 Okay, I'm really honoured and pleased to speak on such an event, on Samsung's 60th birthday conference. And I think we know each other for almost 40 years, I would estimate. And I always wondered why Samsung was such a visible personality in our community, in our scientific community. I decided one of the reasons is that not only he knows how to take care of the people, but he also knows how to involve them into various adventures. So here on this picture which I took in Châtelet, the theater of Châtelet, he gives a talk on the Dao art event, where he was one of the main instigators, one of the main persons. Having organised it, he gives scientific talk and he invited also us, asking to give really scientific talks. And only during the talk, little by little I understood that we were just some exhibits on this exhibition. So it was quite an experience. So this is one of little adventures in my life, which was interesting. And as concerns the science, we sort of always worked with Samsung in parallel worlds, which were very close to each other. It's the same keywords. Gauge theories, string theory, conformal theories, integrability, matrix models, but sort of rarely discussed unfortunately. It's exonsober. It's exonsober, I remember very well. I just mentioned this problem and Samsung solved it in a few weeks, I guess, the correlator and it's exonsober integral. But the life is not over, maybe one day we'll have some collaborations. Okay, so I will speak about the subject which I'm working on like for four and a half last years. This is conformal fishnet theory, CFT. This is the abbreviation, which you know, in any dimension. So the main question why it was interesting to me was the following. Say N equals 4 super young mill theory is an example of four dimensional CFT with phenomenologically very interesting properties. It is very well defined. It is strongly interacting CFT, well defined on all scales from infrared to UV. And also having a rich modular space. And this is quite a standard phenomenon for supersymmetrical, but unfortunately supersymmetry is not observed in nature so far. And the interesting question is do we have non supersymmetric CFTs with these properties in four dimensions? One example which I started studying in detail was gamma deformation of super young mill. Susie is completely destroyed, but if you want to keep it conformal, unfortunately the unitarity is broken. So you have to pay by this or that way for conformality and also what is nice in this theory is integrable even after gamma deformation in large and limit. So we can calculate the spectrum of anomalous dimensions by the method which is now called quantum spectral curve. I will speak about it a little bit. But integrability is very much related to ADS CFT duality, but both stay very mysterious. We don't have any proof even in this sort of emblematic example of ADS CFT. We don't have any proof of either integrability or ADS CFT correspondence. The proof which would be even physical proof, not even mathematical. So I wanted to find some simplified example, maybe some limit in this theory, where the integrability becomes manifest. This joke I want to make, I'm sorry. Once I was giving lecture in Newton Institute, Alan Kohn, who worked here, Alan Kohn, was organizer in sitting in the first row. And I said exactly like that. Physical proof, mathematical proof. He stood up and he stopped me and he said, Samson, I tell you the statement is either correct or wrong. And this physical and mathematical, just get it out from your mind. Is it correct or wrong? And we know the statement is correct. Okay, but we don't understand the reasons for it, which is already a deeper question, yes? Really, we don't understand deep reasons for the theory to be integrable for any coupling. Okay, then the proposal was to special double-scaled, double-scaling limit in N equals four young glills, which combines strong gamma deformation with imaginary parameters and also weak coupling. And the integrability will become manifest. Namely, the outcome of this limit is this fishnet CFT, which is the main subject of my talk. And it appears that it dominates by regular fishnet planar graphs. So sort of this Feynman graph, you take fishnet, regular square lattice. You put propagators, say, between points, neighboring points. It will be J minus XK to the power dimension over two and you integrate in each vertex. So it's like a fragment of big Feynman graph, which is called fishnet graph. After the logic of paper, who showed actually that this statistical mechanical system, this is an integrable model. And the underlying integrability is S or D two comma D conformal spin chain, corresponding to conformal symmetry of this graph. Okay, but let's start now explaining how this model, how the fishnet CFT can be obtained. And first we have to gamma twist the young mills. The Lagrangian of young mills is presented here. It contains three scalar fields, complex scalar fields, three complex fermions, gluina, gluons, and with somewhat very familiar interactions here. You have commutators of model squared of phi, here commutator is fermion, so there are commutators everywhere. And what is the gamma twist? Since all these fields are matrices and color matrices