 OK, so that was the title. So as I said, it's probably more biased than historical. And I will describe a few properties or construction of params from the observable. And the good thing is that Yurg is in the audience, so then he will correct me about the history. I'm sure he will make many comments. No, he's very honest. Yes. Any more mistakes? Actually, there's a safe correct. It's everyone should not view fermions or paraphermions as observables. I use. OK, parapherm- fields? OK, better. But for me, an observable is a map from the space of configuration to some space. Then it works. OK, so first I will describe spins. These are the fermionic fields. So I will describe the construction in a statistical model, in a lattice statistical model. And more or less what I will follow is the construction by Kadanoff and Sevan from the 17th. So maybe it's not the first construction of fermionic or paraphermian field in a statistical model. Before, there was a Jordan-Wigner construction where maybe the first. But I think this is the first which introduced paraphermian. So what I want to describe, besides the construction of, explain what are the these are there on paraphermian field. I want to see the connection with group symmetries. And for instance, one of the echo of this construction is that there will be an analog of the Gauss flow that you all know in electromagnetism. There is a Gauss flow, which reflects some symmetries. And here, there will be analog of the Gauss flow in statistical models, which is linked to the existence of concept of currents. And I want to explain why these fields are called paraphermian. They are called paraphermian because they satisfy some breading relations. The physics we call them commutation relations, which are more general than those for fermions or for bosons. So they satisfy some breading relations. The breading relations code what happens when you exchange permeate in some way these fields, observables. And that will be linked to what we call the Young-Backster equation. So that's what I want to explain in this part. Then, so all this construction will be done on the lattice. So that will be, maybe the lattice can be infinite. The domain can be infinite. But there's not too much problem in defining the object. And then I will go to field theory. So some echoes in our decimal construction in field theory. So I will jump a little bit on field theory without defining completely all the object I will deal with. And mostly what I want to describe first is the construction of the analog of paraphermians in the case of U1, which is due to Mandelstam, which is called Mandelstam, vertex operators. And that's also in the 7th bit later than Kadanoff and Seva. And that's the point I want to make is the connection that comes from this construction is some property or some of objects which generalizes operators, which is called vertex, also vertex operators, but in conformal field theory. So I want to explain how in some way the vertex operators that are pairs in conformal field theory are, in a way, paraphermianic observable. And they satisfy so the breading relation will be related to the monodromy of the collision function. And I think the echo of this construction is the formulation of generalized vertex operator algebra, which includes paraphermianic algebra and more general infinite-dimensional algebra, which generalizes a finite algebra. So that is an notion of vertex operator algebra. So the name associated to that are probably. So the name associated to Young Backstairs is Young and Backstairs, let's forget, which is in the 60s. Here we can associate the name of Zemochikov and Patef. And they define what is called the Zn-paraphermianic algebra in conformal field theory, which is just a generalization of the Z construction on the lattice. So on the lattice, you can define some paraphermian associated to the group Zn. And if you push this construction to the continuum, in field theory, you get an operator algebra, vertex operator algebra, which is related to the group Zn. And in math, maybe you can associate Frankel, Katz, Epovsky, and so on, which use vertex operators to find representation of the algebras and find the algebras, and then which define the setup to discuss what are vertex operator algebra. So that's one first design application construction in field theory of vertex operator algebra. Another echo is this is in conformal field theory, which is a field theory which is invariant by dilatation, conformal transformation, which has no scale. In massive theory, which is a field theory with scale, there is different echo of this construction. One is the notion of tau function, the monodromy deformation. So that was used, for instance, by the Q2 group to study, to de-compute and characterize correction function in the massive ising model, the ising model, away from the critical temperature. So the name, I swear to that, is Sato, Jimbo, Miwa. So that's in the 80s, probably. So that's what I will discuss. Without explaining, I will just discuss how it's related to the para-firmware fields. I will not discuss how to construct this tau function, neither the vertex operator algebra, I will just discuss how it's related. Then the third point will be back on the lattice, where I will discuss relation between integrability and quantum symmetry and para-firmware fields. So I will discuss the relation between the Q. So in the first part, as we will see, that para-firmware fields are related to group