 Так и так произошло, что мы делали долгое время с Самсонами очень различные вещи. Так что он был mostly a kind of field theory, a real field theory man and I was doing integral models, but at some point it so happened that it got closer. And so there are certain things which are very similar now and so I will show them and this is one reason to talk about on this subject which is not very complicated. Another reason is that it is related to what we are talking about is related to Louisville and everybody likes Louisville. And so the third thing is that I will give some exact answers for some quantities but my formulae are just conjectures which I will check. And I really don't quite know how to prove them and probably somebody will be able to do it. And so this is a paper based on the joint work with Zoltan by Nock but also it goes back to our joint papers with Stefano Negro here. So, Siege Gordon model is the simplest integral model, so that's why I am talking about this model. It's the simplest integral model, so here is Lagrangian density. And so what I will be doing finally, I will compute the matrix elements in this geometry. There is some local operator here and so some eigenstates of Hamiltonian here and here, so in the finite volume. So I will use also, so the coupling constant is called B. I will also use this Q which is B, applies B minus 1 like in Louisville. Always we do and P which is like that. Just convenient notation. Sorry? I think Lagrangian, okay, so say Euclidean action. Okay, so, okay. So there is a duality like that. And then, so the dimensionless mass scale. So this combination is sort of dual. And then the mass of physical particle is given by this kind of awful formula. So, as usual, so all this kind of formula are due to allosomological. So, and they are actually very important if you want to have some, if you want to make some numerics. And so this is the simplest, as I said, this is the simplest integral model. So in a sense that, so if you think about it in Minkowski space as in terms of scattering theory, so there is only one scalar particle and the S-metrics is the simplest possible, so you can imagine. So that's why, so this is a good laboratory for checking things. And so I will use for, instead of, instead of this radius of my, so I will be on the cylinder, as I said. And instead of the radius of my cylinder, I will use this R-small, so dimensionless combination. And what I want, what I did not say, so here I draw this contour, so this will be, when I go to UV conformal limits, so this will be important. So now, how do you, how do you find a spectrum of this model? So there is something which is called Q-function. So normally, so once again, so already from here, so we start something which is not quite neat, I would say. Because, so say in, there are physically more understandable situations like, or mathematically I don't know, a sign Gordon, where this kind of relation, so even this kind of object Q operator can be really defined. And Sting Gordon, it cannot be really defined, as far as I understand. But so let me just say, but nobody doubts, nobody doubts, what I will say now. So there is some function Q, which is an entire function in the interplane. And so which satisfies this quadratic relation, and then different functions satisfying this relation, enumerates, enumerates the spectrum of the theorem. This Q behaves asymptotically, with some additional conditions. So it behaves asymptotically like that, yes? So then if theta k are zeroes of this Q, then it is easy to see, or it's a definition, if you wish, that Q is given by the following formula, if we define this second term here as e to epsilon, we just put it into the right-hand side, and then write something trivial. So end, end, end in this way. It's not sure not definition, it's relation, because the exponential of minus epsilon is Q squared. This is the definition, so this is the definition. That's the definition, but that's a relation. So this equals that, the formula above, right? Q equals that. Yes. And epsilon of theta is... You shift this guy by pi i over 2, and you shift it by minus pi i over 2 plus i0 minus i0 bar, you get this, right? So this is the topology. But so from this topology, you get, since you, we wrote this Q, we immediately get this equation for the function epsilon, yes, which characterizes spectrum, and with this kernel. And in addition, so I have this, so this is kind of, you can think about this epsilon as a density of continuous spectrum, and there is, there are these particles, which are kind of discrete spectrum. And so in addition to this relation, you have finite number of equations like that. And it is, we suppose that theta k are all real. And so if I require in addition that all theta k are real, then I get the spectrum of this theory in... And sub k, what is it? Some integers. So that, I, yeah, good question. So I am saying that the spectrum is parameterized by a set of integers. Okay? Yes. These are parameters, huh? No. No, no, no. Positive. Negative. Huh? I think so, yes. Yes. And usual, yes. So, and so I put here minus s under logarithmo arguments, because so s of zero is minus one, and it is more convenient, so not to have jump. Right. So this is an equation. This is a system of equations, which you can put on the computer and solve immediately with great precision. So then the eigenvalues of integrals of motion. So we have... Oops, no, no, no. We have infinite number of integrals of motion with two correlatives. Not correlatives, but so... Anyway, so e s and e s bar. So in conformal limits, this will be indeed two chiral sets of integrals of motion. And the eigenvalues on this... So from the fixed set n, you find epsilon and theta k, and the eigenvalues are like that. And so there are these funny constants somewhere, which are due to... So integrals of motion are normalized in such a way that they coincide in conformal limit with... So that's in conformal limit, they have some simplest possible forms. Some rational respect to k. It's not j. Sun. Okay. That's quite possible. Yes, I think so. But they are close