 Я видел Самсон более than 30 лет назад, первый раз в Ленинграде. 35. 35, да, и в Ленинграде, сейчас в Питербурге, да? И у него очень... У него был контакт в Радгерс 92, 91-92, да? Много дискуссий, много... Ну, я aprendил много с Самсона, это... Во-первых, много проблем, мы говорили в этот раз. Я еще думал об этих проблемах, но, unfortunately, некоторые из них были absolваны. Но... Да. Это было великое время. И это мой пляж, чтобы участвовать в этой конференции, в честь Самсона. И я бы хотел сказать, что организатор для этой великой ситуации. Окей. Я буду говорить сегодня о recent work with my former great student, Глеб Катаусов, и мы... Я буду разговаривать с certain aspects of integrable quantum field theory. И это звучит integrable quantum field... Conformal field theory. Conformal field theory. Это звучит... Во-первых, это немного странно, потому что, в общем-то, conformal field theory не дает много integrable structures. Ну, conformal structure, может быть, какой-то extend conformal symmetries, но... Но я... Я буду focused on additional, some additional input, which is sometimes possible to introduce for the conformal field theory. So, here is outline of my talk. First, I will discuss, most of the time of this, would be kind of introduction, discuss the integrable structures, what does it mean, at least, what's meaning of this. And then we... Among the problem, which occurs in studies integrable structures, problem with the organization of certain set of commuting operators, usual integrability, we have a commuting set, infinite commuting set of operators. And the problem is how to diagonalize it, how to construct the common spectrum. Then, the central subject of my talk is reflection of operators, and it can be introduced in many integrable, in many conformal field theory, but we will focus on some particular simplest possible, simplest situations of so-called KDV integrable structures for conformal field theory. Well, and some application, maybe a few words. All right, so let me start with a rather general introduction. So we are considering that we are doing the two-dimensional integrable quantum field theory at this point. So integrability means the existence of the continuity, an infinite number of continuity equations of these forms, the T. It's the tensile densities, and since we are in two dimensions, that makes sense to introduce the use the Lorentz invariance, introduce the light-con variables and the Euclidean versions, the Galmorphic and Galmorphic coordinates, and then these equations, continuity equations, because some are simpler. So this is local equations, but assuming there are certain boundary conditions imposed in the system, simplest setup, it suggests system with the space coordinate computerized on the circle radius r, so the our worksheet geometry is just a cone, just a cylinder, right, then having these continuity equations, assuming the density of periodic functions along the space directions, then clearly the such combinations, such integrals taken over the over the time slice would be integral of motion. You can freely move the integration contour along the cylinder, right, and it will not change the value of such an integral. So this is integral of motions, but this is not enough for integrability, because having just a classical Louisville integrability, it makes sense to require that this operator mutually commutes. So and then then but this all this picture works for conformal field theory, but now the idea is, let's assume that the ultraviolet behavior of this conformal field theory controlled by the certain conformal field theory, so originally we have a theory with the correlation lens, let's say, with the mass and then consider the limit, so the only dimensions parameters here and the problem is length is the size of the cylinder measured in the unit of correlation lens, so we assume that this dimension parameter goes to zero, then the equations original equations this one, the conservation equations turns out to the condition for condition that the field CT and this density became galamorphic in this limit, because the left-hand side actually vanished in this limit, so and we end up with a set of Riemann Cauchy-Riemann equations for the densities and for the densities to the infinite set of densities as labels the s plus 1, it's Lorentz spin of the density alright, and this way this way these equations actually shows that the infinite number of integral of motions if, for example, we expand from these conditions assuming that the t is periodic functions we can expand in the Fourier modes and these conditions mean that they are all expansion coefficients integral of motions, so a lot of integral of motions we can build this way but they are not commute nevertheless we can construct the commuting family by taking the limit again taking the limit of the integral of motion from conformal field theory quantum field theory and to consider the limit we get something which would be not only conserved but also commuting alright, and in these situations so the problems in this way became rather mathematically well defined problem the problem of finding common spectrum of these operators I call it local integral of motions local because we have integrals over the local densities and this operator since the conformal field theory can be classified the space of states of conformal field theory and also