under trace, so the order of the fields is important. And you simply assign a particular factor to each order of the fields. Say if you have order AB for two fields and B, then if you interchange them, you get a factor QAB, which is exponent of antisymmetric combination of charges of r symmetry of this field. You have SO6 symmetry, r symmetry, so you get here these carton charges. And gamma mu is a parameter, are three twist parameters. So you have three more couplings here. And of course, the practical way to implement it here is simply to substitute commutators by Q commutators. By this formula, so this Q factor appears in this way. And then you simply just write letter Q at each commutator. And of course, this procedure breaks completely r symmetry to U1 to the cube, SO6 to U1 to the cube. But the conformal symmetry stays there and supersymmetry is completely broken, of course. Sorry, do I understand correctly? So this deformation, basically that's the carton of your symmetry group which gets quantized. And then the rest of the symmetry group does not get quantized, just the carton. You just point a vector with this coordinates in r symmetry group. So it destroys r symmetry up to the rotations around carton generators. And that's what you have. But conformal symmetry in principle stays. And if you tune well the parameters, which you get still integrable non-super symmetric logarithmic CFT, I will speak about new parameters. There will be so-called double trace couplings here. I will discuss it. So what is this my double scaling limit, fishnet limit? I send coupling, that young mills, coupling to zero. Then the parameter gamma, this parameter gamma, to I infinity. In fact, instead of a unitary factor, I get some big or small q. And the product of these two factors is fixed. I call these new couplings xij. So in this limit, interesting things happen. For example, from the commutator you see already that if q goes, for example, to infinity, and then one or q to zero, you drop one of the terms of each commutator. That's more or less what happens. You drop halves of all these commutators, or sometimes all commutators, for example. Gluena goes away, decouples, the gauge fields go away, because there is weak coupling, but nothing to enforce the strength of Gluon interaction. So you stay with three fermions and three bosons, essentially, and with some of their interactions. I show you now the whole action, as it comes after the double scaling limit. So you have bosons, three bosons, three fermions, complex. And then you have these kind of interactions, which now are chiral interactions, which are non-unitary, because the whole deformation is non-unitary. And you have some Yuccavo terms of this kind. So it's Yuccavo plus five four? Yuccavo plus five four, but with very particular orders of fields under the trace, so that they become chiral. And here I discuss the underlying Feynman graphs, the perturbation theory. And already it shows a remarkable lattice structure, regular lattice structure, not completely regular. So it consists of three types of lines, three colors of fermions or bosons, green, black, and red here, which cross each other. The lines can move, so you can take this line, move it through the crossings, etc. It changes graphs, but that's the only way to change anything. And also you put dotted, some dotted pieces of these lines, which correspond to fermions and solid, which corresponds to bosons. And also, and then everything is fixed, because if you have, for example, the crossing of bosonic and fermionic line, you have to disentangle it in a very particular way by Yuccavo couplings. So everything is now unique. Once you draw this skeleton graph, it's a unique way that you decorate it by, due to chirality, it's a unique way. So here it's this picture. So interestingly, this lattice system is another integrable system that we know for sure, because the whole Young Mills, N equals 4 Young Mills gamma deformity is integrable, and as Samson says, it's either false or wrong, but we have many, many proofs that it's actually false or true, and this is true. Anyway, this is a strange lattice system with sort of, which has some dynamics. You can move these lines, but otherwise it should be an integrable spin chain. Of what kind? We don't know. But if we stay with only one couplings, suppose we put xi1, xi2 to zero, stay only xi3, then all fermions go away, because they have such couplings. Then these interactions go away. You have only this interaction, and you have the model, which is actually the fishnet model. It contains two out of three scalar fields, and this interaction, phi1 bar, phi2 bar, phi1, phi2. Now, what is the perturbation theory? You have two types of propagators for phi1 and phi2. They're conformal, of course. It's in four dimensions. It's in four dimensions so far. They are massless propagators, and the only type of the vertex, which has specific chirality. If you draw it by double lines, as Hoft taught us, then you should just draw it on the plane without turning around. And then the arrows, which go from phi to phi bar to phi1 bar, phi2 to phi bar, for example, if you go from blue to red, you go clockwise. And the vertex, which corresponds to anticlockwise orientation is