symmetry and linked via the breading relation to the young back-style equation. So the quantum group are groups which are intimately related to the young back-style equation. And if you impose this quantum group symmetry, you produce models which are integrable. And you may wonder what are the consequences of these symmetries in the same way that the first part we will see that due to the symmetry, we have para-firmware field. So I will show what are the analog of the Gauss law on the concept currents in the case of the quantum group symmetries. And that will be linked to Smirnov's currents, para-firmware, as well as the construction of Cardi on the computation of Cardi which relates to the smirnov para-firmware to the integrability of the ON model at the critical point. So that was done in the 90s. And that's 2000 something. So first, these are the para-firmware fields. So what we consider is a spin model on the lattice. So I will take the square lattice and delta. So we have spin value. So on each side of the lattice, for a site on the lattice, we have spins. Let's say we call it S, which take a value somewhere. So it's I, which belongs to, I don't know, some set discrete. And we will define the Boltzmann way by saying that there is a local interaction between the spins. So we give a local interaction, S, S prime. And the interaction will be between spins which are adjacent on the lattice and the Boltzmann way for a configuration, S. So the configuration is the data of the spin variable on each side of the lattice. We associate the Boltzmann way, WS, which will be equal to exponential of minus I, S, J, where I, J is a link on the lattice. So that's a link. So they are neighbor. So I, J are neighbor on the lattice. And the probability of a configuration is just the Boltzmann way divided by the partition functions. Partition functions normalize the probability measure, OK? It's exactly like Hugo defined this morning for the rising model where the spin was taking two values plus or minus. And we assume, so the property of the disorder operator on the para-fermion will arise when there is a group symmetry in the model, which means that the interaction will be invariant under some group. So we assume group symmetry, which means that H. So there is some group, which acts on the set of S. And we just spin take a value. And we assume that, so there is an action of the group on that set. And we assume that the Hamiltonian is invariant under this action. Is it clear? So what I did not by G times S is the action of J on the set here, OK? So for instance, G can be Z2. G can be, I don't know, SU2, V8, whatever, OK? So that's defined a statistical model, which is invariant under some group J. In the sense that the probability of a configuration on the probability of another configuration, which is the second configuration obtained from the first one by the action of the group, are the same. So then the spin variable, so I should keep track of that. So what we call spin observable. For me, an observable is a map from the configuration space to, I will take, the real number. So it's a map. It's sigma, which is a map from S to the real number, which associates the value of each stack. So it depends on I, for I on the lattice. And it picks the spin at the position I, on a saturated function, which I call sigma I. So I just show the I here. So that's I, and that. And then that's a function on my configuration space. So it's a function on the space of event on which I define probability. So I can compute expectation of this function, or product of this function. And that's what we call correlation function. So I can compute, so if I will denote that, sigma, I pick n point, sigma 1, sigma n. So that will be simply 1 over partial function over all the spin configuration of w sigma of sigma of si, s1, si1, si10. And z is the partial function. So z is some normalization factor for the probability. OK? So maybe you call it E. That's the same thing as E of sigma I1, sigma In. OK? And then you can choose to, and we will need that. So this will work for any function s on the space s. But we can select, or we can maybe decompose the space of function on function which are covariant on the other action of the group. So we select function sigma s which are covariant on the other action of j, which means that if we transform the function sigma of g minus 1s, so this function, which is a new function, which is a function obtained by transform, which is a new function on the space on which spin takes value, which is a function to transform by the action of the group. It will become simply, so this function may take value in some space. So it takes value in some space, vector space v. And vector space v can be a representation of the group. So this one will transform as where r, we set to g, r of g, which belongs to the bottom of this movie, is a representation of g. OK? So the function is covariant on the other action of the group. So it means that we will call such representation sigma r. So it's indexed by your representation. Then we can choose to look at the correction function at all the spin variables. So for example, if it's j is gn, if you can give an example, g is the n, the action s takes value on the n2. And I can just multiply that, define my action of the group. And I can label the representation by an integer. So I will have sigma k of v, which is sk for k belongs to what you do n. And if it's su2, I take a spin take value in some representation