in terms of irreducible representations of the conformal algebra maybe other algebra of extended symmetry then this operator can be thought as operator acting representations of such an algebras and the problem of diagonalization such an operator becomes well defined mathematical problem can be formulated just in problem in representation theory and Sir, how did you define IS there? IS? No, the next one This one? No, no, no, it's at the holomorphic in the Z part and it is holomorphic in Z So, how... Yeah, we actually I was starting point what this integral of motions and this limit of this integral of motions you have in the limit when the correlation less goes to zero it became the conformal integral So, this is a formula just what this part actually integral dissipates proportion in the in the in the mesless limit and only we end up only the integral like this so Because I didn't understand this This one? This equation, how does it give the second integral of motions? You see Since we have we assume the pn variance we can always replace z by z bar So, clearly if you have this series with this integral of motions that you can always flip the directions space directions which is replace the spin minus spin so Okay Okay, so that is So, and let me just my basic example would be the since Gordon model So, this is classical this model, the actions given by this famous formula and this is classical integrable systems и и и у у у у у у у у у у для первонатрифийного интеграции, э1 и э1-бар, дает гомильтонию и отличается от момента. Так, это пример. И теперь вопросы. Это теория массива, но вопросы мы интересны в формате. Это немного тряпит, потому что мы... Ну, мусор кажется маленький, В то же время, когда корреляция линейка идёт в 0, то линейка идёт в инфинит, и муэль будет маленькая. В первой линейке мы можем неглядить эти тёмы, а в третьей теории, но это не quite true, потому что это потенциально слабо. Почему-то нет. Нrek, да. Но не так да. Это неглядно так. Это неглядно так. Это неглядно так. Это неглядно так. Но это не очень вероятно, потому что в конфигурационной сфере в филе, когда есть демо, или в филд файл, получается ладж, и это не важно, как маленький ты выберешь мю, этот тронт может быть важным. Так, в частности, это означает, что мы должны... Во-первых, письма выглядит как это, как квантамиканиковые письма, и в филде квантовой эффекты здесь не очень важны для понимания, что происходит. Так, потенциал всегда... Если в конфорте лимит, потенциал почти нет в каждом месте и в определенном сегменте, который зависит от этой рейсерии, от диаметра параметра, но потом, потому что экспонентальный тронт, мы не можем неглать этот тронт, и мы имеем два симметричных экспонентальных тронт, но в конфигурационной сфере, где файл и сладжи, мы не можем неглать потенциальные тронты, но мы можем их располагать с экспонентом, и в результате мы получим ливильные акции. Так, в этом смысле, лимит утравалит синджи-гордон-моделл описан в том, что ливильная теория. И ливильная теория, это, на самом деле, это компонентная теория. Это процесс энергии и момента тензора. Это галаморфик, если вы имеете эту известную форму. Это из-за эвакуации, из-за эвакуации, из-за эвакуации, из-за эвакуации, это галаморфик, и мы можем располагать эту функцию, потому что мы рассмотрим периодические полноценности, хотя бы для энергии и момента тензора. Кофеции генерают известную веростровую алгебру, и, как я сказал, каждые из этих тензоров Итак, в принципе, каждый генерат, в том числе, в том числе, в том числе, это консорт, но они не коммьют. Это не коммьют, а сетка коммьюта. К сожалению, как я сказал, мы можем, используя эту генератую алгебру, построить операторы, которые коммьют, в форме мучили-коммьютона сетки. И как это работает, чтобы построить такие операторы, мы можем рассмотреть лимит или интегло-моции в Синч-Гордон-модел, и в результате, мы получим следующие операторы, которые построены из энергии-маментом тензора, галаморфикой компонентой энергии-маментом тензора, так что они выглядят как тетки, динамичная, первая динамичная, очень простая, тетки-сквад, или определенная композиция, построена из энергии-маментом тензора. И если вы, это дает вам сетку мучили-коммьютона операторов, и в лимитах, когдаPHZ заводится к инфинитации, это… В лимитах C заводится к инфинитации, помите, как С involve Q, и В involve B AKA B, то есть C заводится к инфинитizing, то есть B commercial 0. Это brokersтоĩный лимит. Этоועнимся сет, в exting conflicts с Kampf absent, carriers till blindevcret Это не удивительно, потому что они классически КДВ и Синч-Гордон моделировали и так же интеграбль-хайрохимы, так что это довольно предыдущие. Но в любом случае, у нас есть эти операции, и, как я сказал, у нас есть универсалонвелтный алгебр, но, тем не менее, может быть, это выглядит, может быть, little bit confusing, потому что, может быть, у нас есть проблема, в том числе, в infinitesum, когда мы делаем операции. Но в этой ситуации, в принципе, это не проблема, потому что в следующем. Теперь, если мы consider the representations of the vorossoral algebra, the highest weight representation characterized by the conformal dimensions delta, and introduce the basis, for example, introduce the basis built out of these modes, its standard definitions, construction of the basis in the highest weight representations. Then in each level subspaces, in level subspaces, we mean that it's a graded space, so the