 Yeah, thank you very much for organizing the conference. It's a great pleasure to be here again. It's always fun to come to Trieste, so it's a very nice and productive. So this is a work in progress, and the work which has been published and work in progress with the several people listed here. And then, yeah, I'll mention contributions from different papers, which are some of them I already in archive, and some will appear there soon. So I will discuss large end limits of supersymmetric gauge theories, and what we can learn from them, from them by studying various dualities, or what can we learn for mathematics, if interested in some mathematical applications. So it's essentially a physics talk, but I hope that mathematically oriented part of the audience will enjoy it as well. So it's always, it's hard to satisfy everyone, but I hope we'll manage to do some of it. So we know from many years of studying gauge theories that if say you have a gauge theory with a gauge group of rank N, and then the front becomes large, then the theory admits a different effective description using different degrees of freedom, and normally in this limit theory, it's easier to understand, because those degrees of freedom are somehow combined together of different phenomena that theory possesses, but when N becomes large due to various simplifications from both physics and mathematics, we have a better control of the theory. In addition, when the theory is supersymmetric, we even expect to get even better understanding of the theory, so there's a hope to solve it exactly. And of course, there are many examples in literature, in two, three, four, and in higher dimensions, when large and limits are extremely useful. And we'll view this paradigm of large N theories in the context of three different branches of modern mathematical physics, so the study of integral many-body systems, which I guess we discussed early in the program. I haven't looked at the slides yet, but I'm pretty sure that people talked about all three different corners here. So there's a triality between the interval systems then their descriptions using geometric representation theory and sort of physics of unacoustic gauge theories, which sort of break through in this, in unacoustic theories in early 90s, some kind of encouraged development both here and here. And interestingly, the large N asymptotics is manifest in all three descriptions, and I'll try to convince you that this is true, and hopefully we'll be able to see it ourselves today. So in more detail, here we have, we'll be studying some exactly soluble systems we have with N particles, for example, collager model or its generalizations, and then in the one number of particles becomes large, it admits a thermodynamical effective description in terms of hydrodynamics, in terms of velocity fields. This hydrodynamics is famed with the integrable and it's called intermediate long wave system. So that will be one manifestation. Then on the sort of mathematical side, we'll look in various algebras and their representations. For example, double affine Hickey algebra for GLN, which was introduced by Terednik and studied by other people. At N-Gouston-Shinty, it turns out to be a different algebra, which is also known mathematically, and it's called elliptic whole algebra. And it has connects to many things, like BPS states counting and quantum K theory of N-Kazimov of quiva varieties and so on. And then it will admit also some deformations both here and here, and most of them are not even known mathematically. And then in terms of physics, we'll have a stable, large, and limit of, so we'll be computing some supersymmetric observables in the certain five-dimensional theories with possibly topological, with possibly defects. And we'll see that at N-Gouston-Shinty, they'll admit some nice description in terms of a different theory. Or N is already gone because it's infinite. It shouldn't be there, but it will be some E1 5D theory. So let's start with the top corner of the triangle. So we'll look at, we'll focus on N-Gouston-Gage theories, which admits cyberquit and description in the infrared. Again, I'm pretty sure that we discussed, I'm just gonna summarize it for consistency. And thanks to original computation by Nikrasov, in early 2000, we now can compute observables in ultraviolet and study their properties all over the RG flow, including the infrared. So we can get the well understanding of the macroscopic description of Gage theories from the ultraviolet description. And then, so Nikita did it for four-dimensional theories with the simple configurations. And then since then, his work has been greatly extended to different supergravity backgrounds. So he did the computation in the so-called Omega background, whereas of course, you can do it in Alpha spheres, and as many of you in the room know, you can take not a single Gage group, but many and combine them into quivers and still compute the cyberquit and curve for quiver Gage theories. You can go in the dimensions up and down, so you can take a theory in our four-course Sigma where Sigma is either a point or a circle or a elliptic curve. And then you can go to the dimension down. So, and my illustration today will, oops, my illustration today will have two examples, the three-dimensional theory on the, see as a Euclidean three-dimensional theory on a complex plane with prime, or make a background prime, three-dimensional one times the circle of radius gamma, and five-dimensional theory on our four-course, the same circle, with the two parameters epsilon turned on. So I'll start with the, so this is a sort of canonical example in this approach, which is the n equals to star quiver Gage theory on C cross S1. So let's first have done, yeah, so let's not think about this epsilon as, let's say epsilon one is zero for now, but we consider the following quiver Gage theory. So it has a, its Gage content is U1 cross U2 cross N so on across UN minus one. So the circle means it's a U group, a unitary group. And the last node has a UN global symmetry. And to start, means that it's, it has the same matter content as n equals four theory in three dimensions, but the supersymmetry is less because the r symmetry is deformed and the deformation of the parameter of the r