 Okay, so thank you so much for the invitation. First of all, I'd like to thank all the writers. It's a great pleasure to be at the ICT and also to ICTP. And today I'm going to talk about integrable models, integrability, versus some kind of fuselage, or gain scale. And of course there are various different differences, and I've talked to so many differences in the literature about... Oh, I see, yes. Yes, and then turn it on. Okay, it's better? Yeah, hopefully you can hear. So there is a relation between integrability and gaze series, and there are various proposals in the literature. So people use the integrability to try to solve the gaze series. It's something like there is a gaze series, it's complicated things, but maybe you specialize to some particular sector, or you take some special gaze series, topological gaze series, etc. And then there is some integrable structure there, so you try to take advantage of that to solve the gaze series, for example. So that's roughly in this direction trying to use the integrability to solve gaze series. Which is a very interesting thing, except that mostly what I'm going to talk about is the opposite direction, so namely I'm going to use the gaze series, or quantum field series, as a tool to understand whatever is known about various integrable models in the literature. So that's the direction I want to go. And well, let's see, so then for the aspect of integrable models, I want to understand. So today I want to say that I want to understand minimally the Jan-Bakster equation, which I call YBG, with the spectral parameter. Right, so this is what I want to understand. Well, there are other ingredients, but I think it's often the case that one of the, this Jan-Bakster equation is the definition of integrability. Although there are several different definitions for integrable models, but this I think is one of the canonical ones, I don't understand these particular equations. Well, so what I understand means that, well, sometimes you might want to construct new solutions, but also sometimes people find solutions for certain cases, but people haven't found a solution in some cases. So we want to explain the pattern of what is known about Jan-Bakster equations in the literature. So these are the things I want to do, starting with gaze series. Now here I emphasize the fact that I'm going to think about the Jan-Bakster equation with a spectral parameter. And that's important because that's really a crucial part of the story. So it's likely that most people are very familiar with the Jan-Bakster equation itself, but just to remind you what Jan-Bakster equation was. So Jan-Bakster equations, so there is some operator R, so it's the endomorphism of some vector space B. So in a sense it has four indices, say I, J, K, L, for example. I, J, K, L, if you choose the basis of this vector space, say it's a vector space is going to be in some representation of some group, for example. Say SU2, spin-j representation. So this is vector space B. And then, so this is the operator. And then I consider the three tensor products of vector space V times V times V. And then on this space I could define the R matrix. Let's see, so I could, so this R itself is the endomorphism of V times V. But, for example, I could take R12, which is the R acting on the first two times the identity on the third component. So this is the endomorphism of V times V times V. And similarly I could define 1, 3, 2, 3, for example. And the celebrate of the unboxed equation says that we have this equation. Well, it's important, so let's write here so that everybody can see 1u. Sorry, okay, I'm going to write this afterwards. And R23 is equal to R23, R13, R12. So this is identity, so this is the element of the endomorphism. So you can take a product, compose it, and this is the equation. Now, so, well, this is, so in some cases people are really interested in this type of equation. But I need a little bit extra ingredient, which is a spectro parameter. Oh, there's a color choke, so I'm tempted to use it. So, in my case what happens is that there is one parameter, far middle of R matrix. R matrix is a functional parameter. And this is the perspective parameter. I like it z, for example, sometimes, or u. So, and then there is the equation here. Well, in general, actually it depends on the two parameters. But here in post-sync condition, that it depends only on the difference. So that's one parameter if you care about such details. But anyway, for the most typical case, it's just one parameter. Now, then there is a parameter here, u. u plus b, mb. And then these are u plus b, u. So that's an interesting equation where there's a particular pattern in the argument. And so this is the n-box equation with the spectro parameter. And it's, well, so one interesting question is whether you can find a solution to this. And if you don't know anything about the integral models, the most brute force way I would do, it's just like the older components of the matrix, for example. Say, for example, if ijk runs over two indices, for example, that has 2 to the fourth power, 16. Well, let's see. Anyway, there are many components. So, and then write down the components and just write down the equation and try to solve it. That's the most nice thing to do. Except that then you immediately realize that it's not easy to find a solution, because this is an over-constrained solution. Well, for example, if this index i, jk, et cetera, runs from 1 to n. This gives order n to the sixth power equations. But our matrix itself, it has only four indices, n to the fourth power, for example. So it's highly over-constrained. So it's very hard to find a solution. And if you do a linear