number of modes, the sum of these modes actually is fixed by the level. And in spite of the infinite sum here, once this operator acts on the level subspaces, this sum actually truncated. And so the operator, each of these operator acts irreducibly, acts invariantly within the level subspaces. So in other words, at each level subspaces, the i, it's just a matrix. The size of the matrix is equal to the dimensions of the level subspaces, which is in general a number of partitions of integer n. So that is, for any fixed level, this is just a finite matrix. Maybe luch, of course, if n is luch, then of course number of partitions is very luch, but it's still matrix, and we have a problem of, we can explicitly construct these matrixes using definition, and we have a, so this is just an example illustration, so this for level n we have the only one state, so there's no, a matrix is one dimensional, the space is one dimensional, then at level one we have also only one state, it's again the matrix is trivial. But in the level two, the first non-trivial example, it's a level two, when you have two states, l minus one squared and l minus two, the integral of motions, it's basically, as I said, for calculation of this matrix on the second level, we can drop all the terms, which involve l minus three, higher than in this sum, in this infinite sum, they survive only the few two terms, and all other actually kill any state at this level. So this is just a two by two matrix, and you can diagonalize this matrix, and so here's two agent vectors, you see, and two agent values. So the agent values, as you see, the agent values are kind of algebraic functions of the central church dimensions and dimensions, right? And now, so immediate question secures in this point, so what, how do you diagonalize this matrix for the higher level, because matrix is very large, and so that is clearly a problem. And so the questions, the first questions, natural questions, so for given integrable structure, how to calculate the spectrum of local integral of motions. And I should emphasize at this point, so we consider the particular integrable structure of conformal field theory. In conformal field theory there is no integrable structures. And the one which we discussed, which appeared, which appeared through the certain integrable structure, originally introduced the massive theory, and during the special, the short distance limit of the theory we get some particular basis of state, which diagonalized of the integral of motions. But the matter of facts we can introduce in the same, in the same, in the same high state representation, we can introduce different integrable structure. For example, another well-known example of the integrable system, integrable field theory into dimensions, it's so-called bullet dot model, which looks slightly different than the synergy Gordon, because here we have two exponents, but with the different exponents here, 2B and minus 4B. And again, this is integrable systems. Again, we can consider the conformal limit, and as a result we get the set of commuting operator, which acts in the same space. But now the set is different than the set of KDV equations, the quantum KDV integral of motions, they look like this, you know, even spins, well, spins are different, so you see integral, the density, this density is true, but in KDV we have density T squared, there is no density T squared, only the T cubed appeared first. And so this is what I should emphasize, that this operates in the same space, in the same space of conformal field theory, but it's another, it's a different integrable structure. And for this problem we also have, we can address the questions of the generalization of that set. But you fix the representation, right? Yeah, we have chosen to consider the... We fix delta and C, right? Anything special happens if they generate one? Yes, of course. Yes, yes, there are some, but at this point with the integral of motions, but you see it's algebraic function, you can, in principle, well, at least for the... It appears, of course, in the context of the field theory, on the particular field theory you're talking about. Here we're talking about just some basic property of representations, one representations. Because all the periodic acts on the levels are space invariantly. We can forget about the right coloralities and all other stuff. Okay, so this is some principle. So what we should... The main message from this, my intro, it's actually almost half of my time, but there are a variety of integrable structure, not just one. So we just mentioned PDV, bullet dot, there is another KNS, which includes the famous non-linear Schrodinger equation. There is another so-called paperclip or the sausage integrable structure. But what's important, then once you choose these integrable structures, then with the presence, because of the presence of infinite-dimensional algebra of extended symmetry, the problem of the diagonalization of local integral of motions can be... Admit the mathematical satisfactory construction of formulations in terms of representation theory. It's a well-defined problem. Okay. And now I just explain how to solve... how the problem of diagonalization of calculation of spectrum of local integral of motions can be solved. Actually, I explained the solutions. This solution was given by... I now work with... Sasha Zamolochikov and