symmetry is controlled by parameter t, which is an exponentiated mass parameter over here. So I'm, I'm skipping the dimension sort of, I'm assuming that the radius is scaled to one here. Now, and let's do the, do the simple localization computation with this theory. Now the Lagrangian depends on several parameters. It will depend on masses of the, masses of this n hypermultiples and on the Feyyay-Leopold's parameters, which are a billion Gage couplings in each of the Gage groups and let's look at some simple example. So let's just take n equals two. So it's a U1 theory with the two hypermultiples. So, and then the relevant part of the partition function, which is not even the full partition function, but yeah, anyway, relevant, relevant for my discussion is given by a Q hyper geometric series of, so it depends on the rate, on the Feyyay-Leopold's parameter of this U1 or equivalent to on the Keller class of the Cp1 because we can think of this as the, as an academic group of variety, this is the cotangent bundle to the complex, to the complex projective space. And it has three parameters that T, so this ABC in the hyper geometric series, they depend on the two star mass and on the fundamental masses of the, of this two hypermultiples. Now, if you sort of, and the Q is the exponentiated omega background parameter. So, this is a hyper geometric series. In other words, it's called a vortex partition function. So, it's a sum over all topological sectors with all number of vertices starting from zero to two infinity. And a remarkable thing about this function is that it satisfies, it's a eigen function of the integrable system, which is called trigonometric or is in our Schneider, Schneider system. So, in other words, if you take the operator, which is a shift operator in a failure plus parameters, which are parameterized this way, and weighted by some rational functions of T and tau, or trigonometric function of, well, log tau and the mass, that's why it's called trigonometric, although we can write it as rational function, acting on this B, on the vortex partition function, gives the sum of the masses of the hypermultiple. So, in other words, the integrable systems, the first space of the integrable system is the parameter space of the vacuum manifold of this gauge theory in three dimensions. And this is, you can generalize into more generic, so you can consider more generic for the full, what's called T-unclever, whose geometric description is the cotangent bundle to the complete flag variety in n dimensions. The same formula is true, so you have a family of operators, n of them, so then corresponding to n different Hamiltonians in the integrable system, they act on this vortex partition function or homework block, how some people call it, of the gauge theory, and the eigenvalue is the VEV of the Wilson line of electric line operator in the corresponding representation where the degree K corresponds to the skew symmetric power of the fundamental of U.M. Right, so it's also been known for some while now that those Hamiltonians, they can be sort of quantizations of certain vacuum expectation values of two vortex loops in this three-dimensional theory. So if this is a Wilson loop, then on the left, it's like a two-flop, so it's composed of magnetic degrees of freedom and sort of, this equation shows that there's a conjugation, if you hit with the sort of magnetic and electric variable sign in some sense canonically conjugated to each other, so one of them can be thought of moment and the other can be thought of coordinates, and there is a very precise way to make this happen. Additionally, the statement itself can then be understood as, so we can think of the three-dimensional gauge theory that we have here as a sort of as duality wall in four dimensions, and then if you can take a four-dimensional theory with coupling tau on the left of the wall and coupling minus one over tau on the right of the wall, and then you could just require that the three-dimensional theory that lives on the interface has to be exactly this guy here. And sort of many nice properties of the theory follows from this construction, but I will need them at the moment later. So mathematically, what we have found is that there's an additional ingredient which is not present in gauge theory per se, but sort of if you can construct it with the additional input that the Hamiltonians of this n-particle trigonometric model, they form a commutative subalgebra in double affine Hegge algebra for GLN. So I'll give you an example on the next slide, but essentially there's a subalgebra in Dachau which is formed by those Hamiltonians, and then you can study representations of all Dachau just by inducing representations from its commutative subalgebra. So let me be more specific on that part. So let's consider the n equals to start gauge theory on Earth three-crosses one with gauge group UN, which is exactly the four-dimensional lift of that theory, because we are adding extra direction and you'll see why we need that. Now, the modular space of vacuum of this theory is characterized by vacuum expectation values of line operators that trap the compact circle. And of course, there are several. There's an electric line, there's a magnetic line, and there are various dynamic lines which sort of have both charges turned on. So let's see what this is. In this case, equivalently you can think of it as the modular space of flat GLNC connections. C means the complexity. On the torus with one puncture, and this puncture has to be simple, meaning that it's a monodromo matrix has only one distinct tagging value. And the claim which was made by Alexei, Oblomkov is that, at least for rank one, like for SO2, the deformation quantization of the space, in the same sense that I described before, namely that if you declare that magnetic variables are canonically conjugated to electric variables would be obvious sort of quantization condition. Then this quantization of this, deformation quantization of this modular space would be spherical double affine Hickey algebra for GLN, for AN. So let me show you the example. Let's do SO2. So just trace out the GL1 part. Now, we have SO2 flat connections, so there are three generators in the algebra. It's the polynomial around the A