algebra, for example, unless there is some miracle and there is some linear dependence between these equations, you won't find a solution. But that miracle happens, and then that interesting question for i that miracle happens. And the question is to understand that miracle from gauge theory. Well, of course, this is not the only approach. People try to take advantage of symmetry. For example, one approach is to take advantage of symmetry, and people use the quantum groups and other symmetries, which I sometimes come to. But I'm working a few series. I want to understand hopefully everything in gauge theory now. Now, why is this equation important? Because it's related with the existence of conserved charges, which is another characterization of the integral model. So suppose that I have this R matrix. Then I could define a statistical mechanical model. Well, here I define a classical statistical mechanical model defined on the two-dimensional lattice. So there is a two-dimensional plane. And there is a statistical lattice. And, for example, for the simplest case, let's take the square lattice. And let's also assume that it's a torus, for example, periodic boundary condition. Sorry, this is supposed to be on the boundary, for example. So, on the left is supposed to be a lattice, for example. And then you can define a statistical partition function by associating a factor of R to each factor. So what you do is that you associate this vector parameter. Well, okay, first of all, R is dependent on a spectro parameter. So you could associate, for example, the same spectro parameter u and... Let's see, for example, I could associate the same spectro parameter u here, et cetera. And then once you see this intersection, you associate this R matrix here, which has four indices. And then when you connect these four-valent vertices, you're going to sum over the corresponding indices. So, namely, if you have this, for example, I would say that there is a Boltzmann weight ijkl, for example. But now suppose that there is a PQRS, for example. And then I'm going to also associate PQRS. But when we connect these things together, I'm going to make an identification that s equals j. So this s is replaced by j. So take a product and sum over the j. So you associate the R matrix to each vertex, and you're going to connect everything together, and each time you connect it, you sum over the corresponding index. And by repeating that, you obtain something, so eventually you contract all the indices and you define the partition function. Yes, in this case, it's a torus. And you can also try to have a fixed boundary condition, et cetera, on the boundary. But in those cases, you have to make sure that the integrability is preserved on the boundary condition. And there is a reflection, for example, K matrix. There is a similar equation involving. You have to take into account the effect of boundary, and then there is a boundary generalization of the unbacked equations, for example. So this defines the two-dimensional statistical lattice model. Once you have this statistical mechanical model, so this is just the definition. But what's crucial here is that this unbacked equation guarantees that this, the commute, well, let's see, should I explain this? Okay. Well, maybe I just state the fact. So from here, so this is the partition function. But for example, you can, instead of computing everything but at the vertex, you can try to first compute this part, and then combine everything together to form the whole two-dimensional lattice. So, namely, you might want to first compute this thing, namely, single layer, where because of periodic boundary condition, it's identified. So this is a transfer matrix. And then suppose that there's a spectro parameter here. So this is known as a transfer matrix. And the partition function, then the power of this thing, where this L is the length in this direction. And the crucial point is the unbacked equation means that these two transfer matrices commute, two transfer matrices with different spectro parameters commute. And hence, you can expand tu in u to, and because everything commutes for arbitrary values, u and v, you can expand. And the coefficient should also mutually commute. So if you expand, like t, t, tn, un, for example, then they should mutually commute. So that's another characterization of integral models that they are mutually commuting charges. So for this reason, what's crucial is that you need a spectro parameter u to do this expansion. If you have a solution to unbacked equation without this u, you do not necessarily have this conclusion, because you simply cannot expand in u. So this is the well-known stuff in the integral models. But the reason I spent some time explaining this is it's actually, although there are a lot of literature about trying to use Gay's theory to understand some aspect of integral models, there has been a long-standing question of how to understand spectro parameter in Gay's theory. I'm going to explain a little bit more later, but before coming to details, first of all, we have a unbacked equation with a spectro parameter, and there are several different approaches to this. And I have tried several different things over the past five years or so, and what I'm going to talk about today is one approach which uses a four-dimensional Gay's theory. So this is somewhat like a Chan-Simons-like theory, but in four dimensions. And this approach has been first proposed by Costero in, I think, 13. It is a very interesting paper. And so we are trying to take advantage of the full power of his understanding. And so what I'm going to talk about is mostly work in progress, which Kevin Costero and Robert