actually missing here due to this specific limit. So it shows already that the model is not unitary, but once you accept the non-unitary world, you are in the paradise because the only graphs which are left are precisely these kind of fishnet graphs. You simply draw, for example, a piece of a big graph, and you cannot reorient this. You have only these kind of structures. You try to turn, for example, this line of propagators to turn back. You immediately encounter this kind of vertex. So this is with appropriate boundary conditions. You can only draw regular plane of lattice. And in the big graphs, it will be always regular square lattice. So this more complicated picture becomes a much simpler one. And all this mass of n equals 4 Young-Mills diagrams goes away. You have these nice graphs. There are very few of these graphs in each other, sometimes zero, sometimes one, sometimes two. It depends, of course, on the physical quantity on the boundary of these graphs. The boundary is defined by physical quantity. And it's remarkable that, like in Young-Mills, the coupling psi-square doesn't run. The beta function is zero. But, for example, this fact is important because, for example, you can see that you have no mass renormalization. This is the graph which renormalizes the mass. But, of course, you have one chiral vertex, but another one, just by topology, should be anti-chirals. So you discard this graph, and this property propagates, of course, to higher orders. But it's well known that you generate so-called double trace vertices of this kind. For example, this graph is not Lanner, but trace-trace still survives even in the large n-limit. Like n-trace is of the same order as trace-trace. So we have to include this graph. I'm sorry, just one point. Considering this dynamical picture, could you have a dynamical analog for the integrable spin chain structure with the dynamics on the sides? I could, but I don't know what are these spins. You mean about, you speak about this? Yes, yes. I wouldn't say a space of running couplings and then the better function would correspond to something. No, here, once you have a graph, I mean, this couplings, you speak about double-trace couplings? Yes. They appear only in very specific graphs, which can be cut into two along this vertex, double-trace vertex. Otherwise, they don't survive. They actually rather serve here as regularizers, in some sense. I have no time to speak about it, but the only thing I wanted to say is that a careful choice of these couplings leads to the critical point and the theory stays conformal. It's true also for the gamma-deformed n-equals form. You generate these kind of terms, but you can tune to the critical point where the theory stays conformal and integrable. Internet divergences? Ultraviolet divergences. Of course, you have ultraviolet divergences, but they sort of regularize them, and then everything looks nice at the end. Then, actually, you have to add them, of course. All these alphas depend on Xi. It's lines, critical lines, not just critical couplings. Xi-dependent couplings. What is interesting, this theory even has a space of flat moduli. How does it happen? As usual, we can give an average to each of two fields, vacuum average, and look at the fluctuations. Then the effective action for fluctuations, the effective action will contain the original action, depending on these averages. Then plus one loop correction, where m is the mass matrix. It's just corresponding to one loop contribution with this mass matrix as inserted. L here, of course, contains both signal trace and double trace terms. Then it happens that, actually, there are no other corrections here. The action here is exact. One loop Coleman-Weinberg potential appears to be one loop exact. You can see it, for example, trying to draw other loops of two kinds of fields. But you see, the circles should cross each other, both in chiral and in the chiral direction. If you have a chiral, allowed vertex, then there will be another one, anti-kiral, which is forbidden for planar graph. Is the sign correct for Coleman-Weinberg? There are two cases, kind of, right? Upside down Coleman-Weinberg and the right one. I wouldn't remember. I hope so. I should warn that everything is complex already. So it doesn't matter. You are in a complex world, so this question doesn't make sense. Okay, but it's nice. The game is still nice. So the flat vacuum must obey, of course, the condition that you are at the extremum. Again, for a complex vacuum, it could be minimum, maximum, whatever. And one can give examples of such vacuum. For example, for phi 1 field you take 0, for phi 2 you take a diagonal matrix such that its trace of square is 0. The matrix is complex, so you can, of course, easily fulfill this condition. Then you get this factorized mass matrix. For example, for the choice when you have this kind of breaking of the symmetry, you have four similar eigenvalues which repeat many times and over four times. You impose unit modularity on this, guys. You impose that classical Coleman-Weinberg is 0 and you also impose that quantum part. This is 0, which is this condition. Then you find a nice solution. You can always find a solution. Somehow this theory inherits the vacua. And one