cycle, polynomial around the B cycle, and you can either take the polynomial around the puncture, or we can take the polynomial around the A cycle and then B cycle again. And then polynomial around the puncture will be a linear combination of these three guys. And if you work out the algebra, yeah, so then here X would correspond to the VF of the Wilson loop, which in my previous notations was something like mu plus mu inverse. If you say that mu one is equal to mu and mu two is equal to mu inverse. Then the operator Y is the trigonometric Reusner-Schneider Hamiltonian. It's also known as symmetric magneton operator. And the classical, the modular space is this cubic. It's a cubic in C3, so given by single equation in three-dimensional complex space, it has dimension two. So for rank one, the dimension of the modular space is two. And V corresponds to the polynomial around the puncture, so its eigenvalues are V is basically T and T minus one. Good. Now the quantization, now it's, now basically declare some commutation relations, basically saying that in this case, quantization is really obvious. We just take X and Y, Q commutator is equal to Q minus Q inverse times Z. And then plus cyclic permutations of this relation. So there are three commutators and they all look like this. And you can clearly see if Q becomes one, then everything becomes commutative again. But this is not the full story. So you need also to specify the Kazimier, well, it's not really Kazimier, but in case of Dachau, but something that is central. So you can show that the, basically, you can see that this omega, which is again cubic and X, Y, and Z, can be thought of as a certain quantization of the left-hand side, left-hand side of this formula. I might have messed up the sign, I think there should be a minus in front of the cubic term. But essentially you need to put Qs in the right powers in the right places and you get the Kazimier. It turns out it's the generic story. And the Kazimier has to satisfy some relation, which I'll connect to later, but it has to be essentially, in general, it depends on the eigenvalues of X, Y, and Z, whatever they are, but then something nice would happen as we specify the various Kazimier on this locus. So this is very nice, how does it appear physically? So it's intuitive, you can understand it very clearly. So again, quantization in physics language corresponds to introducing omega background. On a previous slide, here there was no omega background, it was just R3 cross S1, and we got the modular space of flood connections. Quantization supposed to take in R2 inside of R3 and then introducing the equivalent deformation for this R2. So you get R2 epsilon cross R cross S1. And it's useful to think about this R2 is the cigar. So there's a tip and there's S1 action at infinity, which rotates the cigar and Q, again, is E to the epsilon one. And now we can reduce along the action of the circle. So the cigar becomes a ray, so it's, we live in a half-dimensional space, ignore these Bs for a second, so think of it as the kind of infinite line and then there's a half-plane above and this is the world volume of our theory. And the circle is not shown, so there's basically half-space times R, half-space times the circle. And now those line operators, they wrap the circle over here and they're forced to sit at the tip of the cigar because of the way omega background works. Now, which means that in this picture, they all have to sit at this bottom, at the real line, which is on the boundary of the half-plane. And now, because they sit on the boundary, you cannot move them around anymore. Basically, when you have two operators colliding and you want to understand what it means to sort of permute them, there's a price to pay, basically you need to add some power of Q and introduce the Q commentator. So this is a sort of very generic approach of see how various affine-hacky algebras appear in this deformation quantization story, questions and so forth. Yeah, and then what about those brains that if you're going to study representations, you have to put some boundaries. So it replaces R by an interval or by a half-interval. Then you can have some left boundary and the right boundary. And the sort of the representations of this data of this algebra would be certain, can manage certain sort of home spaces from various brains, like from this BCC brain to B brain, which lives on the boundary, and brain corresponds to a certain boundary condition. I'm not going to go further into this. It's still work in progress, but essentially to understand algebra, it's enough to live in an infinite dimensional space. For instance, representations, you have to introduce boundaries. And then different boundary conditions was corresponding to specifying the weights which come from the monodromyagin values. So this is a story about how it appeared. Now, we need to generalize our setup to elliptic models. So we take, not to describe more generic system, we can introduce extra parameters. So before we just had a Q and T. So T was the exponential of the mass and Q was the omega background parameter. How about introducing one more? So turns out there is a very canonical and nice way to do it using defects. And the trick is to take our favorite theory from the first slides and gauge the global symmetry at the last node by the gauge group of a theory in five dimensions. And we match other parameters as well. So the two star deformation of this theory corresponds to the mass of the hyper-multiplet in this theory. So it's basically n equals two in five dimensions. So it's again, it's this theory with the maximum supersymmetry and we deform it by half by turning on some mass parameter. So we have, in other words, we have a five-dimensional theory in C cross C cross is cross the circle. And then there is a defect, three-dimensional defect that lives in the first factor and the circle. And it's a point at the origin of the second factor. And the sort of, once we've done that, we sort of, we can think of it as the deforming the, deforming the algebra of chiral observables in three-dimensional theory. Basically, if you have a BPS correlators between two operators, denoted by this green dot, there are two, if you think about it, if you think about it