Diagraff and everything. So that's what I'm going to talk about. And before talking about that, let's also mention that there are several other things I have tried. Another approach to this, in fact, that's what I talked about here in this exact same room three years ago, in fact. But I know that there isn't too much overlap, so I'm not going to assume anything. And also, I'm now writing a review about this whole stuff which is supposed to be very pedagogical. So if you're interested, please have a look once it comes out. But anyway, so another approach is here, I'm going to use the four-dimensional, but not necessarily four-dimensional. The other idea is to use supersymmetric Fieber gauge theory. This approach I call the gauge YBE because it's a relation between the answer to the equation of the gauge theory. And I intentionally wrote the 2G gauge theory somewhat vaguely because there are actually several different versions you can do with this. The original story I worked on was 4D N equal 1, for example. But you can also go to 3D and then 2D, for example. 2, 2, for example. And, right, so that's, and then, well, what's interesting about this story is that since, well, first of all, the main idea here is that there's a YANBASTA equation. So what is the counterpart of that here? Well, the simple answer is that it corresponds to some duality, which I call the YANBASTA duality. So it's a duality among... among quantum Q series, 3G-Qieber gauge theory. So the YANBASTA equation promotes it to a duality. But this is very nice because once it promotes it to a duality, you can compute various different partition functions, for example. And then you can obtain various different answers, for example. It's just that one answer that sometimes, for example, in the difficult case I worked on, is that the original I started working on S1 times S3, but they're also supersmitted for localization function, which is sometimes computed with Francesco and also Tatsuma Nishioca. So I also computed these observables and then we obtained different solutions of YANBASTA equations. So it's actually, you could obtain new solutions. And the new solution is, it's supposed to contain all the known solutions of the star-to-angle relation with the positive Boltzmann weight. And it's a crazy generalization of the Ising model with many parameters, continuous spin, discrete spins, et cetera. And it involves, for example, well, function-wise this involves some special function or the elliptic gamma function. And you can take various degeneration limits. There is a quantum dialogue, a fork, a symbol. There you can go to gamma functions. Well, a gamma function, in case it's a little subtle, but that's okay. Yeah, sure. There is an expression involving gamma functions, theta functions. So there are various special functions appear. And so there is a very nice structure here. Well, so I'm very fascinated by the story, except that by talking with people, I realized that this is not necessarily the type of the integral models people mostly encounter. So, namely, these are solutions, typically, not quasi-classical. So what is quasi-classical means? Well, quasi-classical means that there is some parameter, exponential parameter, like h bar, for this r matrix. So if you have this r matrix, r u, for example, but suppose it has an exponential parameter, h bar. And then suppose that it starts its identity, and first order, let's write it as small r, plus for the h bar squared. So in some cases, there is a nice exponential parameter around which it starts its identity and then expands. It expands to the first order and the second order, et cetera. And if there is such a parameter, the solution is called the quasi-classical, for example. But many of the solutions are not quasi-classical, whereas many people in the literature encounter quasi-classical solutions. So I'm not completely producing some of many of the things people, most standard stuff people are discussing. Now, here in this approach, I'm going to... Well, these are particularly useful for seeing the... quasi-classical solutions. And in fact, it's useful to see the classification of the quasi-classical solutions, as we will see. Or maybe I'm going too slow. Any idea why? Well, from the viewpoint of integrable models, for example, I mean, from the viewpoint of integrable models, it's very hard to see that because it looks like a completely different integrable model. Unless you know that underlying gaseous is very hard to see. But except that there might be a different answer of the different type, which is like, algebraic structure like a quantum group might be the same, for example. For example, these are associated with scratching algebra, for example. And, well, I don't really know this 3-mode gamma case, but given aspect scratching algorithm, you can try to change the representation, for example. So maybe it might be that changing representation might correspond to changing the geometry. Or maybe we are identifying the new... even new structure, for example, the new form of how they might be new algebraic structure, for example. That would be more fascinating, but I don't necessarily know because, well, I definitely have to see that quantum group or elliptic group that structure here, but not necessarily directly so far in this approach. Yeah? Yeah, yeah, yeah. You're right. Well, so you're talking about the one by Odeski, for example. Well, sorry. But Fielder doesn't necessarily have this R, for example. Yeah, yeah, yeah. I think so. So I don't know, but I do suspect that this might be a new algebraic structure underlying behind that. So this can be defined for SLN, and it has two deformation parameters. So these are continuous