in a jk. So, not very far. So... But if these guys are normalized in the best possible way in conformal limits, then there are some constants here, which are due to Barzhanov-Lukanov. The Molochik for particular energy resistance momentum is that. So now, of course, so with this TBA kind of equation, it's very easy to check that for large r, you will have Lusher corrections and so on, so forth. But it's more interesting to think what will happen in the ultraviolet limit when r is small. So everything essentially depends on this constant, yes? And when r is small, so the potential becomes like that. So it's long, long, long. So what happens is... So for zero mode, I'm talking about zero mode here. What happens is that you have reflections from these two boundaries, right? And so Barzhanov-Lukanov's conditions and the reflections are Liouville ones. Because here you have these walls are Liouville potentials. And the Liouville reflection amplitude is known very well. So this is the Barzhanov-Feldt quantization condition. So this is exactly so. This is the integrals of motion I expressed in terms of this P, like in CFT. So you see, this is for the first integral of motion. So this is P squared minus 1 over 24, central charge is 1. And plus m, m is for one kerality and m bar for another kerality. And so delta is used later. The central charge is like that, yes? So... And then we identify... So I will not go into detail how exactly this... In general, how these sets of integers are identified in ultraviolet limit. So with CFT states, but so with Liouville states, but so... For example, this one, minus 2, 0, 2, corresponds to the primary field. So here L, there are primary fields, which are counted by 1, 2, 3, and so on, yes? So this L is the primary field number 4. So the spectrum of primary field is quantized because of... Yes, like... So this is... The ground state can respond to L equals 1. But so now here this is rather high, so L equals 4. But still for R, small like that. And I don't remember what I took for B. So the ratio of integrals of motion, which I compute numerically, and exact values from CFT are rather reasonably good. So which means that with this zero-mode consideration, I arrive at rather good precision. So now here is the identification just of these sets of integers and conformal fields. So L are different primary fields. So then here are descendants. So this is simple. So here I have descendants on level 1, and so they are always unique. Here I have descendants on level 2. I will have to... And even 3, I will have to distinguish them somehow. And they are not distinguished, of course, by the first integral of motion. I have to go to third and so on. I probably will talk about this. So now what is my goal? My goal is to compute the diagonal matrix elements of this type for any local operators. Local operator. There is a one-to-one identification between local operators in ultraviolet conformal field theory, or Liouville in my case, and in my CFT. I will not explain much, but I will count these operators as I do in CFT. So there are primary fields. Primary fields are just exponents of phi, and so these exponents, so these primary fields, are normalized by Lukyanov, the Molochikov. One-point function. If you don't know what is this, forget about it, it will be later. But this is an important thing. And so now the main point, which I advocate for a long time, actually, is that when you are doing integrable model, and you compare it with the conformal field theory. So in conformal field theory, the fields are counted by Verasor algebra. But of course, when you make an integral perturbation, this is broken. So then you have to think how to count local operators in the integrable model. And in this integrable models, which are close to Liouville, okay, Sainte-Gordon, Siege-Gordon, this kind of guys, and more than that. So the operators are counted, so there are some primary fields, and on primary fields some fermions are acting, so there are two of them for each chirality. And so the... then so I don't know this multi-indices here like that, yes? So I take these guys with the only requirement, total number of biters is equal to total number of gammas, and I claim that this... give me a set of... give me... enumerate me... okay. All the descendants of this phi and Verasor descendants of this phi. So there is correspondence between these guys and the Verasor descendants of this phi and some shifted phi. And so which is... the relation is exact. And that for this... when I count operators like that, then this one-point functions or this matrix elements are easy to compute. So let's see. General rule is that I have... Let me see. I have this, yes? But if the number of elements in M is the same as number of elements in M, and consequently for another chirality, you have just conformal... UV limit, you have just conformal descendants of this field. If they are different, then you have conformal descendants of this shifted field. And so here to be precise, so this is... this is the conformal descendants of the shifted field. And it is defined like that. And there is some definition of... so how I understand the operators with negative indices. And they have, of course, under relation to operators and like that. So I will not need these formulas in generality. So let's skip them for the moment. So, and then the claim is that a matrix... that's the ratio of matrix element of this fermionic descendant by the matrix element of the primary field and then by determinant of... so these are sets of integers, yes? So determinant of some matrix omega mn with some simple addition. And this omega mn is... so the only physical input, so all the rest as you see, more or less combinatorics, which you need to compute the one point function in this model. And so now... and so here I must say that this case I cannot derive this formula. But so I just guess it's kind of a little bit out of the blue by analogy with other... my procedure is I guess this thing by analogy with other cases and then I check the limits like conformal limit and... What is it? Omega, what