perturbatively, there's exchanges governed by the Kaplan constant of 3D theory in the T and also by the Kaplan constant of the five-dimensional theory in the P. There's like exchange through the bulk, which you have to take into account. And luckily, there's again sort of, using the construction of ramified instant ones, it can be done very robustly. I'm skipping many details here, so that we did it with a paper with a Hitchel and Matt Bulimor a couple of years ago, which was based by work by Satoshi Navata and then used some results from Aldi at the Chikawa and also Okunkov and the Braverman, the Schindelberg and the Kajima. So it goes quite far into the math literature, but we did explicit computation for five-dimensional theory. And the generalization of the integrable system was very straightforward. So the operators, so the 5D, 3D partition function, which is now a function of all these parameters, including the five-dimensional gauge coupling, is again an eigenfunction of the elliptic integrable system, whose operators look similarly to what we have, except that instead of trigonometric functions, we have elliptic functions. And the ellipticity parameter is controlled by the 5D gauge coupling, such that if you kill it, then you go back to the trigonometric version and then in terms of this theory, it means that you completely decouple 5D degrees of freedom. And the eigenvalue again is the same vacuum expectation value of the Wilson line corresponding representation. And to make this equation work, you have to send epsilon 2, the parameter in the complex plane, which does not belong to the whole volume of the defect, to zero. So that's where the integrability happens in the necrosis of the Schrodinger limit. If you go away from necrosis of the Schrodinger limit, you get something more generic, which was described by Pest and Akimura, but if you have time, you can comment on that too. Good. Questions? Yeah, so just to give you a glimpse of what this function looks like, now, the ED, so when P is zero, we have a single series, so yes, it's a double series in kilo parameters on the defect or the failure pulse parameters, and in the sort of inverse kilo parameters, and sort of they're weighted by the gauge coupling of the five dimensional theory. So you can see that if P goes to zero, we go back to the single hyper geometric series that we had before. And similarly, the Wilson line, so here, so what stands on the right hand side, it's again a certain equivalent localization expression, so here we have, so the partition function stays on the bottom, it's a normalization, and then to compute the Wilson line, we introduce the fundamental equivalent character, which accounts for the heavy BPS particle that propagates on the circle. And again, it's expression is basically the three dimensional version, the sum of the masses, and then it's multiplied by P series, which sort of accounts for instanton corrections from the bulk description. Just to summarize what we have discovered so far, there's a table of dualities between, yeah, and then this model is called elliptic-orisonar-schneider, so the trigonometric-orisonar-schneider and elliptic-orisonar-schneider, RS. So in elliptic-orison we have N particles, positions tau with interaction tau, and the plant constant, which is the shift prime to Q, elliptic deformation P, and it's mapped onto the, well, it should be P here. Yeah, I don't know why it didn't show, it was showing it on my slide. Anyhow, we can see it here. So the number of particles is mapped to the rank of the gauge group, or the flavor group in three dimensions. The FI terms are mapped to the particle positions, and the mass is mapped to the coupling, Q is the omega background, and the elliptic parameters, the 5D instanton parameter. And again, eigenvalues and eigenfunctions are given by Wilson-Lupes and the partition functions. Now, I promised you to show something about large N. Now, to understand large N, we basically need to take separately the large N limit of the eigenfunction, and I like large N limit of the eigenvalue, and the operator to be precise. And that has to be done separately because I thought I convinced you with the Daha story that this brain picture with Daha is that algebra and the Hilbert space, they go separately, so you need to understand limits in each sort of unit. I understand large N limit for each of them. So let's start with the states because it's somewhat easier. So let's consider a partition lambda of number K is just less than N, and in this approach, I assume that the P elliptic parameters turn off. And then I'll just show you the answer when it's turned on. And now let's make a specification of the mass parameters. These guys are, as I told you, weights of Daha representations eventually, to the following local. So it's Q to the lambda A, where lambda A are columns of the partition lambda times T to the N minus A, okay? And so this will be for this flag, for the flag theory. And interesting thing happens. So recall that the Q is the E to the H bar and T to the E to the mass. And then if you remember the expression from the Kazimir in Daha, you can see that the specification of this masses, as so, would somewhat remind you the expression which you had for omega there. So in fact, it's exactly the same sort of locus where representations of double-fine Hickey algebra, which are in general, infinite dimensional, they truncate. And then you can try to classify them what are the all possible conditions of truncation. So it's, again, it's a nice geometric representation theory story. But anyhow, so just we don't need it at this slide. What happens if you do this specification is that the partition functions, the vortex partition functions become polynomials. And not just any polynomials, they become McDonald's polynomials. Because it's not surprising because the operators are McDonald's operators and just looking at again functions within different Hilbert spaces with different boundary conditions. So if it's a polynomial function, it has to be a McDonald's polynomial for the partition lambda. So let's take an example of k is equal to two. There are two choices. We can have a column, a row and we have a column. The exact form of course depends on n. So for example, for