parameters, like a Q of the quantum group, plus integer, R, for example. Yes, so that's a discrete parameter. And maybe it might be a new, completely new algebraic structure. That would be very fascinating. And to some extent, we already know the R metrics, given the R metrics, you can define algebra easily. But there's R metrics, the complicated R metrics, whose indices are continuous parameters, for example. So the standard method doesn't necessarily work. But anyway, I think there is a lot to understand, even mathematically, because these are concrete solutions. It goes through quantum field theory, but the resulting solution is very concrete. And well, I should say what the spectral parameter is. So in this approach, the spectral parameter becomes, in this approach, as we will see, a position, a geometric position, on surface, which I like to see. And here, in this case, it's actually related to some U1-R symmetry in supersymmetry. And so, namely, there is a U1-R symmetry in the UPR could mix with the Freba symmetries. So there is mixing, and there is a parameter space, and that type of thing, yeah, that type of parameter. Well, in fact, sorry. U1-R symmetry in the IR, I should have said, is related to the spectral parameter. And, well, so there are some indication questions about how to relate this and have some thoughts, but I won't say anything today. And, well, of course, U1-R symmetry, typically in the brain realization, is a rotation in the transverse direction. So that's also geometrical, so it might be relations. And in fact, since I'm saying this, there is another thing I tried, which am I like A because it's sort of a different particle. So another approach is to try to use the scattering amplitude of Prana scattering of Prana scattering amplitude of 4D N equal 4 and 3D ABGM theory, N equal 6 ABGM theory. So in this type of approach, whenever there is, for example, well, that's exactly the case in ABGM, I really regard it as a crossing of particles, for example, gluons. And then there is some interesting spectral parameter. There is a nice structure. And in particular, the r-matrix, the one consequence from this is that r-matrix is written as an integral, as a grass-moneyon integral because people are trying to write this scattering amplitude in terms of the grass-moneyons. So interestingly, some of the r-matrices for a young young for PSU 2-2-2-4 or SP 2-2-2-3-6, 6-3-2-2 has nice grass-moneyon formula, for example. Although in this case, unfortunately, the minimal spectral parameter is like deforming the helicity and the physical meaning isn't completely clear. So let's try to describe this theory, Kostero's theory. And Kostero goes through a series of chain of arguments like getting one super-milk and then do the twist, giving a mass to fermions, except that, after all, the resulting thing, what is crucial is a very simple and, well, let's consider four dimensions. Well, first, let's write down the action and then try to explain. And then this is the sigma times c and then there is a dz which shan-symons a. So this is a very simple action. So let me explain the notation. So first of all, what's crucial is that 4D geometry is always of the form d-man surface times d-man surface. So this is where the spectral curve lives. So I said the geometrical surface, but it's actually c, c and automodear. So this is where the spectral parameter lives. And this is another surface which I could take, for example, lattice model lives. And let's take the coordinates to be x, y for the first sigma, and then there is a complex parameter which I was writing z here. So this is a complex parameter. So let's write the real part and the imaginary part, t minus t and theta, for example. I might not use that, but z, z and z part. So in particular, I have assumed that this is a complex manifold. And well, it actually turns out that the theory is topological. So we need a homomorphic structure here on this d-man surface. As you have already here, I already needed that. And here, the theory is topological along this surface. I didn't use any metric here for writing down this action. Now, this itself is already actually very interesting because typically in Chan-Simon theory, you want to do the Chan-Simon theory. So let's start with the 3D Chan-Simon theory where people are more familiar with. And in that case, you wanted to define a 3D Chan-Simon theory on arbitrary 3-manifold. And that was actually crucial for understanding the theory because there is a 3D covariance. So the original motivation of wisdom was that there is a Jones polynomial, but that's defined by projections. And somehow, the theory could see that it's independent of the way of the trivial in Chan-Simon's formulation. And that was the power of the 3D covariance. So that was very crucial for application of the node theory. Here, I'm using a different thing. So now I don't necessarily have the covariance in 3-manifold or 4-manifold in this case. And I just restricted the geometry to this case. Because that's a little bit of a trick which by hindsight is very simple, but... We need to change the mindset a little bit in order to discuss integral models with our Young-Bak's equation with the spectro-parameter as compared with discussion of node invariance. And then, so what's this A? A is... Well, what's also there is funny is that this A is a connection, of course. But we actually don't use some of the components of this connection. A z bar d z bar. So no A z. Well, no A z doesn't mean that we are fixing the case so that A z equals to 0. That would break the case symmetry. But a lot of what I'm doing is that the value of A z is unspecified and I'm not really using that to write down this action. Now, the Chang-Simon's A is the standard Charles-Simon's term action. Chang-Simon's action So if you like, you can integrate the parts. And I think there is a factor 2, I guess. Like z f h a, for example. f h f, sorry. So this is by integrable parts. So this is like a set angle. But the set angle depends on the complex parameter z. And also, for example, the parameter here, h bar, 1 over h bar, low to h bar. So this h bar obviously plays a role in the Chang-Simon's term action. But it's not necessarily quantized. So in the 3D Chang-Simon series, it was quantized. That was the level. And that was because one way to explain that was, in fact, we did similar operation like this. So in that case, there is a Chang-Simon's term on the defined number 3, 4. 