of su2, and I act by this value. So that corresponds to local observable. I just have representation. I insert the spin at some value here. i1, i2, in, and I compute the n-point correction function. Is it OK? So in the case of easy model, for example, you don't get much choice. No, g is su2. So either you take the sigma is identity, or just the action sigma is some z to z, to s to s. So that's the spin observable. So there are local observable on the lattice. And then because there is group symmetry, you can define the disorder operators, which are non-local. Non-local. So the local spin observable are indexed by representation of the group. So the disorder operator, which I will call mu, are indexed by group element. g belongs to the group. And the local observable are indexed by the point on the lattice site. And this one will be indexed by gamma. And gamma is a contour on the dual lattice. Oriented contour. So what it is? So I take a contour. That will be gamma. And for instance, I orient the contour counterclockwise. So the contour will cross the number of links. So each link cross by the contour, I will actually have to wait for formula. So if the contour cross a link from here to here, so the contour is good like that. So I call it s minus s plus. So mu of j of gamma will be the product over all the link cross by gamma. So let's say it belongs to gamma. And what I do is that I will. So I insert these factors, the inverse of the Boltzmann weight. So I'm going to turn it with a plus minus h. This means that every time I cross a link like that, I replace the Boltzmann factors by the Boltzmann factor where I twist the spins inside the loop by j. So this means that here, when I was computing this correction function here, I had to sum over all the configuration with the spin inserted over all the configuration. Now, so it means with a weight which involves the Boltzmann weight for each of the links cross by gamma. So I change this weight just by shifting the spin, which is here. Now this one is multiplied by j. This one is transformed by j. OK? So that's a contour. So that's the definition. You just change it for the links which connect to the outside of the contour. You don't change it for the inside. I just change it on one side of the contour because it's invariant. If I were changing both spins, it would do nothing. But this s minus is in heat. It's all the contour. It's a product of all the links. So if I have many links. So I do it for one links. But that's the link. Cross by l is the link cross by gamma. So all the links which I cross by gamma, I change the Boltzmann weight. So s plus is minus. Refer to the link if you want that one. This has to do the link. And the link is this is the link. This is l minus. This is l plus. And if prioritization is gamma, this is the direction. OK? Is it clear? So it's safe avoiding or I can intersect itself? So what happens if you cross it several times the same edge? Do you allow that or not? No, I will not. But probably you may, or then you will learn. I have a more stupid question. Is it for purpose that it is closed? Yes, up to now, it's in purpose. OK? So then I can compute. I can insert here this correction form. I can look at correction function with many local spin variables. And say one of the spins gamma. OK? I can do that. This one is non-local. So this will correspond to this configuration. So I compute the expectation value where there is many spins, the contour, some of the spins are inside, some of the spin are outside. OK? And then the first property is that this is the same thing. There's not much to do as a correction function of the spins without the contour. But then you act. So the correction function with the disorder operator inserted is the same thing as the correction function without the disorder operator inserted. But you have to rotate all the spin variables. And you rotate according to which to the representation which is associated to the sides. OK? So that's the property. Since I am not a mathematician, I never write theorem, but just property. So where does this follow? So when you compute such correction function, you have to sum over all the Boltzmann weight, all the spin configuration with the Boltzmann weight, and with this variable inserted. But since the contour gamma, the disorder operator, inserted defect all along the contour, you can renown the spins inside the contour. By S is the renown G minus, G minus, inverse of G on S. OK? That's a bit, you assume it's a big action. So then you do the sum, and that will, then you will undo the defect, but you will transform the spins. OK? So it's important, as Hugo's question, that the contour is not self avoiding. Probably, because you need an inside and outside, right? Yeah, but if I had many contours, so I would define it. If it's, I will do for, that will correspond to two contours, gamma 1 and gamma 2. That's an exercise, you should take the rotation number or something like that, defect. So I will be able to decompose on each of the connected components. So the effect of the contours is just to rotate the spin inside, OK? So that's what we call in a field series, this is a wide identity. In field series, that's what we call wide identity. And that's the same thing as a go slow, because what does a go slow? It tells you that the flux of something is equal to the charge, flux inside the ball is equal to the charge inside the ball, OK? So that's exactly the same thing. So why? Because for