can actually show that the vacua of original n equals 4 gamma-deformed Young Mills are continuously connected to this vacua. So there was, of course, kinetic term also, right? Kinetic term somewhere. Yeah, but it gave this part, for example. Yes, you integrate it over delta phi's. And it also has kinetic term. Okay, you consider you do this decomposition. You start integrating. You look at Feynman graphs and all of them disappear. So after all, you found the full effective action. Later you can expand around this effect. No, I mean that does effective action has a term d mu phi squared. Derivative term in phi. Phi is a space time dependent object, right? Of course, you look for the vacuum. You look for translation invariant vacuum. My question was d mu phi squared kinetic term. Also, loops can get a correction. It can be d mu phi squared times a function of phi. I agree, but that's what I say. You carefully write the whole action as function of delta phi. Then you start integrating. Then you observe these graphs and you will see what happens. Of course, at the end, you have to put them translation invariant, not depending on if I understand your question. So I think it's all quite orthodox, this kind of consideration of what is flat vacuum. This variance for z are exact or this is approximation? Approximation, yes. You solve this system of equations. You mean these exact numbers? These are not approximations? No, of course it's approximation. This is an algebraic system which has, of course, it scales. You can put 1z4, for example, to minus 1. This is approximation. Yeah, you have. It's approximation. It's on computer. But this is a unique number. These are unique numbers. Solutions of this system with z4 equals to minus 1. Okay. Now, what kind of physical quantities we consider here? Various, in this biscalar theory, various local operators and their correlators. So this is the most general operator containing fields, their Hermitian conjugates in various powers, the derivatives, and then you can, of course, make permutations under trace. So pictorially, for example, without derivatives, it looks like single trace, the fields phi1 under single trace and a couple of fields phi2, for example, which we call magnones. It looks like a spin chain picture. So phi1 creates a vacuum and phi2 creates, there are two magnones, two excitations. For correct choice of these combinations, for operators diagonalizing the dilatation operator, we get particular dimensions, which we want to compute. This is the spectrum of the theory, spectrum of anomalous dimensions, and we want to compute it. The simplest possible operator is just trace phi1 to the vacuum. In this theory, the vacuum is not protected, no supersymmetry. So the renormalization of this operator is given by this wheel graph. So the center is this operator, and then according to the fishnet rules, since we have particular orientations of propagators, this is the only graph at each order of perturbation theory, which we can draw. So everywhere, apart from the center, everywhere it's a regular square lattice, as it should be. Then if we compute this graph, we have some, in epsilon expansion, we have some divergency. The power of divergency corresponds to the number of frames here, the highest power. And by standard rules, we can recalculate it to the dimension of this operator, which is a combination of these coefficients. And in this way, we can generate... No way from epsilon, no longer from epsilon? No, no. Never. Never. It's especially done to avoid logs, yes, epsilon expansion. Okay, then there are interesting generalizations. This is sort of a vacuum. It's called wheel diagram, but there are also kind of multi-spiral or spiderweb diagrams, as I call them, for when you have a couple of magnons, then they start turning around because it's the only possible kind of vertices. So it's multi-spiral configuration, which is also integrable. Okay. And in principle, the dimensions of this theory can be computed by this machine, integrability machinery, which has been developed in Enneco's force super-ragmules, including gamma deformation. But you have to go to this special limit, double-scaling limit. And I wanted to take a little bit of time to remind the general integrability, the result for integrability of spectrum of anomalous dimensions in Enneco's force super-ragmules, because in my opinion now I need two transparencies to explain this. So now it's about the quantum spectral curve, the method which we developed after long development of integrability in Enneco's force-ragmules in ADS CFT. The final method we developed is called quantum spectral curve, including gamma twist. And here is how it looks like, because maybe for mathematicians it should be an interesting object, I feel it. It is CFT-4, right? No, 5, everything is 5. It is CFT-4, sorry, 4 of course. And it was for enneco's four-dimensional. Sorry, sorry, sorry. It is 5 for CFT-4. I probably copied it. So quantum spectral curve is an object which involves a set of finite number of so-called Baxter functions of spectral parameter U. These functions are labeled by specific multi-index. And here how they look like for JLM symmetry. For example, JL2, you have Q empty set, Q0, that's called, which