is it? Yes, it will be here. I guess it, yes, and I did not... I will... Yes, some formula. So, let me see. What is it? Yes, so this is exactly... this is exactly what... Why? I said that there are some... there is a similarity with what Samsung and Nikita were doing, that if for some... in some case alpha equals zero, this function omega... this omega mn is nothing like a Gaussian of the matrix of second derivative with respect to higher times of... more or less like that, yes, of this young... young, young action. Let me remind the scores of the better way function. Not exactly, because... you see, this is an interesting point. Because... Look, what is the action of young and young, yes? It depends on the better roots, yes? And so when you differentiate with respect to better roots, yes, and make the determinant, so you get your norm of the better function. But it can also depend. So this is out of shell, yes? Because you differentiate with respect, yes? But suppose now that you have external parameters, possibility, for example, in your spin chain, yes? And you compute on shell and then differentiate with respect to these guys. And this will be... It's not the best thing. Well, spiritually it is, because in some big space it is... those two things are different. In big space everything is the same, yes? Depends how big the space is. OK. So... You remember, I had this function, this function F, yes? When I characterized the... the discrete spectrum, equation for discrete spectrum. Now, using this, I derive... I derive the derivative with respect to the radius of this epsilon function. And it looks like that. And motivated by this formula I introduce the convolution, which is kind of normal from the... I don't know... There is some... So you have functions, which have a continuous spectrum and discrete spectrum. And you have a measure, which is a part of measure is continuous and a part of measure is discrete. So there is a convolution like that. It's a positive, the square of... G, G, G will be positive. Put J equal to H for the positive. Yes, I think so, yes. Yes. So, and then this omega is a simple thing. You take just a kernel like this, 2 by 2, not 2 by 2, but so discrete spectrum and continuous spectrum. Kernel like that, and this composed of these guys, which deform the kernels, which you have in TBA. And so then a vector like this, so these are for discrete spectrum and this is for continuous spectrum. And so you sandwich the resolvent for this kernel with this measure, yes, between these two guys M and M and you get my omega. This is a differential formula. Not so bad, yes. Okay. So, so now I started this work with Baynok, because these guys in Budapest they like very much this kind of matrix element. So, and the right for them formula of the type of Lecleron-Moussard, which are good for writing the series for large r, yes, but and so then this, of course the formula are complicated, but they can be compared with this simple formula which I explained with this omega and they give the same, of course. But, so, for me this is not so interesting. So, I want to see what will happen in the ultraviolet limit when the radius of cylinder is small and when I should be in touch with Louisville. So, now I can see the ultraviolet limit and so Louisville on a cylinder. So, of course for when I am a conformal limit so I would probably map this cylinder to a sphere but so I want to speak such a language which is well for conformal and deformed case. So, I will still stay on the cylinder so then you remember, so I had this picture so I have the energy momentum tensor and I take the coordinate on my cylinder so cylinder is my, what is the joke? So, so I take that which belongs to this and so either I define local one which is around z equals 0 where my local operator lives or global ones which are around the cylinder, yes? And so, this defines me l small and l capital and now I am interested in matrix elements of conformal fields between two fields between two states Psi and these Psi are descendants created by these guys and these all are Verasoro descendants of my primary field So, the dimension of the field is like that in my notation Normalization is the same Normalization is the same Normalization is the same for the for this thing for the primary field for the primary field, yes for the primary field, yes this is looking of the malochik of normalization, yes exactly In CFT you will limit it's CFT and now you say in CFT from CFT point of view it's a full correlation function or it's a three point function two full full no, no, no full, yes it's not so easy so, yes yes, yes so, and here I write this transparency you can check if you wish so so, for example here this is descendant L-2 descendant of primary field and so between between two primary fields so, delta plus one means the first descendant so the first descendant is unique so I allow myself to write this not very nice not very nice probably notation yes, and and so now what I promised to say and so there is a there is a we can see the UV limit of my of my local operators created by fermions and the claim is that they are so I write here in the simplest cases but I can compute as far as you wish so they are related and, yes, I did not say one important thing that of course I compute everything modular descendants of by local integrals of motion because so descendants of local integrals of motion they do not contribute to the one this one point function because right I have the same thing, yes so this means modular action of local integrals of motion so and then so the formula are like that for the simplest case so these are for one chirality so I create L minus two by these two fermions I create L minus two square and L minus four fermions and so you see there are two descendants modular local integrals of motion there are two descendants and I take descendants modular local integrals of motion as being created by even verasora generators and so you see so from here I can find L minus two square and L minus four if I wish and in the case one number of operators so we call chirality coincide