Volodya Bajanov a long time ago. So descriptions of the spectrum based on... Remarkably, it's related to the spectral theory, which Leon discussed today. And I'm lucky that he gave a nice introduction in the spectral theory of this operator. And actually, many... many I'm discussing here it's actually deeply related to what Leon said. And so basically, the key ingredient here is differential equations, Schrodinger equations. Right? Of this usual Schrodinger equation by the potentials ux. First, if you... What's important in this potential? Forget about this term first. Clearly, this potential is just an harmonic 3-dimensional oscillators. Parameter alpha is in the case alpha is equal to 1. We have a potential like x squared plus L L plus 1 over x squared which is centripetal potentials 3-dimensional oscillators. Right? Another case we can consider here that alpha is equal to infinity. For example, then the potential will look like well, infinite well potential which is also an example which Leon mentioned in his talk. But anyway this alpha... Anyway, so this is... this troms is clear From the mathematical point of view we have differential equations with one regular singularity at x equals 0 and a regular singularity at x equals infinity. But I also would like to introduce some additional singularities but very special way, right? So it will contain so clearly there would be set of singularities dependent on the set vA and we import certain conditions, restrictions of the singularities. And the condition looks like this set of algebraic equations and imposing this equation the meaning of this equation is such that once this position satisfies this equation then the solution any solution of differential equation actually be single valid in the vicinity of the singularities. It's very strong restrictions you see once you introduce the singularity in the potential then in general you should get braiding some non-trivial monodromy property of the solution if the potential has a singularity even regular singularity the functions the wave function is not single valid in the vicinity of the singularity. But imposing this condition that the singularity is single valid we fix they cannot it requires the fixed fine tuning of the parameters of the position of such singularities and this is equation so of course it's this equation which described the condition that singular are apparent sometimes such a singularity is called apparent singularity. You see the delta here it's related to L momentum term well and the next step it's a spectral determinant which was a subject of Lyon talk so clearly for the potential like this so this is ux so if you consider the potential like this the potential looks like this so we have only discrete spectrum you can clearly formulate the problem e.g. it's a spectrum it's an energy spectrum for the Schrodinger equation well, this is without this additional term including this term should not change It is self-adjunct, right? I think so No, no, no for with these terms With v? Because you want to pay attention For complex p For complex p well for real L of course by the conditions that certain real values of L certain domain would be self-adjoint but actually the spectrum I'm not quite for general the v is complex numbers here it's not possible for self-adjoint but anyway the problem can be introduced and because of the presence all the singularity apparent singularity the monodromy we can forget about this the additional insertion still formulate the same type of problem like an usual in Germanic oscillators so the central object would be the spectrum determinant well and actually these products converge only in the case of alpha greater than 1 but if alpha is not greater than smaller than 1 they need some regularizations but again the most important would be the trace identities which Leon mentioned and basically it's useful to consider the latch e asymptotic I consider the energy goes to latch actually it's negative at least as was mentioned in Leon talk the spectrum determinant meet the expansion of this form and the coefficients of this expansion the first coefficient, leading coefficient doesn't depend on the presence of apparent singularities actually the numbers then if you focus on subleading terms of these expansions general structure looks like this then we should actually focus on the coefficient which appeared in certain powers of the expansion of the spectrum determinant inverse powers and these quantities actually can be for such a quantity the trace identity can be written and they can be calculated in terms of systematically in terms of using the WQB approximations some and other coefficients here but we are actually at this point we are interested in these coefficients like this because the reason we are interested in these coefficients because they are exactly the coincide with the general value of local interval of motions so the statement which has come from our paper is that if you take these algebraic systems that and identify the the same delta as highest weight representation the central charge related to this parameter alpha is to alpha so this is related to the the potential and n it's a level number of apparent singularities it's a level subspaces then we checked explicitly for just the solving of the equation for for for small n 1, 2, 3, up to 5 I guess that the number of solutions for