3, 4. But for that, you need a trivialization of the gauge view. It doesn't necessarily exist. So you do integrable parts. If you can do the integrable parts, you can write it in the boundary of 4-manifold. But it depends on the choice of the 4-manifold. So that's why this level was quantized. But here, all the fields are already defined on the 4-manifold. So that's this Lagrangian. Is there anything else I should explain? DZ, Chang-Simon's. So I'm going to consider a slightly generalized version of this momentarily. But this is the simplest case. So it's extremely simple action. Now, the question is what does it help? Well, just like ordinary Chang-Simon's theory, we attempted to compute observables. And the good observable is the Wilson line. So in the case of the compact group, the usual Chang-Simon's theory, the Wilson line gives nothing variance. So we attempted to do similar things. But first of all, we have this product geometry. So we have to think about which direction we want to go. Now, we have to be careful with the fact that there isn't really az. So in the usual way to define the Wilson line, you have to do the parallel transport using the gauge connection to compute the horonomy and then take a trace. However, there isn't really no az. So you cannot really transport in the direction of z. So what you want to do, at least I'm going to do, is put the Wilson line here on this frame. Let's take this to be T2, for example, elliptic curve. So remember, this is supposed to be where the lattice model leaves. So depending on what kind of boundary conditions you want for the statistical mechanical model, you might have to think about different surfaces. Say it might be a disc if you want a boundary condition. Then you have to think about the boundary conditions of this. But let's say you have a torus, for example, periodic boundary conditions. And let's do the similar. And then what I want is the exact same picture which I wrote 10, 20 minutes ago, which was this picture. But now, the meanings are slightly different. Well, first of all, maybe I should orient this. So these are the Wilson lines. So each of these becomes the Wilson lines. So this is the sigma, which is the topological, along which the theory is topological. So I'm not going topological plane. Now, there was a parameter. Well, let's see. First of all, if it's along, then it should be a particular point on this surface. So the whole geometry is this time the surface, C. Well, the curve C, except that in this particular case, this surface C is taken to be a complex plane. Z, the complex coordinate, for example. Well, I'm going to comment on the generalizations momentarily. But this is the simplest case. So this is the simplest case. So there is an infinity. But, and then it does a fixed point. So, for example, this used to have a spectro parameter, for example. Let's see. Parameters like Z, say, let's say W, for example. And these are located at point Z and W. Well, you can try to change the position, if you like. But these are located at different positions. So, the idea is pretty simple. So there was an unknown parameter, but that was turned into a coordinate of the extra dimensions. And this Z and W should play the role of the spectro parameter. Well, except that there could be various objections to this. So, well, okay. So the statement one might want is that I consider the expectation value of these views on lines representing the statistical lattice and then compute the expectation value. The statistical partition function of the integral model. That's the type of things. But there are many things. Well, first of all, why it's integral? But even before coming to that, it's not even clear why this is the statistical mechanical model in the sense I may explain. So, namely, in general, well, in the statistical mechanical model I mentioned, everything comes from the local interaction at the vertex. So, namely, there isn't a long-range interaction between the variables here and between the variable there. So, you have to make sure that whatever is your computing here factorizes into local contribution from here. Once you know that, it becomes closer to the statistical mechanical model I was talking about. Now, let's see. Now, there's one peculiar, not yet another peculiar feature of this theory. Which is that this theory, which I tend to recall in this theory, is very interesting. This is power counting unrenormalizable. Because if you don't have this in three dimensions, that's what the usual Chang-Simon's term, but you have this extra coordinate dz. So, well, it depends on how you normalize this. But if you keep the canonical dimension for the gate field, there is a dimensionful parameter here. This makes the theory power counting unrenormalizable. However, well, this is a nice theory. And in particular, due to the equation of motion, all the counter terms, which you can think of, gauge invariant counter terms actually