gauge theory on a non-commutative space, which I'm not really too familiar, but there is an integrable deformation that corresponds to it. So I would say some models of this should exist. Maxime, I have just a very stupid terminology of question, but most of them from the very beginning. I'm not an integral system, but what I heard about Youngians, it's horribly complicated. Out of this core product, it's not liable at all. And when he said it's excellent for doing it really core product, was it? Oh, yes. Yeah, yeah. I mean, OK, I tried to make this non-technical, but I should have maybe this. Well, OK, I was here. This is the core product that you need, because here you just have a local action, and this is the core product of the Youngian. And second question, if it's not liable, how it can add some space of classical solutions? I didn't say that. I did not. No, it's the symmetry of the classical theory. I didn't say it acts on classical solutions. Equation of motion, serrated sets of equations of motion. It transforms. Yeah, I mean, it would be interesting to discuss what this actually means. How can it act on something completely? Well, the point is that you need to look at polynomials of the fields, and then the ordering within these polynomial matters. But I'm not sure you can make it act on solutions. It's a good question. I just don't have a very, I think we should talk about it and find out the answer. Thank you. You just want to. You restricted your considerations to SUM. Yeah. Could I not have considered that SUM? You could probably do that, because in the large end limit, there is not too much of a difference, and it would just somehow project out half of your states or something like that, which are invariant under this automorphism. I think that there have been investigations of that and still works, as far as I understand. Yes. But it doesn't change much about the interval structure.