instance, we can take g, g group. And that's also what I want to say, that's more or less the same thing as a neutral current if you want, neutral CORN in field. You know that in classical mechanics, every time you have a symmetry and you have a concept of current, which goes to a neutral current. So here we have a symmetry, and then we have an assuade current. We will have an assuade current in that case. So for that to have a current, we have to consider a Lee group, OK? So if the Lee group, we can look at group element close to the identity. So we take j is equal to 1. The 1 is a group plus epsilon x. x belongs to the algebra of the group, OK? So then when I do the expansion here, I will have something which will be close to the identity, OK? So then in that case, mu of j of gamma will become something like 1, first time, plus something of order epsilon, which will be linear in x, OK? Which will be a sum over all the link in gamma. Something which I will call g star x, I say to the link, OK? Doesn't matter what is g star, it's just what you got from the expansion here. It's linear in x, and it's assuade to the link, OK? And then what this identity says is that I want to give a name, def. I will call it integral over gamma, discrete one if you want, of g star of x. It's a sum over the contour. So then what this identity becomes to? So for g equal 1 plus epsilon gamma. So I expand that to first order in epsilon. This one to first order in epsilon. And then I got sigma i1, r1, rn, integral g of x on gamma. This is the same thing as the sum. When I expand that to first order in epsilon, I got the sum over all the sides, i k, which is inside gamma, a of x, r k of r on correlation function. So let's go slow. And this is the way we write, usually, the y identity is in field theory. The flux of what I will call j star is equal to the sum of the charges inside the domain, the surface. That's the analog of the neutral currents. So it means that. To deduce this, you need a continuous group, a loop group. If g is plus minus 1, like in the discretizing case. So what you can do, so you have this statement here. What you, but nevertheless, what you can do when it's discrete is that you can see that you can deform the contour. So here, you can see that you can deform the contour. There is no obstruction to deform the contour as long as there is no topological obstruction, it's the sum that you don't cross the speed. So that's true for any group element. So remember that. Here, what you see is that when there is only group, you can deform the contour. So this form is closed, actually. So away from charges or spins, g star j of x is equal to 0. That's what we call a conservation law. So it's a flat curvature is 0 if you think that this is a connection. So here, you have the same thing. It's a transport. You can view it as mu of j of transporting a collection along gamma. And this connection is flat. You can deform it. So that's true for discrete, yes. Well, that will be true for quantum groups, too. Is it clear? What you're saying is that the contour cannot cross the speed I1, I2? No, the contour is on the dual lattice. So it never crosses the contour is on the dual lattice from the middle of the face to the middle of the face. On the spin around the vertex. OK, so now, paraffinions, so they are non-local, are just, so I call it psi g r of i. So they are indexed by a group element, j belongs to j. This is a representation of j in some vector space in v. And that one takes value in v. And i is a lattice site. OK? And it's just defined by sending one of the contour to infinity. Sorry about that. Otherwise, it will depend on two points on the final lattice. So I take i. So that's i. And that is a representation associated to it. And then, which color I choose? And then, I have a j here, OK? That's my disorder operator. I will put it until here, OK? j. And then, this one will go to infinity. So you may choose infinity to be one of the points on the boundary if you have a boundary, OK? And then, from what I said before, as long as you don't have insertion of other operator, you can move this string as you want. So the j was the contour? No, j is the element of the group. So you have, you put an effect here. And it means that you change the Boltzmann weight. And you insert a spin here. Attached to that, there is a representation. This is like taking a contour that is actually just a path? Yeah, exactly. On the infinity, the point on the boundary of a physics is far away, OK? And then, that string, you can, if there is no insertion here on the lattice, you can move it from what we said before. So it's a flat connection, OK? So that's a parafermion, OK? So it's exactly as from, it goes lecture this morning. This was the analog of the winding number that we introduced. But it was for the n, not the n, exactly. OK, so the point I want to explain now, it's why they are called parafermions. And the fact that they are parafermions is that they satisfy some breading relation, which means I want to take a product to compute the sum correction function, whereas there is two parafermions inserted on the lattice. Imagine I want to compute some correction function, whereas there is this operator inserted. And then, before doing that, you can try to say, what happens if I want to compute here? So let me, maybe something simpler, I want to compute some correction function, where I insert sigma on my lattice, OK? So then I have, that will be I1. And imagine I