the number of solutions of this equation coincide with the number of partitions of integers n up to of course the action of the symmetric group so this is an equation for what for apparent singularities for v right and these conditions just guarantee the potential for the apparent singularities and so and that's actually the statement that number of solutions coincide with the number of partitions was proven just recently by David Mazaira ok and it means so the number of number of solutions coincide with the number of states at the level subspaces and we can use these sets to label the states in the level subspaces and so the integral of motions can be considered as the functions of these sets or of the solutions of this set and the statement was that this coefficient exactly gives the eigenvalues of local integral of motions possibly one more stupid question so it continues one of the questions as before what if your variable module is degenerate so if delta is special then what happens is not right then probably you need a small number of solutions no solution would be the same but something different would be maybe I will mention but the number of solutions remain the same but something of course interesting happened but it's a different story so here some explicit formula for example e1 and by the way each of these as a function of these sets it's a symmetric polynomial it's symmetric polynomial for order of m-1 so for the case of the first integral of motions the order of the symmetric polynomial is zero so it's just the number of i3 would be just a linear symmetric polynomial it would be linear functions of v for the next integral of motions would be quadratic one this one linear quadratic terms and so on and so in other words without so in order to diagonalize in order to find at least a spectrum of the local integral of motions we need to solve these systems and then it's done so symmetric polynomial will give the agent value of course this can be checked with direct calculations by diagonalizations of matrix but together with local integral of motions there is another interesting integral of motions with another integral structure another structure which is included in this integral structure it's called non-local integral of motions was discussed a long time ago again with Sasha and Volodya and the simplest example of non-local integral of motions is so-called reflection operators it's commuted it's among the commuting families commuted with the local integral of motions and it has many applications and what's interesting that these reflection operators reflection operator for the KDV integral structure is related to the reflection as matrix in linear theory and the matrix which was discussed in many works before but probably the best explanations of this object was given by Sasha and Alyosha in 1996 this correspondence that you had before works for any central charge? yes it works for any central charge actually it's algebraic equations and you see when you diagonalize your matrix it doesn't matter what the main it can be applied for any values of central charge dimensions and that's the beauty of this because you don't need to worry about the main of your parameters and let me remind you how the Liouville matrix works so again we return to the Liouville theory energy momentum and the energy moment we have the this is our representation in terms of field phi but now let's think about this again once you finish all these things and you diagonalize everything at the end you get back your model yes we are talking about I should emphasize actually the problem at this formulation is a pure representation theory we are not going to glue these different corollities we just focus on one corollity actually one level when everything is well defined problem in linear algebra so let's think about this little bit what happened with the Liouville theory and in the case when the phi again we consider the domain of configuration space where phi becomes very large but negative if it's large and negative then the storms it's negligible and we have just three actions the theory of free both fields which means the function d phi it would be a galamorphic function which can be explained in the Fourier series of this form and from the action we read the canonical commutation relation for this oscillators creating and negilating oscillators and so in this asymptotic domain the energy momentum tensors can be expressed in terms of the creating and negilating operators and it looks like this it's a famous basinization relations basinization formula and we can force the asymptotic basinization formula for the virus or algebra but remember that but we can build representations of this algebra now it's a folk space and as a folk space this formula actually provides a structural representation of the virus or algebra in the folk space but this folk space actually as a linear space it says amorphic to the verma models or the virus or algebra and the zero-mode momentum of the folk space related to the dimensions of delta but now it's interesting the important feature that delta is a function of p so another word we have two choices which correspond to the same delta with the folk space with positive p and the negative p and clearly this can be interpreted just in the spirit of quantum mechanics we can interpret it as each of these the basis of the folk