is usually just one normalization. Then you have two Baxter functions, which are usually the solution of second-order Baxter equation. And then you have Q12, which is the Voron skin of these two. Then if you go to JL3, you have three neighbors of Q0. And then you have, say, you continue from Q3 to, and from Q1 you get Q13, which is the gain of Voron skin, et cetera, et cetera, up to the opposite corner of this cube. And then for JL4 it will be a hypercube with maximum four indices. This is a single hyper-index notation. Because usually you want the same ij, i-slot-j, so that you can find the full correspondence to which which it goes. Here everything is prescribed by these labels, I would say. It's quite usual notation for Haase diagram, the ordered sets, where you have no, in any index you cannot have equal digits, yes. But so Haase diagrams for JLN groups represent an hypercube. Now we also have some structure on these Q-functions. If we draw main diagonal from Q0 to full set, then the Q-functions are related by Pluecker relations. So if you have, say, a phase here, then on this phase along this direction, the product of these two functions is the Voron skin of that two. Voron skin with these shifts of spectral parameter. And in this way you can, knowing, for example, single index functions around Q1, Q2, Q3, for example, neighboring for QMT set, you can restore all other Q-functions. There are determinant formulas which restore all of them. This induces a Grasmanian structure on the set of Q-functions. And so this is more or less the description, the full description of the algebraic structure, except we need JL8 more or less, where you have 256. So you need JL4 slash 4. You'll see. So we start from this JL8 system where this red axis shows where how these Pluecker relations are directed, the flow of Pluecker relations. But if you want to super-submitrize it to make JL4 slash 4, you simply, I mean, it depends on where you take your boundary conditions. For spin chains, for example, you take Q0 equals 1, and this Q is u to the length of the spin chain and say single index functions are polynomials. That's enough to completely fix beta equations. But if you want to super-submitrize, you simply turn, you simply start from this to that. I mean, say, for 4 slash 4, you take this vertex, you fix it to 1, and here you fix it u to the length of the system. And this is enough to formulate the supersymmetric beta equations. So you just turn it. You mean Baxter equation or beta equation or Baxter? Beta equation. Baxter, I mean, everything. You just write this Q, full set, is equal to u to the L. That's enough. Imposing polynomials, I mean, it's for JL8. But JL4 slash 4, you do the same in this direction, and it's more complicated to relate this to that. The relations, Pluecker relations. Equations are the universal. This is a boundary condition. You look for solutions specific to equal 4, right? Equations, what equations? Whatever equation you write in... Pluecker structure, I mean, this Gerspanian structure is universal. And only the sort of the boundary conditions, how to call them, boundary conditions, initial conditions are different. So you have to interact with some relation between Q's. That's true. So you impose polynomiality and you impose that, for example, this guy is equal to u to the L. And that fixes all better roots, etc. You can restore all beta equations. Okay, but this is also a universal picture. It also should work for sigma models, for example, quantum field theories in two dimensions, and that way we can apply it to ADS 5 string. Okay, we describe the algebraic structure. Let's go to analytic structure. And it's also quite simple. You have specific 16-Q functions which have nice analytic structure on the physical sheet. So I go back to this Hasse diagram. Then first thing we fix, as I said, G4-4 means that you fix here Q functions, but you fix both of them to one in this game. So this one thing. And then you take single index functions on one side. I mean, I blow up here the vicinity of this pole. So you have four plus four functions. We call it P and Q. And also on this opposite side we have also, they're called Hodge-Duel, the diametrically opposite functions with upper indexes. And we know about them nearly everything. We know that the large US asymptotics on the physical sheet is defined by carton charges of the state you are considering, namely P functions which actually describe the dynamics on S5 sphere, the r-symmetry dynamics. They have this exponential factor which is twist. So the twists explicitly are introduced here. And also you have a power like factor which for various B's, it depends, you have different combinations of sides. Everything is defined by group theory here, of course, by charges. And also, you know, the other thing we know that there is only one single cut on the physical sheet which goes from minus 2G to 2G where G is the Young-Mills coupling. So the only place when Young-Mills coupling appears in the positions of these branch points and the only place where the twist appears is in these asymptotics. And then we have also Q functions which describe the ADS-5 dynamics. They have asymptotics related already to cartons of conformal group and they