put I2 on the same line, OK? That's I1 and I2. No, that's I2. I don't have a problem on I1, OK? So I have two spin observable inserted at this position, I1 and I2, which are on the same line. And then I have to introduce my desired operator, J. And then I have two possibilities. Either I do that, or I do that, OK? So before, it was the same, because I can move it. But now they are not, a priori, the same, because there is a topological obstruction to move the strings, OK? So that I will call sigma2, and then paraphernalia, because I define by convention that this is the order direction in which I order the observable, where this one will be sigma times psi. Because this is first psi, and then sigma, OK? Now I do something for this product. So this product I will have I1, J1, I1, and I have I2, R2. And then here I do first, shit, I will do the opposite. I will do first one. Is it possible to explain how the end of the path is restricted? I mean, does it have to be in the box next to I? How is the end of the path restricted? You have the freedom to put here, here, here, here, here. You have freedom. For the icing model, you can do both. And that will define four fermions, which will be the four fermions. Changing the end of the path does change a bit. So if I do that, I first act with the paraphernalia number 1, and then with the paraphernalia number 2, which means that the second strings, which is G2, act afterwards. And then I can move it as I want upstairs. But now this string, I can move it so I can write it as. And then I recognize the first paraphernalia number 2, which act first. And then the paraphernalia number 1, which has been dressed by all these strings around here. And what it does, as we said, this is a close contours with the spin inserted. So you will rotate the spins. And here, we will have the product of, because we go, there is a direction, we conjugate G1 by G2 for I1. So you see that, oops, there it is. That's what we call a commutation relation. We view it, we order the things. So we define ordering of observable. And that's a commutation relation. If it was G2, it will be just phase. The end will be just phase. If it would be G2, the minus 1 group is abelian. So it does not show anything. If it was G2, it would be just plus or minus 1, this one. So it will correspond to fermions. So that's called breading relation. And this answers the question, why are they called paraphernalia? Because it satisfies relations which are more complicated than just to pose a fermion commutation relation. So again, in the case of plus or minus 1 spin, then everything breaks down to plus or minus 1. Yeah, exactly. How could that give so many information? I will tell you because of that. You can reconstruct completely the operator from that commutation relation. And because you are breading something, actually you, so you see when you bread something, you are permuting position 1 and 2. So you will get a representation of the breading groups. And that's what we call Young-Baxter equation, which are key ingredients for integrable model. But here we get it as a compatibility for the breading relation. So we have a representation of the bread groups to the solution of the Young-Baxter equation. Because what we do here, we start with V1, which one was first, 2. So we have V2, some element which belongs to the representation space of 2, G2. We have V1, G2. So V are a vector in the vector space due to the representation R1. And V2, a vector in the representation space due to R2. And what we do is that we exchange the two. This one becomes R of G2, R1 of G2 acting on V1. G1, G2, G1 minus 1. And this one becomes this left invariant. So that's what we define on these objects. If you want, this is group lj1. That's a vector. Yes, you're OK. And then we can do with three points. So we have 3, 2, 1. We can try to break all of them. Then we will have break everything. What you can check is this is the same thing as reading further to 1, then reading the rest. I'm reading again the two. It is. Sir, you should decide whether you want to have over or on the front. Yes. I was careful here, but not here. So here you have to be careful. This one is always above. And then I have to be careful. You have to decide. But here I am careful, here I am. So then you have this relation, and that's a young backstay equation. Excuse me. In your braiding at the top, then, could it say G1, G2, G1 inverse? Yes. G2? Or is it not supposed to be G2? Ah, OK, yeah. G1, G2. I have a tiny problem. I am dyslexic, so I mix the letters and numbers. OK, so I did the first part. And there's five minutes left. So I can explain. So probably I can explain. So what I want to say is that maybe I will discuss why you get so much information for icing. And then maybe that's why they asked me to talk. That's to discuss more about quantum group symmetry. So here I did a sequence of invariants under a group, or a Lie algebra. The group on Lie algebra can be deformed. Then if you assume that's called quantum group, then if you assume that you have an invariant under this deformed structure, then you have an analog of the neutral current. So you have current, which are conserved. But they are non-local. And this, so that was the construction we did with Giovanni Felder in the 19th, showing that every time you have a quantum group symmetry, which occurs in any integrable system, you will have infinite family, which can be infinite if the group is big enough of non-local concept