space is the space of the asymptotic states so for positive p consider it as a propagation of like a mini super space approximation consider the propagation of the wave which scatter on the financial barrier and this folk space would correspond to the in state and with the opposite sign is a negative p would be reflected out state once we have such two spaces in and out spaces we can introduce this matrix and so it's a reflectionless matrix it has this form, some structures it contains some normalization factor which is a famous normalization factor which contains a lot of information about the legal theory but for us it would be important how this matrix is operated to be precise it's a twiner it's between the different folk spaces and it's interesting and important which was emphasized by Sasha and Alyosha that the centered twiner meets the first of all factorized on the right and left corallity in absolutely independent operator S bar and S and each of these operator acts in between the centered twiner between folk spaces carol, carol part of the S matrix we normalize such a way to make this one then all the normalization goes to this factor and what's important that this carol reflection operator carol S matrix is fully determined by conformal symmetry it's quite evident because we, as I said in this theory the energy momentum tensor the issue of these components is conserved charges so in classical just think about the classical that this quantity would conserve and principle what we need to do it's solve in a classical level to construct this reflection as metric we need to express the oscillators a in terms of ln inverse this relation, this formula but in other words we should solve this equation with respect to phi it's basically Riccati equations and the in quantum level this term can be formulated at the statement that these operators unambiguously defined and the constructions and the next idea that having this if you have this S matrix reflection S matrix then we can build the operator which commutes with the local integral of motions and it works the following way let's emphasize first that these operators X between the different space different for space it doesn't have any sense to diagonalize these operators it's intertwiner but instead of but we also can introduce the intertwiners another intertwiner which X similarly but very simple way just to flip the the sign of oscillators this sign so if you think about this picture with the Liouville returning to the Liouville theory and see what happened with the wave which is we first forget about the left barrier we consider the scattering on the right barrier then under the scattering we get the phase which is Liouville S matrix dictated by Liouville X matrix that then we should go to the left barrier again and find this way the total scattering if you describe the whole scattering we need to include the scattering on the two barriers but after the scattering because of identifications of space we need to introduce this additional set conjugations because it's a flip the sign of phi but anyway that principle that the combination like scattering C this barrier then we should identify if phi of minus phi and phi should insert the reflections so it's a conjugations then scattering about this barrier give this this operator and then back so the result we may expect that such combinations of operator which X now X in the full space can be viewed with the local integral of motion sorry just a question this is already an extended carol space no no it's just a carol space because we are focused only the one carality and we focus on one carality and build this operator well you see this is a square of operators clearly it's a square for this reason I introduce Cs and actually for historical reason I put minus 1 this operator it's now X in the full space given full space in this operator I call it reflection operator so it's additional operators it's very it's a matrix very similar to to the reflection matrix but we need to flip this sign somewhere okay so construction of reflection operator is very simple indeed it follows the following way representation formula like this but we can have a Verasor algebra with the same central charge but if you flip the sign of the of the background charge q if you just flip minus here but of course it would be different oscillator bases in other words in the full space we can introduce 3 different bases Verasor bases then Verasor the one basis Fock bases built out of operator AM another base built out of operator AM bar but of course this is a base in the same space so there is a matrix omega which relate the Gazingberg bases and Verasor bases this one there is a certain matrix and similar for the tilde bases labeling what labeling it doesn't matter you know it's just a basis let me choose this this is a set of the bases in the space okay this is the basis that you choose for the Verasor generator block you see it doesn't matter actually what would be interesting for me it's a basis which relate this matrix relate the base Verasor one Gazingberg operators and but also using the same matrix but also applying the 6 conjugations I can relate build the matrix which relate the bases the tilde bases with the Verasor one and what I'm going to do just because they are the same I need the relation between in and out the tilde and until the base if you want