have also the same two branch points but long cut going around infinity. And that's it. And then we have, of course, to know something about monodermy around these cuts. And the condition is also quite remarkable. It's more or less here. You relate various Q functions from here to Q functions there by complex conjugation. Say Q function in this part is related to Q1 on the upper sheet. On the upper half plane related to Q2 bar which means on the lower half plane. And these are gluing conditions. And they are more or less Riemann-Hilbert conditions which fix everything. I describe the whole scheme. If you fix charges, you will find a discrete spectrum of anomalous dimensions. And there have been many... It's already quite heavily used this scheme. The quantum spectral curve. There are many results obtained. Very precise numerics for anomalous dimensions for essentially all interesting couplings. For example, for Konishi and similar operators. Konishi operator is like 4s equals 2L equals 2 here. Strong coupling expansion. And Konishi dimension perturbatively. I mean it's known up to 11 loops but only the size of the computer or the computer's strength is needed to go further. There was zeta with sub-enix. Is this the value of Riemann zeta? Yeah, everywhere zeta functions here. Everything is expressed through Riemann zeta numbers. And there is a... He was asking why is there only odd arguments? Only odd arguments because even zeta functions are trivial, like pi. There is also no name for gernambles, you know? That's true, that's true. I mean real numbers, not all of them are integer. But here, there are only odd functions. That's true. What do you think happens to XXX correlations? Yeah, we can remember it, yeah. And also, for example, this beautiful picture of analytic continuation of this operator with respect to the spin obtained by Gromov and Kovoters. And numerics here for quantum spectral curve works like you can get 50 digits after the coma or 100 digits in no time. It's really exact. Still, I still have a question in this long history. Do you have z-capital with three digits? Oh, I forgot. It's some... Also some zeta numbers, but I forgot how it's defined. It's... We should ask the authors of this 11-loop calculation, but it's in the same class of... Also, it's kind of generalizations of riemann zeta, but I forgot... There is zeta with two arguments, multiple zeta values. You do have, of course, multiple zeta values here everywhere. This is multiple zeta values. But zeta means something else, right? Yeah, it's something else, but I forgot what I should have. Do you include a huge number of digits and then you use some program to rewrite it as a combination of... Not here, not here, actually. Here it's really a quantum spectral curve. You generate these numbers. There is a way to generate. But there is also this dirty method like fitting these integers with zeta with the whole basis of zeta function. It's not meant to be seen. I should have done it smaller, huh? Topological recursion. Not that I know, but maybe it does. Like in matrix models. It's slightly more complicated, the object, but it might have something to do with it. At least Inar always says that it should have, but I don't know how. Okay, and already some results in the fishnet limits. I mean, this is exact for the n equals 4. But for fishnet limits, for example, for this operator, we compute directly the graphs. What is nice, since at each other you have one single graph, you compute directly Feynman graphs, very complicated Feynman graphs using spectral curve. So essentially for this operator, we obtain second-order Baxter equation by scaling, double-scaling the quantum spectral curve and works I, the coupling, this fishnet theory coupling enters here. And then there is these gluing conditions that boil down to some quantization conditions between two solutions, which are called q2 and q4 by some historical reasons here. And that's, and also you have asymptotics defined by charges. And you solve, you can solve it numerically or generate the perturbations theory in Xi and it gives exact values of these graphs. Or rather, it's called periods of these graphs when cells are diverging. So here are some results. Also a lot of zeta numbers. And what was known before quantum spectral curve is the first line. This can be computed by hand. This one was quite a complicated calculation by Panzer at all. But then you see you can have 12 loops, for example, in no time using quantum spectral curve on computer, of course, here already. Okay, and numerics is quite precise and it shows, for example, the anomalous dimension, it is real up to some point when it becomes complex. You see, it's real part of dimension as a function of Xi of coupling. This is imaginary part. So imagine it stays zero up to some critical coupling where it becomes complex, which is normal for nonunitary theory. Okay, so actually the generalization to any number of spokes of these wheels is possible, but it's some work which already, I think, is done for four spokes, but would be nice to get some idea for any number of spokes. What are magnons? Ah, magnons, spiral graphs. You remember, I inserted some phi 2, you have phi 1 fields which create vacuum and if you insert phi 