vanish because the equation of motion is basically f equals to zero. So, it's actually doesn't much, too much trouble that the theory is not power counting unrenormalizable. And Kevin Costello has an amazing story of trying to use the battery-invade-Covsky formalism in a really mathematical rigorously perturbative quantum view theory. And he has shown that this theory is finite, except that please don't ask me why, because some of the arguments still don't understand. Yeah, but anyway, you can already see that the usual problem associated with the counter term, it's not there. And also, well, if you're really skeptical about this, for example, there is actually a realization of this starting for four-dimensional theory, and then try to give a twist theory, giving a master freemium, et cetera, and that will give a UV completion to this theory. So, if you like, I could rewrite on that, and that itself is fine. But anyway, the reason I said that this pan-quantic unrenormalizable is that the theory becomes IR-free, so long-distance physics is trivial. So, suppose that I have this complicated set of reason lines and then try to do the perturbation from the field theory, and what I do is that while there are gruons here and there are gruons here and there are gruons here, et cetera, I have to think about all sorts of gruon exchange, and that there might be loop diagrams, et cetera, which you have to think about. Well, except that the fact that this theory is trivial in the long-distance means that you really don't have to think about the gruon exchange from here to there, which is far away. In fact, it should come from the local intersection between these two reason lines. So here on this surface it really crosses. So there is a gruon exchange, and you go to higher loop order, there are more complicated loop diagrams, et cetera, that will be a more complicated story, but still the fact that there is a local contribution associated to this vertex is fine. So that explains why this partition grew together, while entangled together in a very complicated manner, factorizes into local contributions from this vertex. So this is a very interesting explanation. In fact, in the other story of supersymmetrical theory, I talked about similar localizations come from supersymmetrical localizations, both terms of fermions, cancellations, et cetera. So one loop is up. So that's the explanation there, but here we have a different explanation. Sorry, that's a comment maybe I shouldn't. Now, once we have this thing, then you can try to compute it. Simply, just do the loop computation, one loop, well, three-level one loop to loop, two compute R matrix. And then we call that these are located at a point Z and W, so that should give the R matrix, and it turns out this only depends on the difference. Well, which is not too surprising here, because there is a translation symmetry. So in this case, it depends only on the loop computations. That should keep this one. But it's still a hard problem, as we know from the field series, because loop computations are hard. So the first thing we should do is to try the simplest, namely, to do the three-level. So which is this diagram? So what is the three-level answer? So at three-level, so it's the first leading order in H-bar expansion. So, namely, if you have this R, H-bar, or Z minus W, but then I might write this identity plus H-bar, or Z minus W, and what is this? Oh, R itself. Well, so that's going to be the correlator of two Wilson lines. Yes, at an angle here. Yes, right. So the point is that here there are a lot of different observables, and they might be focused each other, but we can forget about the interactions among them, just concentrate on the sort of factorization of the correlation function of the Wilson lines. So what is this going to be? Well, you still have to do the integral. You write down the propagator, and then do the integral. So there is a rule on and there are positions here, and there you have to integrate over all the possible positions. Well, except that, but you don't know, but here let me explain the slightly different ways. So, maybe do I want here, let's see, i, j. Okay, either it's fine, but let's see, okay, it should be fine. Yeah, so sure, for example. So let's try to compute this. And suppose that there is an index i, j, i, j, k, l, for example. And we knew the law from the fine diagram, which is that whenever there is a vertex, there should be a structure constant. So if there is a, if I double this dummy index by a, for example, then what you should have is that there is a structure constant, i, j, i, k, a. Well, maybe I should write it here so that the correspondence is super clear. And then here I have a, well, already say I anti-symmetrize every single example. Things like this. So the answer at least, without doing much complicated computations, but the whole computation, that's not complicated, but oh, oh, sorry, yeah, sorry, yeah, okay, I didn't say explicit, but let's take the mortgage group G. Right. Yeah. And in fact the point is that it's, it's general. Yes. Oh, yes, that's right. Yeah. Well, well, okay, sorry, sorry, sorry, yeah, okay, so maybe I shouldn't use the notation a, a, a, or anything right? Yeah. Yeah, yeah, yeah, that, that's right, that's right. Yeah, okay, yeah, okay, right. So, so these are in some representation. So this is in representation r and representation r prime, for example. There could be different representation, for example. Yes, a is the joint, right? And then what you obtain is that, so this r, same, but thank you for asking. I didn't, if I'll say explicit, yes. So you can easily see that this should take the form of z a, t a, for example, t