current. And then if you do that for the six vector models, which reduce to the O-N model, you produce all this power fermion, spin-off power fermion. And the reduction was proved by Icliffe and Weston. And then you can also reverse the logic that asks whether you have, in which case, you have a conserved current, which is sometimes olomorphic paraffin, but they are actually conserved current in the sense that you have contour integral vanishes. Then this forces the Boltzmann way to satisfy the Young-Bachster equation. And that's what Cardi did. He reversed the logic and asked, when, under which condition on the Boltzmann way do you have a paraffin unique current? And the fact that he imposed this condition imposed that you have the Young-Bachster equation, so you have an integrable system, is just the fact that if you impose a conserved current, you are imposing a symmetry. If it's non-local, this symmetry has to be a quantum group. And if it's a quantum group, then you know that to be symmetry, it has to intertwine the Boltzmann way, the air matrix has to be the Boltzmann way. And so it has to be integrable system. So that was the third part, a bit difficult to explain. Two minutes. Then I would just. Excuse me, this conserved current is supposed to be an element of the quantum group, or? Yeah, that's the universal element of the universal enveloping algebra of the quantum algebra. OK, then I can explain in private, if you want. But the construction is more or less the same. Once you understand where the can of several constructions and to move to the quantum group, construction is simple. So maybe what I want to explain is why a few consequence of the paramilitary construction or what you can think is the consequence of that. So that was two because in a physics. So maybe I start with the massive case and then on massive. So what we call massive is that a way from criticality. So if you are at criticality, you have macroscopic object, like Hugo said. So then you can take the continuum limit. If you are close to criticality, you can still take the continuum limit because you have macroscopic object, but you have to scale the way you approach criticality in the same way as you scale the lattice. So assume that we are in this scaling regime. Then you have a field theory. So why you can this give information is that there is two remarks. So the first remark, which is general, which is this breading relation are purely topological. So they are independent on the fact that you are at criticality or away. So they are invariant. In physics, we will say they are invariant under the renomination group. They apply also away from criticality. So breading relation, still valid, are valid at and away. Then the remark two, which is true for ising. So for ising, you have G is SU2. And so we can take the only note. So there is a spin observable, which I will call sigma of x spins. You have the disorder operator x, disorder. And then you have what I call the paramount, which is now a fermions. Just for when G is a non-trivial element in G2, and R is a non-trivial representation in G2, paramount. And this satisfies some relations at psi of y, sigma of x, mu of 0, psi of x will be the sine of x, psi of x, mu of 0. Because why is there this relation? It's because it's by construction. Is it this one I want to? Maybe it's sigma. Sigma is better. OK? So for example, if x is bigger than, so maybe it's, so if my paramount is here, so if this is x for the paramount, that's 0, they commute. There is no obstruction. So it should be minus 2 minus x. If x is negative, that doesn't matter. If x is bigger than, I have an obstruction. Because I have spin here, 0, or the string here, x. Then I have an obstruction. So now I can, I'm physics. So I have sigma here, so I can just multiply by the inverse. But this observable satisfies anticommutational relation. So this means that sigma at 0 is the automorphism of the commutational relation of the fermions. So that means sigma of 0. What we, in physics, we call it Bougolois transformation. But here, it belongs to gl infinity. And you can construct it completely from the fact that it induces automorphism. So it will be, then there is an explicit expression of, it's an element of, it's a group, which, an element of the group which preserves the commutational relation. It's a specific element, which is specified by this, how it transforms the fermions. So then you construct it completely, and then you know the correction function. That's what is called the expectation value of these objects are an example of tau function. And then the Kyoto school generalize it, and they find what is that. The tau function are orbit of these groups. So that's the tau function emerge from the study of the Ising model. And one of the basic point was this one, that you preserve that sigma is the generate, the automorphism of the commutational relation. Isomodromy deformation, which is another way to obtain the Ising correction function from the fact that when you transform the temperature, you preserve the commutational relation, which means that you preserve the monodromy of the correction function. So that was one of the consequences. So the last consequence I want to discuss is vertex operators. So as it was said this morning, if you look at loop model at n equal two, critical point correspond what is a special, and it correspond what we call to free boson in a critical temperature. And when you have free bosons, you can construct