2, so this picture is already not, this wheel picture anymore, but spirals. And for spirals, for magnons, you also can, it's also an... It's phi 2 insertion, right, magnons? Yes, yes, but the graph completely changes to this multi-spiral. You can compute this multi-spiral graph. Okay, now the theory can be generalized to any dimension. It doesn't have any more, it's supersymmetric mother theory, but simply directly you write this section. You put here phi bar, Laplacian phi 1, but Laplacian is on power delta. For another field it should be d over 2 minus delta to make it conformal with the same direction, with the same Feynman graphs, and also conformal theory. So d k with all this dimension of the spacetime? Delta is actually any power. If you take delta over 4, it will be... Any real number? No, but better to make it between 0 and g over 2. That's small. Yes. On the equivalent spin chain it will be the spin on the side, on each side, the value of the conformal spin on each side. So you have this kind of propagator, the fishnet graph, and actually if you want to solve this problem you can start from writing such graph in terms of, say, for periodic boundary conditions, in terms of so-called graph building operator. So this is a product of propagators, and here is the picture. You have propagators going in this direction, of field, say, phi 1, and in this direction of field phi 2. This is an operator which projects the coordinates of the dimensional space here to the axis, to y's, sort of spins. Is it d over 2 there, or d has to be even, or the power has to be... It can be even continuous. It will be 1, for example? For 1, yeah. You can have 1 in principle. It makes sense, also, one-dimensional model. It was used, actually, the similar model was used in SYK. Two-matrix model, quantum mechanics. Two-matrix is one dimension. The answer is no. Two matrices. Your expertise. Two matrices. Quantum mechanics. Yeah, but it's nonlinear quantum mechanics. I said that for many physical quantities you can compute something in this theory. What if quantum mechanics you solved it a lot longer ago, right? No, but it's not the same. There are many different quantum mechanics. With two fields, with two matrices, it's not so simple. Not one matrix, not sequel one, but two. That's true, but not every two-matrix model is solved. This one, I don't know. Maybe you have to have different methods to solve it. But it's integrable, for sure. Any D is an integral model. Okay, then you can then build the, for example, the periodic cylindrical lattice by taking the square of this operator, simply integrating over intermediate vertices. Oops, sorry. Then you can take the next one. It will be the cube of this graph-building operator, et cetera. And what happens? It appears to be the conserved charge for conformal SO2, G spin chain. So, in fact, this operator, this graph-building operator, is simply one of the conserved charges of Heisenberg spin chain with this non-compact group. And the spins are in principle series representation here. An interesting limit is delta going to zero and D equals to two. It actually boils down to the Li-Patoff model of Regi-Ein's glons, famous BFK. So this is one of conserved charges of BFK, of Li-Patoff's Lagrangian. So things are interrelated. Five minutes. I will, there have been many questions. I will take slightly more. Okay. Okay. Now it's next to the last topic. We managed to construct for this spin chain for this conformal spin chain. We managed to construct the thermodynamic bit-unjuts, which calculates fishnet anomalous dimensions in principle for any operators. But let's start from the simplest ones, from these wheels. So, again, we take these multi-magnon operators. We have M magnons and L the length of the spin chain. These are the representatives, the multispirals. If you have magnons, this wheel operator with zero magnons. And you see this picture repeats in each sector. If you take these propagators drawn by bold, if you repeat in all sectors, you will build the wheel. So it's another graph-building operator represented by this object, which I've drawn here, I've written here. And if you diagonalize it, so if you have this operator, you simply have to take the power of this operator. So if you diagonalize it, you calculate the corresponding configuration for the Feynman graph. So let's try to diagonalize this operator. And if you have only... n is the number of these propagators. If you have only one, like n equals 1, it's very easy to diagonalize it by just conformal properties. It's a power times some spherical harmonics. And here is the eigenvalue, which already depends on arbitrary parameter u, which will become very soon the spectral parameter. So you just diagonalize this kind of object. You diagonalize it by wave function integrating here and this kind of picture. Now, if we go further for n equals 2, everything becomes more complicated and it's a real challenge, because already without integrability you would never solve this problem, probably. You diagonalize... I mean, suppose it's only 1, 2 and there is no this upper part. So this is the image of this graph-building operator. We managed to construct, using, of course, some interesting papers, two-dimensional papers