yes yes yes yes yes it will be soon so this is as you say correctly so this is here I take the case when the number of operators coincide and so moreover only one chirality of course I could put here they don't interfere yes and so these are our favorite functions so we are very proud about these ones so and then so then I compute my function omega I don't know when and how they became capital I don't remember but so these are my functions omega and so what I am saying is that but in the limit of when r goes to zero they must be given according to everything all I said above they must be given by these expressions and so this thing we know so I mean was written and so on so everything is known here this can be checked so and then we come next question so if I take for the simplest case when I act by beta star and gamma gamma one bar so this is allowed yes then it shifts it shifts the primary field so that's what happens and so then what I have to the ratio of this of the three-point function for this shifted primary field over three-point function shift one is given it can be read from Dorn Dorn Otto Dorn The Molochikov Of course I read from the Molochikov and so here is F which is the ratio of these two Molochikov functions a mass like that and we so this must be this must be the UV limit of this omega one minus one already it's more interesting because here I have here I have something fractional power and so on more exciting yes so now how do I do I check numerically so and I check numerically for r equal 10 to minus 3 a is for some reasons like that b is like that so I don't want them to be you know avoid all possible resonances so I don't want them to be similar and for simplicity here I start with I consider the case when no descendants with respect to other kerality and so here is numerics I don't know what to say something coincide with something yes this is CFT for example omega 11 so this is numerical which I give from my equation and this is CFT so here so this coincides but this is then the descendants are getting nastier so here for example even no descendants so this is the primary field number 4 and so here it is not so good but more or less alright so good so that's why it seems that my conjecture is correct and so then and then I want to consider just I have some time so I want to consider the simplest case why L-1 with degenerate L0 so on level 2 I have 2 operators 2 vectors so I L-2 L-1 and now the first integral of motion which is essentially L0 do not distinguish between them and I have to think about the next integral of motion which is I3 so I3 is like that so easy to compute because easy to compute from the fact that the density of I3 is T squared so now this guy acts on these 2 vectors it gives me some matrix with this with this eigen values and these eigenvectors and I must identify them with what I get from from my numerics and so I compute the integral of motion from this TBA and so I take this once again this and so this this is what I get in CFT I think and this is or vice versa and this is what I get in by TBA and so this is how I say that this guy corresponds to this one and this guy corresponds to this one and in natural notation so now I have these 2 vectors so L- I don't remember who is first, who is second L-2 is first and L-1 squared is second so I have this matrix of 4 yes and so then L-2 of this guy given by this one and so with this matrix elements so this is I believe me it is true and so now this thing omega 11 plus minus in this case is given by this formula where I use this vectors psi for eigenvalues of integral of motion of degenerate and so these are the eigenvalues in this case for the minus guy and for the plus guy for the descendant and for shifted primary field and you see that probably they are not as good as before, but so alright from safety point of view it is a generic situation right? you are not sure? yes, yes, yes, yes so oh yeah, yeah so and so this is the end of the talk almost so and to finish I want to say that with so for a long talk time with some song and then it it comes, it returns so we were discussing some very nice Georgian film which was after poems of some Georgian poet whose name was Vaja Psavela and so I do not know where the British translation exists and I cannot write or read in Georgian so that's why I am just I give you some some Russian translation by Tsvetayev Tsvetayev was a great poet another guy by the translator and the point is that there is another translation of the same poem by Mandelstam and this is complete so I looked at the same piece completely different even the Gregorias he is a more expert now can you translate? no, no do you recognize the original? I don't know spirit spirit this is when one guy was attacked by another one his wife tells him that we are poor because you have to rob somebody because otherwise he does not want and then he is attacked by somebody he confines him and then they drink together at some point these are very high mountains What do we get out of this for Lyubil, something interesting if you start going other way around for Lyubil I don't know so actually I would say why would you be interested in this diagonal matrix elements one question one answer is that if you want to make a one point function on a torus then you will start summing up these diagonal matrix elements but the question is whether it is possible using this technique to write this somehow to simplify the expression for the one point function on a torus because this is not easy even for a conformal case so this probably will help I don't know One point function also cause organization and kind of smell one point function why one point function because if you want to compute the general correlation functions using the OPE OPE is a perturbation you obtain OPE okay so not so easy but so more or less you obtain OPE from perturbation of conformal field series and then by OPE you reduce your multiple point function to one but then it's not enough to know one point function of one operator or two operator but you have to know for everybody yes so that's what I'm doing