vertex operators following Mandelstein. This construction originate from string theory, but it's a direct application of this one. So CFT construction, so Mandelstein. So you consider a boson, so it's a field, which is x and t. t for me is the direction which goes up when you do field theory, classical or not, you have a conjugated momentum, which satisfies canonical commutational relation. And then what Mandelstein defined is something which depends on alpha, x and t, and which is plus minus, maybe it's plus minus, e alpha integral from minus infinity to x of pi dy times exponential of i alpha phi of x. So again, this is an object which is infinite strings, with the spin, that's analog of the spin observable, local one, and that's infinite strings. And the group here is a u1. So that's the object which was introduced by Mandelstein. They satisfy breading relation, so you can also, because if you try to compute the commutational relation, you will find some breading relation because of this commutational relation, maybe there is i here. And if you look at, if this is a massless field, you will see that this is analog actually to, if it's a free boson, that's correspond to a special conformal field theory. Well, that's one of the examples of vertex operators in field theory, which are holomorphic. So if I wrote something like x plus i t, you will not be happy, but so this, in the massless theory, so in CFT, this will become vertex operator, which are holomorphic. And in general, vertex operator, so if you look at correction function of such object, because this is a commutational relation, this will have monodromy property. Okay, you can look at, when you turn the, you look at endpoint correction function, it will be on the sphere, some remand surfaces with end puncture, and you can try to see what happened when you move the pointer on the, on this puncture on the surface, you will get phases. The monodromy property reflects the breading relation of the operator. More generally, you can look at breading relation of vertex operators intertwining conformal theories, which satisfy many property, for example, I did many things on that, and all these breading property are, in a way, a formulation of the breading relation of the paraffin. So all vertex operators in conformal theory can be viewed as vertex operators for some group, as to some group or quantum group symmetry. Okay? So that was the notion of a vertex operator algebra, monodromy property, which are consequences of this paraffinic construction. Okay, so I have been long enough, I think. Now we stop here. Thank you. Yeah, I apologize, but I want to return to this question, why Dyson model is so special. If I start with something else, right, then it looks like I have a representation, not of commutation relations of the thermos, but those of pattern thermals, right? So what's, I'm not familiar with that, but what's the problem? They're not well-studied, they're just wild. First, on the lattice, as it was said, you can define different paraffin, depending where you put the strings. And that's how you will define, so you will need to use different set observable. And which, at the critical point, will become the two components of the mass, the direct, the massless thermals. Okay, otherwise you don't have them. So that's one special property. And then, Dyson model is also particular because it's paraffinic observable if you look at, so on the lattice, it satisfies linear equation of motion, that if you look. No, this I know, but somehow your argument was, I mean it looked like algebraic, right? Yeah, but you know, and you gave this argument, it was no, nothing said about this linear equation, so it looked much more abstract. Yeah, so maybe it's why you get, maybe there is hope to do something. So in practice, yeah, you have commission relation, and you know on which space you have, so this satisfies some commission relation, which means you have some algebra, and you know on which space you can represent these algebra, and then you have an automorphism on this, algebra automorphism on that space, and you know how to represent this automorphism. That's, you don't know for the end paraffinion, you will not know how to do that. Okay, so that's what is special about the, I mean. It's the equation of motions, I mean it's truly much, I mean I think it's generality, right, that there are some linear equations for probes of skins and sorters. Yeah, so they don't, if you want in a way, I think model is a question of motion closed on a finite set of object, which has a paraffinion. For the zian model, they don't, when you look. For example, if you look at this current, they don't close, but nevertheless, for example, that was one question I wanted to ask. So you have, in general, for the when model, you know that you have it, you have a series of infinite set of quantities and from another work done by the Kyoto school by Jimbo and Miwa in the 90s, so Jimbo and Miwa. So they use this quantum symmetry, so they don't, so that's what's two parallel work, but they were able to use the quantum symmetries, which is an infinite dimensional quantum symmetries, the six ver test model, to compute correction function in some regime. So even if a question of motion do not close, you know enough about the representation theory to compute correction function. So I think that's one open problem for the probabilities is to do the parallel construction in probabilistic term of what Jimbo and Miwa did.