to say that you have to order them oh yes yes of course this should be the basis but finally what base you use it doesn't matter because finally what having this now making the quality here and as a result I get the matrix which relate the tilde bases and you know the matrix finally it's hidden in the summation indexes here and so the lesson is that reflection operator is expressed in terms of single matrix what you need to do is calculate this matrix choose some basis calculate this matrix and then build the reflection operator in this way yeah this is the ordering here well it should be just a basis and it's not difficult to show now that this period indeed commutes with the local interval of motion the main observation base if you write the energy momentum tensors it looks like this t for example it contains the quadratic terms which is invariant with respect to flip of the sign but it contains also the terms but this linear term but this terms is second derivative which is actually dissipated so all integrals of motions all of them when expressed in terms of field phi actually even function of phi so they invariant with respect to this flipping the sign so it doesn't matter the form of the integral of motion doesn't matter what fields you use phi or for tilde and this means actually the operators which is relate the phi bar fields and phi actually would commute with the local integral of motions all right and here is explicit formula so just illustration for this and it goes this way so for example in the first retrieval level it's a second level in the second level we have two states a salutary state a minus 1 squared, a minus 2 and in this basis in the Virassura base we have L minus 1 squared and L minus 2 and we can express this state in terms of the salutary eugenius optimization formula so this gives us the matrix omega and once you have the matrix omega you build the matrix of reflection operators and this reflection is this matrix indeed commute with the local integral of motion matrix of reflection operators so this way we have this problem of that since it's commute it's equal to 0 they can simultaneously diagonalize the reflection operators with the local integral of motions and since we label the states by this apparent singularities then immediate questions what would be the spectrum of what would be the agent value of the reflection operators yes, what would be the agent value of reflection operators and the answer is rather again we should return to the differential equations differential equations which we start with but now the subject of our interest would not be subliminal coefficient but first subliminal coefficient you see in expansion we have something which doesn't depend on state but the first non-trivial coefficients in the spectrum of determinant is the one which is also the one which is probably which is the main subliminal which depends on the position of apparent singularities these coefficients so there is no such some rules such trace identities like for integral integral of motions from the theory of differential equations so it's more complicated problem but nevertheless it's remarkably that well so this problem was for a long time but remarkably very recently this was work of Yuriyovkin-Tarasov just about the theory about the theory of Fuchsian differential equation with the three three regular singularities and any number of apparent singularities and they give some beautiful result about the solutions they present explicit solutions to such as equations and using the result of Yuriyovkin-Tarasov and performing the certain limit we can actually extract these coefficients and in terms and expressions and now it's of course it's a certain symmetric function but well you know the result is not so simple but it is given by certain determinant involving the involving the apparent singularities but of course the formula is not particular transparent but finally we have an explicit formula of the values of reflection operators in any states and again they express in terms of the apparent singularities ok so probably I don't have time to discuss the final probably the final I would like to mention that if you understand what's going on here that you can easily construct the reflection operators for many integrable systems for example for bull or dot what should we do that looks like by the different exponent and in another word to construct the reflection operator we need to take the reflection operator corresponding to this exponent right reflection matrix but now we should glue it with the reflection operator corresponding to the exponent with a different background charge and this combination indeed commutes with the local integral of bull or dot and still can formulate the problem of degenarization of this another example we can consider the model this one this part of the model is a famous dual representation for black hole Euclidean black hole in dual fields and you can think about the Lundredge complex inch Gordon model as a perturbations of Euclidean black hole exponential like a Louisville field and the same story happened here so for this conformal field theory there is analog of the reflection operator the reflection s matrix the s matrix I call it cigar s matrix so then to build the reflection operator for this integrable systems we need to accompany this reflection operator of the Louisville