the column, we have a degree two polynomial which is symmetric in tau one and tau two. And it depends at Q and T as a rational function. Of course, when it explicitly depends on n, it's hard to send n to infinity because you have to compute any n set up by hand. So let's do something more appropriate in this case. So it turns out that you can make a change of variables who can give up symmetry of the polynomials than the SN group action is not manifest, but instead it will be easier to take large and limit. So let's take this change of variables, p sub n. I hope you want the p in the elliptic coupling parameter, we're not gonna, these are different p's. We have this monomial change of variables from there's some from one to n. And then in this formulation in terms of p is the shape of McDonald's polynomials only depend on the partition and does not depend on the n on the number on the degree in the previous slide. So you have this one for the row and this one for the column. And since it's true, it's more or less straightforward to construct the space of states in our system. So we start with some folk vacuum over here and then we hit it with creation operators and creation operators will be built by P lambdas. So we have a table lambda with the columns from one to L then just act with a string of corresponding creation operators on the vacuum to create a state. And because we can construct a McDonald's polynomials arbitrary degree by truncating the weights at arbitrary higher position, then we'll be able to sweep out the full Hilbert space of states. And you can work out the algebra that from the orthogonality of McDonald's polynomials, this operators has to commute as such, so it's a deformed Heismarck algebra which has two parameters Q and T in the right-hand side. And then the relationship is more involved if you turn on the elliptic parameter P. Good, so I just summarized what I said. So in sense the vortex series which comes from the 3D gauge theory, it encodes all the states in this Hilbert space. Of course, if you have 5D theory then you have to basically do the same but then there will be series instead of polynomials. And I'm gonna skip it over here. Question? Right, now to understand eigenvalues we use, we can employ the free boson realization of this integrable system which was sort of developed by Dina Haran and then some follow-up papers. So basically we introduced two currents. One of them is the, so we should call eta, which is a normal ordered product of this exponential. You can also write this as a P exponential as in Eric's talk. Then there's another current phi which has a different function here. And then we have n variables then we just sort of package this phi together. Now, and then, so we create a state which is just the action of this phi n onto the fork vacuum. And it turns out that the state is an eigenfunction of two operators at the same time. The operator which is the first coefficient in front of z to the first power in this current. So after you expand it, do the normal ordering, we'll get some operator. You can see that its eigenvalue is the same as for the McDonald's operator with the corresponding data weighted by this rational function and plus t to the minus n. And now, assuming that the t is inside the union circle, then you can take the large n limit here. So the first factor disappears and then we get a nice stable limit out of the remaining piece. So later I'll show you what exactly this expression is, but the claim is that the limit exists and it's finite and you can compute it for any values of parameters. In the elliptic case, you need to modify the currents a little bit. So I'm showing you only eight apart. Basically you need to take this operator and then take this exponential and multiply it by another one, which depends on p. And again, you do the same game, you take normal order product. The phi is even more complicated, I'm not showing it on the slide. After you have done the same, you can take the large n limit to get, yeah, so then the expression will be replaced by more complicated function involving p parameters. And again, this is the stable limit. So, and then sort of just to connect with the first part of the talk, you can see that the vacuum expectation value of the Wilson loop, say in fundamental representation at the lock with lambda, lock was given by the partition lambda, is mapped to the, with proper normalization, is exactly this expression on the right-hand side. So in other words, so we found the recipe to compute the large n limit of the eigenvalues. So we take the Wilson line, multiply this by this function, by the simple rational function of t, and take the limit n to infinity. So we'll get the finite answer. Now, I don't have much time left, so I thought that would be good. Yeah, I just, I thought I'm finishing at 2.30. So I just want to make a little digression to describe what is the equivalent description of the integrable money body system at large n, and what kind of energies have we just computed? So in other words, if there are infinitely many particles, what else does this do deformally, which I showed in the previous slide described. And the answer is given by the so-called intermediate long wave model, which is a one-dimensional hydrodynamical system. So we have two fluids in the channel. It can be either infinite, infinite, or it can be periodic, so we can sort of put the periodic boundary conditions. And we study fluctuations on the interface between the two fluids. They have different densities, and one of them is bigger than the other. The height of the channel is given by H, and it's all happens in the gravity field. So, and depending on the wavelength of the fluctuations, there are three different regimes, which are quantitatively different and qualitatively. So when it's lambda is much bigger than the height, it's so-called long wave regime, and it's described by famous Cartvector-Riz equation. In the short wave regime, we get this so-called Benjamin-Ona, and in the full generality, when these two prime of the same order of magnitude, we have what is known as the full intermediate long wave system. Quantitatively is described by integral differential PDE, well, PDE, whatever, whatever this guy is. So it's basically