of Dierkachev, Manashov et al. We managed to generalize it to actually find the eigenfunction, which consists of various propagators, which I don't want to explain every detail here. It's just a bunch of propagators of the type modulo x to the powers which are just presented here, which consists of deltas, et cetera. This is the spin here. So we have here already two coordinates and two spins. And the eigenvalue as it should be an integrable system is the product of those eigenvalues. And what is interesting, when you take... for this quite complicated function, I forgot to say that you integrate or these two points you integrate here. So it's a complicated integral. If you study the asymptotics for these points on a very big distance, you recover sort of scattering theory, where the coefficient, the scattering phase, is O-D-R matrix of zomologics precisely, depending on spectral parameters and spins. So already it smells like integrability. And if you go to any n, you can use this two-magnon block to actually to generalize it to any n. Say you can generalize to any number of... These are like spins. So you can generalize to any number of these spins. And lambda will be a product of eigenvalues. And this U1, U2, et cetera, Un are supposed to be separated variables of Sklanyan in principle. Okay, so this we can really... How would you treat the central integral of this case? The last case, how would you treat the center? Because... This integral? Two integrations, yes, no. But in the n-case, arbitrary... In the last case, yes. It will be many, many integrals. It will be sort of a triangular lattice filled by all these squares. So it's more and more integrations. But in the asymptotics, it's supposed to be just factorizable in terms of two particle matrix. Okay, and then we go through standard scheme by Alyosha Zamolochikov, the trick where we treat the finite system in terms of thermodynamic details. Maybe I don't want to go into details. At the end, we write the TBA for TBA and the corresponding y-system. So the anomalous dimension gamma is given through a set of y-functions. Y-functions obey y-system with this specific o, g, o, g, 2 diagram, y-system diagram. And okay, the details are not so important. But what is nice, you can also do the particle-hole transformation here. And this model appears to be due to another model, sigma model, which has Zamolochikov's... Actually, Zamolochikov's d plus 2 as matrix. And it's supposed to describe the ADS d plus 1 sigma model. But the S-matics is the same as Zamolochikov, but okay, there is y-system behind it. But the dispersion relation for one Magnon dispersion relation is quite different. It has this beryllian zone structure. You have sine square p over 2. And instead of this standard Einstein dispersion, you get Gapless spectrum. Then the particle-hole corresponds to filling infinite Fermi C. Instead of this one, you fill the outer parts of this. And okay, now I'm... Actually, I'm almost finished. And we reproduced in this way the exponential asymptotics of large fishnet graphs, which were observed already by Zamolochikovs. Zamolochikov in any dimension. Okay, I don't then speak about four-point function. This is only one transparency. So we can calculate interesting four-point functions and really generate the structure constants and the spectrum of conformal dimensions is given by this formula for exchange operators. Bucks and Dixon type correlators computed explicitly in two dimensions. Bucks and Dixon computed them in four dimensions. And finally, so I think that by saying that fishnet CFT is a unique opportunity to study a non-super symmetric quantum conformal world at any coupling. And it may give a window into the origins of ADS-CFT integrability. In addition, also the physics of flat vacuum of spontaneous symmetry breaking can be studied. And we want to obtain the full quantum spectral curve description of all operators of the theory in principle, compute various structure constants, which some of them were computed by both certain splendid separated variables. And there was a very interesting set of papers which claimed that they found the ADS dual to ADS dual of fishnet CFT of Gromov and Siver. They call it fish chain because they have sort of discrete string which leaves on ADS, which is supposed to be the ADS dual. It's still in discussion whether it's a real ADS dual. Then the last thing is, there are very nice fishnet amplitudes which are just pieces of square checkered paper which you cut out and then the external legs appear and you have explicit Jungian symmetry. You can construct from Lach's operator taking its product around the boundary. You can have absolutely explicit and well-defined Jungian symmetry which we observed some time ago. That's it. Thank you. Maybe a half one. If you take this fishnet here and say you compute the four-point amplitudes, how many channels do you sum over? Is it color-ordered to the S&T channel or do you also have a new channel? It's four dimensions, so... Just because it seems like it's an ordered theory. Ah, yeah. It's of course color-ordered. So it's color-ordered. So all your diagrams are color-ordered. That's true. Probably S&T, yes. Otherwise, you have to consider finite N. And ask a new one.