theory and indeed you can check that reflection operators commute with the local integral of motion for this model and interesting that a long time ago Samsung discovered this model in the representation in black hole and there was some manipulations with the first integrals and finally the conclusion that this model should be somehow equivalent to the Louisville theory that was somewhere but remarkably that finally one of the come out of our papers which I am not going to discuss that this reflection s matrix for this model for the black hole actually can be expressed indeed expressed in terms of Louisville theory there is a explicit formula which relates this to s matrix actually up to some factors they are almost the same this is kind of some interesting problem some interesting long this problem has a long story and now we see that indeed some the signals, the theory somehow very deeply relate all right, okay okay I don't have a time to discuss the implications and so basically so that my conclusion discusses some integrable structures, reflection operators and also all of this actually has interesting application to to study the scaling limit of critical latest systems integrable spin chains and the reflection operator is very useful and actually allows one to classify the state in the latest system, the better states actually there are the states which they can analyze all integrable motions analog of better states on the latest and that gives you very powerful method to track the scaling limit of each individual better states and there are some interesting ideas that maybe the scaling limit of the theory describe the non-compaximum model so that story but it's a different story, okay anyway once again, happy birthday some song questions, comments yes I have possibly a question, how is it on that picture that you showed with the singed garden potential you have some long but some finite distance between those two walls so then I don't understand this formula that we have that the total product of reflection matrices commutes with local integrals is it approximate or basically is it exact good, that's a good point actually that was a principle in my transparency, it's a part of my discussions but you see indeed we consider that just apply this quantization condition for this mode, zero mode and naively zero order of approximation you can replace this potential by the infinite well potential so the quantization condition like the lens or the total lens of the well multiplied by p which is proportional to the integers it's quantization condition for the allowed values of p and then you should take into account the effect of scattering the phases which is gained on the turning points and Sasha and Alyosha Zemolochikov-95 propose the following equations asymptotic equations which involve basically the reflection operators actually there was some kind of mistake in their papers because they say that a matrix commutes but this is not a matrix reflection operator because it's a matrix different spaces but basically the quantization condition for p it looks like this and in principle now we can take the in the short distance to the behavior you can express the quantization condition if you know the spectrum reflection operator you can write this quantization for each level now what next suppose we solve these equations then when actually we should take remember that the Kiddiway equation would be polynomial with respect to p and now we should substitute to solutions of the functions of the distances of the r and what we get we get the value of integral of motions out up to the power law correction so we this quantization conditions actually account all logarithms but the power law of course is out of the some approximations so it's some kind of W kind of W yes it's some kind you know in the case of it is similar it can be done for modified since Gordon equations and Leon told us about the quantization condition in this case but in that case it was exact but here we have a power law certainly it's up to the but accuracy of this approximation is rather good but it's very well because it's not because it's because it's actually numerically it's very good it gives a very good question Professor so in the case of cigar actual they were asking restrictions on numbers to get so it was a specific coefficient parameters were fixed yes yes you see this point as I said if you take a look on the formulas here we should diagonalize this matrix this parameter Q you are talking about it's fixed somehow but you see this is functions actually is a function of Q it's a meromorphic functions you can set any numbers this matters of course some miracle the maps that we did with Martinik were specific to work that is another point because I talk about the general relations and for these particular values I don't know what happened but actually the relation exists for any background charge this is just algebraic relation because it's a relation between the matrices you check at level by level and you see that is I have a question and this is another it's a good point you are talking about болудот болудот like a singe garden this is a complex complex you mentioned spin chain yes I mentioned spin chain certainly I don't know I don't know the answer for this problem but now there is some certain progress in this but the full answer still for me I don't know