the value, the, so we introduce the velocity field, which depends on coordinate and on time, and equation is ut equals 2xx minus sort of Laplacian applied to the integral transform of the velocity field, such that the kernel of the integral transform is the wire stress zeta function. And then again, there's some ellipticity parameter P here, which, as you might guess, will later be related to elliptic parameter-engaged theory. So this kernel admits some nice limits. In one of the limits when, yeah, in one of the limits to get Benjamin-Ona, when the wire stress function is replaced by a rational function, so there's a representation when you have an infinite sum of when the denominators are shifted, and in this limit, only one piece remains. And in the KDV limit, you get some derivative of the delta function. So, and because of the delta function, actually it can do the integral. And that's how you can get the uxxx in the right-hand side. And it's a good exercise to show it explicitly. But what's interesting about this system is that it's integrable for, even in the most generic case. So if you have, if you declare the Poisson commutation relations for equal time velocity fields as the derivative of the delta function, then you can rewrite the, you can rewrite the LW equation as the evolution equation, where I2 is the Hamiltonian, which in this representation is a very non-local operator. So it's integral of the, integral transform of the function itself. So it's genuinely non-local. But nevertheless, so I2 is non-local and all higher Hamiltonian cells are non-local, and they all commute with each other. They're all in evolution. Turns out that we don't need the whole story. We don't need the full spectrum of LW. Rather, we only need that, when we look at the solitons that propagate on the interface. So we'll be looking at the solitonic spectrum. Now, one can impose this n that, which is very restrictive. So we have, if the velocity field factorizes as such, so it's like, it's a sum over from one to n of rational function, which has a pole and position of the pole is time-dependent. So basically there are n particles, which describe centers of those solitons and they move around in the fluid. And then you can take this n that, plug it into the equation, the LW equation, and see that the consistency condition for this n that's to go through is described by the equations of motion for, in this case, in the Benjamin-Olon case, by Collegio-Moser model. So this is basically, Africa calls MA equation for this, for particles in this model. So it's the acceleration of the particle given by the force, which is the basically nabla V, but the potential is inverses quadrat, depends on one over A minus A squared. So the poles describe propagation of solitons. So if you take, now we need to take a generalization instead of taking the LW model itself, we take a difference LW. And difference LW corresponds to promoting Laplace operators to different Laplace operators. So this difference operator will enable us to describe the elliptic model that elliptic gauge theory, sorry, elliptic integrable system that we have started with. So if you, instead of LW, consider difference LW, difference Benjamin-Olon, you're gonna get relativistic models. Like for difference LW, you get relativistic Collegio or trigonometric RS model. For difference LW, you're gonna get elliptic RS model if you do the same procedure. And you can basically pick any model, you can pick your favorite integrable system, and there will be thermodynamic limit, the hydrodynamic limit of this model. In other words, there is a duality, so you can start with either hydrodynamics and look at solitons, and you'll discover some n-body integrable system, or you can start with integrable system and then send it to infinity to get a thermodynamic limit. So large analytic collodger will give you intermediate long wave system, and elliptic RS model, which comes from five-dimensional theory, and it was one star five-dimensional theory, it will become finite difference LW model. Oops, okay. So that was the larger description, the story in the lower left corner of the first slide. So now let's, we need to understand what happens in the top corner. What is the effective gauge theory description of UN theory? So we can get some, we can get some ns inside from M theory, and from topological strings from geometric transitions. So let's start with M theory on the circle, cross CQ, cross CT, cross T star of S3. So this is a deformed conifold with a deformation parameter related to the area of S3. Then the, so NQ and TI are the same Q and T that we have in gauge theory setup. Now, and we put NM5 brains on the circle cross, CQ cross the base of this fiber. Now again, so if the compact manifold again here is needed to describe the transition, to understand some properties of the large limit, as I showed you, you don't need compact manifold, you just can basically replace it by R3, but to get the full story, you have to compactify part of those things. Anyhow, so the, as we know, the NM5 brains will give some, give rise to some gauge theory in a low energy description with rank N. Now let's send N to infinity and see what happens. And again, you can also see that the, because of the CT, the T parameter, which deforms one of the complex plane in the complementary directions, the r symmetry of the theory is deformed. So it's broken to a subgroup. And as you can convince yourself that it's exactly the same pattern of breaking that we need for 5,0001 star theory. Now as N goes to infinity, we go through the Gepakumar-Waffler large N transition and the M-theory on this setup with brains, sorry, with M5 brain, but with brains becomes M-theory on a different background, essentially the same five dimensional space times resolved conifold, which you call Y and there are no brains. So N is gone. And now we can just make a reduction on this Y on the conifold and because there's a, that the base, the P1 in the fiber of this conifold will get some theory with a single vector multiplet with a theory with eight supercharges with one vector multiplet that lives on S1 cross C cross C. So you're going to have some five dimensional theory with eight supercharges. And what we're going to do, we're going to