a prime, i j, for example, i j, let's see. Yeah, in this, sorry, i k, I guess, j a, for example. For t a's are the representative for the generators for the, so a is the joint index, and i j, etc., are the indices of representation. Well, this could be the same representation, but I know the straight and different general expression, where the two representations are different. So there should be such, such groups theory factor. And, and then, then, well, then it turns out that this is a very simple factor, z minus w. Well, at least it's, it's okay dimension-wise. As I said earlier, because the theory is unreliable, this theory h bar has a dimension length, so at least it cancels the dimension. And, so this is a very simple answer. So in fact, it turns out the answer is exactly this. Well, that requires computation, but it's very easy. Now, it turns out this is the argument, this is what we call the rational solution. Well, so here, well, okay, see here the construction, it looks by construction, well, the answer is which I'm coming to, but by construction because we could do the patabasic expansion h bar. So the r matrix is quasi-classical. And this expansion r is this classical r matrix. Let's call the classical r matrix. And in particular, this classical r matrix satisfies some equation. Well, of course, there is a full quantum Young-Bach's equation. So if we expand in h, y, if we expand this way and then take the whole term of h square, there will be a relation. And the relation is that r12u and r3 v, okay, maybe to save time, this is let's type of the sum u plus v plus r13 r23. So this is the Young-Bach's equation, but h bar limit of Young-Bach's equation known as the classical Young-Bach's equation. And people have found the simple solution of this classical Young-Bach's equation is this one. So we have reproduced this classical r matrix. Very interesting mathematics. So this is about the classical r matrix, but here's a very interesting mathematical result saying that if you know the position basically if you know this classical r matrix then by using the Young-Bach's equation you actually know suppose that you have two classical r matrices sorry, suppose that I have two quantum r matrices here satisfy the Young-Bach's equation and let's do the expansion and then obtain two classical r matrices. But suppose that we find that the two classical r matrices is the same and then and it turns out that the full quantum r matrix is actually the same. So this reproduced the classical matrix, but if you could resort to his general statement we actually know about the full quantum r matrix which is interesting as a statement in part of the quantum physics because otherwise you have to do perturbation order by order. But if you take advantage of this structure the full structure full part of the series can be computed from the full r matrix which is computed by universal r matrix. Yeah, yeah, that's right, that's right. Yes. Well, non-part of the corrections I see yeah Yeah, right, so it's a finite series? Yeah, I don't think there are any instant tones and things like that. Yeah, but I don't think there are instant tones, etc. So that's the physical argument, but I believe that if you look into the argument of Dreamfield probably it might be somewhere there in Christendom. That I'm not so sure, maybe some people in the audience might know. Yeah, so this is well so at least we have already derived this simplest classical matrix this was essentially they are already in the Costello's paper from 2013 but here we are using a more elementary approach which is easier to understand for quantum physics. No, okay, that I don't know. Yeah, so that requires to understand the full part of the series part of the corrections to these, for example. And that seems hard, in fact. Yeah, I think so. Yeah, there is a prediction for part of, so I think we are just lazy but you can go to high loop, one loop, two loop, etc. and then this should match, for example. That's a very, yeah, that's very convincing check. Perhaps I should do at least one loop, for example. Maybe find a graduate to do two loops. Anyway, I'm joking. Sorry, yes? Well, yeah, it should be at least order by order. Order by order. Yeah, yeah, that's right, that's right, yes. And, right. But, sorry, but in fact, right, sorry, but maybe I should say that I realize that I haven't told you the most crucial part which is that, well, okay, so you already explained the integral of all the power matrix but why do we obtain the solution of the Janbach's equation? Sorry, I haven't said that. Which is the very most important part. Sorry, I almost forgot that. That's the problem of doing the blackboard top. I forget what I'm supposed to talk about. And, right. So, what is the, well, so Janbach's equation, graphically, means that there are three different lines, for example. And I could change the rate of position like this, for example. And usually, well, you have to go through this continuity coming from here to there because it's a two-dimensional picture and obviously there is a point where everything there is a three-point intersection which is singular. But first of all, here we have the spectral parameters ZW and ZW, for example, and ZW. So, although in this plane, the topological plane, it looks like that they are on the, well, they might finally change it, for example. There is a triple intersection, but actually on this plane, on the other plane, for a morphic plane, where the spectral parameter leaves, but first of all, there are the different points. So, even if it looks like that everything is coming together and everything is singular, first of all, they are still far away in the extra-dimensional