count instantons in this theory. So there's, there's an equivalent description within perturbative string theory, which makes some formulae, which I showed you before, very sort of, sort of, which emphasizes the role of some formulae, including the formula for the Kazimir. So in type two A you can describe the theory as a stack of default brains, which uncooling branch is separated in these directions and then they helically wind it, so they wind it around so they're periodic, and they end on the same single NS5 brain, from, which is from zero to five. And the endpoints of those default brains, they are shifted by the mass parameter of the two-star theory. So once this is going to get the maximum supersymmetric theory because NS5 can be decoupled, but not in this case. And once the mass parameter is arbitrary, nothing fancy happens. You just have a cool branch of this theory, which you can study, which is known manifold. But let's see what happens as we again take the values of the masses in a special way. It's sometimes called Higgs branch locus, or it goes with different names, but essentially it's the same specialization that we did for McDonald's polynomials, and the same specialization that we did for Dacher-Kazimir. And then again, so you can show that this written proper variables, coincides with the quantization condition for Gopakumar-Waffa transition. So there should be a certain relation in terms of string coupling between area of S3 and the values of P1, right? And this is exactly what you need to have. So, and sort of for large N transition, what the transition would move you from this picture to this kind of picture. And this is what happens. The, all these guys combine together, so they rejoin and the default brains, essentially this brain connects to this half brain, so this brain comes to this brain here, and then so on and so forth. And we have a single helical default brain, which winds around all the way from top to bottom. And then once we do that, we can sort of turn on the, we can go into the Higgs branch by changing the radius of the P1, which in this sort of notation corresponds to the length of those D2 brains. And it is new that there are two types of D2 brains. So there is a brain, there's a D2 brain, which is orthogonal to the D4 brain and ends on the D5 brain. And there's also what is called D2 prime brain, which is embedded inside D4, such that the brain charges conserved. And then you need to carefully count how many brains each sort are connected, but basically there's a junction of the two brains and total charge should be zero. If you count brains and anti-brains in a proper way. Okay, so this is it, and the delta is the angle, basically it's angle where the spiral winds, right? And basically, oops. It's given by the tangent of delta, it's given by the ratio of the, basically 5D gauge coupling, or one over the radius with the mass parameter. And now let's do large N limit. It's kind of hard to, again we have to do some tricks. So instead of sending number of brains, instead of extending this picture to infinity like that, we can introduce a scale in such that the height of this spiral does not change, but the density of winding changes. And as you do so, the brains here and here, they'll be sort of come closer and closer to each other, and in the infinite and limit, the whole picture will collapse, and then all these D2 prime brains will become incident. And not coincidentally, the total charge of those D2 prime brains is zero, because sort of they're formed by brains and other brains in the neighboring, in the neighboring D4 brains, there's no total brain charge over here. So once we do that, the NS5 brain decouples, and it can leave the picture, and so you just have D2 brains submerged inside D4 brains. Well, there's some complementary directions, but you can trivialize them away because they're periodic, and in the end, you're gonna arrive to the same setup, which is the sort of modified ADHM data because there's some periodic directions for single, for the U1 instant for U1 instantons, and the number of instant, the topological sector corresponds to the number of D2 brains, which we have there. And again, you see those, we have those lambdas, remember lambdas were heights of the coulomb, heights of the yank tableau, in that specialization, and the size of that partition determines the topological sector. So we're gonna get exactly KD2 brains inside the world volume of D6 brain, which is the ADHM. Okay, sorry for that. So we're off to U1 instantons in the non-commutative setup by sort of Necrasev and Schwartz, and turns out that mathematicians have already looked at the spaces before, and the Heismar algebra that we had, several slides ago, they already knew that it appears independently in this setup. So Necogenma, then Schiffman-Ushiro, and some other people, they studied this Heismar algebra, and in fact, it's a part of a more general, struck bigger gadget, which is called elliptical algebra, and it appears naturally on the, when you study modulite space of U1 instantons, basically. Eric Carlson mentioned some of the results today, which I'm not gonna reiterate, but physically you can think of, to have the five-dimensional theory on R4 cross S1, and then this instanton is like a KK monopole that propagates around the circle. And then as you go, as you wind around, you increase the topological number by one, or equivalently you can think of some large gauge transformations that change the boundary conditions at infinity on this circle cross circle at infinity. Now, the modulite space is given by the ADHM quiver, which we'll think of as a three-dimensional N cross 2 theory on C cross S1. So again, it's the same as one, because it was the same on both sides of the transition, and it's given by the standard ADHM data. So it's a U1 gauge group for UK, sorry, U1 flavor group for UK gauge group with the fundamental and the fundamental fields, and to add joints, and basically, you just look at spaces of all homes from here to here and back, and homes from UK to UK itself, symbolically quotient by the action of the gauge group that will give you the modulite space of instanton. So in this case, it's the Hilbert space, Hilbert scheme of K points in C2. That's right, which is not coincidentally related to the flat connections on the torus. And then we can compute many things about it. For example, we can study it's equivariant quantum cohomology, or equivariant quantum K theory. And using an approach by Nikrasov and Schutterschwilde, we can do it by studying a vacua of the three-dimensional theory with a quiver that was in previous slide. So we can compute the twisted curl ring just by studying massive vacua of the gauge theory, and then sort of cautioning out by the vacua equations, which are simply the extremizations of the effect of twisted superpotential. So in other words, it's just the Jacobian ring for the effect of twisted superpotential. So you can do it like this, you're gonna get contributions from each hyper-multiplet, each carol-multiplet will contribute to the equation. So it will depend on the Fahy-Eliopoulos' parameter of the gauge group, which is governed by the same parameter P. So it counts instant on, it will be the topological number, which will count instantons in five dimensions. And then again, all the parameters here are exponents of what we've seen before. So the Q is the same exponential as before the transition, and then the T now is played by epsilon two, which is the omega background parameter in the second complex plane in the same 5D gauge theory. And the Fi is given by P, as I said. And what we have checked with Antonio, we had two papers like this year and one paper last year. Well, paper this year, yeah, so we have two papers published where you can sort of convince yourself by the computation that the eigenvalues of collager Hamiltonians, which you can write as using free boson realization, they coincide with the quantum, so they describe the quantum multiplication in a covariant quantum cohomology of this modular space. So more explicitly, you can compute a covariant shown character for this modular space, which is this expression on the right. And you can see that for different specializations lambda, the, those characters that take certain values that coincides with the energies of elliptic of the operators, the elliptic resonational operators that we discussed in the first part. And those, in turn, are proportional, again, as I showed you before, to vacuum expression values of UN boson loops, say here in fundamental representation, as n goes to infinity. So in other words, there is a stable limit, which takes you certain equivalent, and then remember that boson loop as again comes from some covariant turn character. There is a stable limit, which takes turn characters of basically you infinity instantons, and then translate them into the different turn character for you one instantons. So the remarkable thing about it is that you can do the limit just head on on the nose and the limit exists. So in full glory, the duality is given by the stable. You have, but there are three different parts. There's like interbosystem part, which is like, well, ADHM part and the, sorry, yeah, and the gauge three part at you infinity. And probably the least trivial line here is the first one. So the coupling constant in RS model, the T, is here was the mass parameter of the two star mass. However, it's replaced by epsilon two, the omega background parameter in the ADHM theory. And again, sort of large in transition kind of explains why this has to be true. Yeah, the role of epsilon one stays the same, and then elliptic parameter counts instantons, both in you infinity theory and you one theory. And again, I can state some maps straightforwardly according to the rules that we had before. Right, so before I end, I want to throw up some mathematical conjectures, which some mathematicians in the room may like if they're still here. So, yeah, like I said, the elliptic whole algebra is the large and large limit of spherical data. It's a recent paper by Schiffman and Wieselow. Then in the trigonometric limit, so we can compute the, yeah, there is another result which I didn't announce, but essentially the gauge theory allows us to compute equivalent quantum K theory of those spaces. So for example, the Cartesian bundle to the flag, there's a very straightforward way how to write down what its equivalent K theory is. And if it's a classical K theory, then, I'm sorry, here it's a quantum K theory of the flag of the star of flag. And turns out that the stable limit of this ring of the subject is classical K theory of the modulite space of all instantons. So essentially we take the direct sum of all topological sectors, and, yeah, we get this invariant object. And in full story, you need to replace it by, well, this ring is replaced by something infinite dimensionals, essentially we need to add this Q or P, some associative algebra, which depends on the P, on the elliptic parameter, and we need to model it by the integrals of motion of the elliptic arrest system. So it's, again, sort of straightforward generalization of this ring, and large and limit of this guy gives the full quantum covariant K theory of the instanton modulite space. I'm sure there are more useful ways to make these conjectures, but I just have to turn and emphasize that studying large and limit turns out to be a very powerful thing to do. Yeah, there are some open questions, like if we can, should be able to generalize the elliptic homology of the same spaces, or maybe K theory of some infinite dimensional spaces, there's a relationship of what I said to not homology, which I believe was addressed to in Alexei's talk. And again, you can still keep adding further gradings in homology. So now people talk about quadruple graded homology. Can you have, say, one more parameter which corresponds to this elliptic deformation that I had? Will we be able to compute something more generic than quadruple graded homology? Then instead of five D theories, we should be able to take six D theory at large N, and should describe by some holographic construction which people studied in M theory. And of course, there should be some elliptic generalization of DAHA, which can be used explicitly in the computations. Good, so since it's the last talk of the workshop and I want to thank the organizers, I guess you all joined me. So I guess you're all tired, but I came relatively fresh. So yeah, thank you very much for.