direction. And besides, but the theory here along this direction is topological. So this is the topological plane. So it actually shouldn't matter exactly how these lines are written. So the topological invariance, topological, the theory is topological on this plane. Together with the fact that they are separated in the complex plane, sorry, the C, the curve for the spectral parameter. So these two facts explain why if you compute these two useful lines, they have the same expression value. And then this was the Yan-Basch equation. So this is somewhat similar to the similar argument in Knott's theory. So in the case of Knott's theory, there are three lines and they change the rated position. That's why they might have three. And the usual argument is that there are 3D covariance and the theory is topological, so you can change the position without changing anything. You don't have a 3D covariance, but you have the topological stills have the topological invariance on the two plane. And the cost we pay is that we don't distinguish between over-crossing and under-crossing, for example. So in Knott's theory, we worry about whether it is over-crossing under-crossing that changes the knot, which is very crucial. But here it's a plane, so I don't have any distinction. So I lose some information of the knot, but at the compensation, I obtain the spectral parameter. There is no distinction between the Young's integral model with the spectral parameter and the Knott's theory. Oh, the time is almost over. But let me just mention so far I only represented this, so maybe some people might need to support it. But let me spend two minutes maybe explaining what the general result is. But in fact, we believe that this framework is very general. There are great groups, et cetera, representation, so hopefully we could produce quite a bit of the aspects of the integral models. And just one result, for example, I can tell you in the two minutes, I guess, is that there is a classification of the integral. The classical armetrics are classification due to Berabian dreamfield. And the precise theorem is a little bit more complicated. But roughly speaking, the statement is that the classical armetrics comes from the rational trigonometric. What this means is that in practice armetrics are written in some polynomial in Z, or Z and X, for example. But there's a polynomial or trigonometric, sine and cosine. And here in this elliptic case, we have theta functions, for example. The armetrics are written in terms of these functions, for example. And correspondingly, the spectral parameter leaves on C, leaves on C star, leaves on the elliptic curve. Now, there's a counterpart of this. Now, I think I only have one minute, but I could tell you one minute, which is that we could generalize the story a little bit. Well, okay, so far, I was writing dZ, which is Chan-Simon's A. But in fact, that implicit assume that that was C, for example. Complex plane. The sigma was a complex plane, so I was writing dZ with the differential Z. See, it's a holomorphic, yeah, holomorphic coordinate. Oh, sorry, sorry, yeah, I was saying the wrong thing. So this was used to be C. Sorry, yes, thank you. But I could take C to be a different manifold. For example, I could take to be C star or elliptic curve, for example. And then I take some differential. For example, in this case, there is a simple differential dZ over Z. And in this case, there is a differential dZ, where these are obvious canonical coordinates here. And I use this in the definition of the action. Well, so this is the one differential. Well, of course, then you can say that why don't you take a higher G than D amount of it, et cetera. Well, now the point is that this is not the complete classification of general R matrix, but classical R matrix. So, namely, this is a situation for oh, sorry, maybe I'm going over one minute, sorry, sorry, let me finish one minute. So this is a situation where you can take a classical limit. So H bar goes to 0 is applicable. So here I have H bar. So, namely, this is a parameter, so I could always take H bar to 0 and discuss classical limit. However, the subtlety is that there might be 0s, for example, of W, for example, of omega here. If there is a 0 of omega, it's 0. So, the semi-classical argument here doesn't apply. So it's an intrinsically quantum theory, at least near that point, and it might be fine as a quantum theory, but it doesn't necessarily have a part-up of the expansion I was talking about, so it doesn't fit with this classification. So, I want a differential but no 0s. And possibly, and it is the mathematical result that if you want a differential, a globally defined metamorphic function on the surface with no 0s, but positive poles, these are only three possibilities. So, 0s minus number of poles is 2g minus 2. And so, in this case, for example, there are two poles, the order of two poles at infinity. In this case, there are one pole at two points. So, in this case, that's number is 2, for example. And in this case, number is 0, for example. And that number goes to minus 2, minus 4, etc., you go to the Hygienic case, which is the only possibility. And that matches nicely with the classical R matrix, for example. Now, the DREAM, okay, so maybe I should finish, but the DREAM field service is more, for example, you cannot really find a solution to well, I don't know, ANK is in the elliptic cases, for example. And there is a counterpart of that here, that bundle module has parameter, etc. But anyway, so this is one small thing, but we are going to do some other aspects, and hopefully we could explain various results in the integral models from this four dimensional gaze area approach.