 Я хочу сказать спасибо очень большое к организаторам. Я извиняюсь. Потому что в этой конференции мы можем видеть наши старые друзья. Вадим был нашим молодым коллегам. В любом случае, я думаю, эта конференция для людей, которые здесь, очень важна. И память, конечно, о Вадиме тоже очень важна для нас. Сегодня я хочу с тобой conversать. Вчера, Самсон сказал, что все модели, все модели, которые могут быть делированы от n equal to super... Симметричная теория, наверное, не все. Сегодня я хочу поговорить про какие-то различные дуалитеты в интеграмных моделях. Почему дуалитеты в интеграмных моделях важны? Во-первых, потому что интеграмные модели, кроме какой-то философии, Лагранжу и Дефинишн, есть еще некоторые дуалитеты, в том числе С-метрич. Это довольно простой объект. Если эта теория интеграмная, эта метрика имеет несколько функций, и они имеют всю информацию о теории. Эта функция достаточно для реконструкции теории. Это дает нам возможность, иногда, показать, что теория имеет некоторые другие дуалитеты. В какой-то смысле, мы можем показать, что есть мэпы между сильным дуалитетом и большим дуалитетом, которая дает нам возможность изучать теории, в более большей части дуалитетов, даже без интеграбилитет, и с различными методами, в том числе пертрубационной теории, и так далее. Но раньше, я бы сказал, что это, наверное, санлювильная теория, которая также имеет дуалитет, санлювильная теория, я бы сказал, что некоторые результаты, которые я буду говорить, были дирайвированы вместе с Сашей и Далешей, с аналогичным и с инриконовым. Итак, что это санлювильная теория? Это санлювильная теория, которая дает нам возможность изучать дуалитет, которая имеет дуалитет, это значит, что это санлювильная теория, и она также описывает санлювильную теорию, и эта санлювильная теория называется сигар, витон-сигар. Но даже внутри этой теории есть дуалитет, это дуалитет между моментом и витон-сигар. Это дуалитет от модели к дуалитету, который называется тромпет. Это не может быть полностью надеванным в 2-дименциальный пространство, но даже в 3-дименциальном пространстве, но visible part может быть санлювированным. Что было в конжектуре, которая была сделана Саша Алёша и меня много лет назад, в 1996-м, эта модель имеет дуалитет, и теория в том числе, если мы переметрируем, это содержит 2 параметра, А и Б, которые связаны с одной эквейцией, так что реально у нас есть 1 параметра в жизни, и центральная цена может быть именно так, если мы переметрируем их по параметру К. Это цена, это центральная цена этой кассетной модели, или цена сигарной модели. Теперь, почему эта конжектура, почему она была база, в которой реальные факты. Так что эта теория есть какая-то пара-фермиония, это пара-фермиония. Эти пара-фермионии не компактные пара-фермионии, которые могут очень просто представить в том числе 2 пара-фермионии. И это возможно, очень легко proves, что эти пара-фермионии работают в сильном смысле, с эти пара-фермионии, которые могут пертурбироваться с нашим потенциалом пертурбирования в нашей теории. Это в сильном смысле означает, что это в locally, или это в халаморфике пертурбирование. В принципе, это дает возможность построить все W-алгебры и все locally fields. Эти fields не local, но их продукт-экспансион содержит local fields, которые получают infinite W-алгебры с infinite number of currents of spin 3, 4 и так далее. Например, в spin 3 халаморфика имеет эту форму. Итак, теперь мы не ... да, спаяс из primary fields здесь очень легко описать в том числе fields in sign-level theory это зависит от 2 номера n and m и n and m can be considered as 1 is the momentum number and other as winding number and momentum for cigar is conserved and not winding number not conserved in principle and here we introduced some dual fields which is related this simple relation with the fields, right? So we see that duality, for example, between trumpet between cigar and trumpet it is duality which changes the number m and n. In trumpet dividing number is conserved because we have no way to move cigar outside this trumpet and another important characteristic of this theory which is important for ultraviolet behavior of the theory or perturbation is to point function or the reflection amplitude It means that if we take, for example, this field and we'll do the transformation of parameter alpha which does not change all quantum numbers all quantum numbers it is the numbers related with the eigenvalues of W fields then of course after this transformation some only some numerical coefficient can appear for this field and this numerical coefficient is rather interesting and important object can be calculated and was calculated then it was also calculated three point functions in this theory and really it is enough from our, at least from my point of view if two theories have the same chiral symmetry algebra and two point and three point functions it is enough to show that they are equivalent but there are still a lot of other papers which contain this more and more rigorous proof but we will stop about this point and now we'll consider the possible integrable perturbation base To find the possible integrable perturbation it is to find different errors which how we can do we can consider the enveloping algebra of our W algebra and to see there some carton subalgebra or the subalgebra of integrable objects which can be called as integrals of motion and this analysis show that at level 3 it can be only one field which can be the density of the conserved charge of course proportional to W3 field and but it is possible to find another field which commute at level 4 which commute with it and of course this object are defined up to the derivative because only the integrals of this charge commute are the charges and commute so it gives the possibility to show what is the perturbation of this so the perturbation term in the Lagrangian which appears should also commute with this integrals of motion not in the strong sense as it was before but with charges, with integrals so the first perturbation can be found and the first Erarch which we found is generated by this perturbation then it is possible to find the second Erarch it will generated by perturbation it will not have the spin 3 current but it will have spin 4 and spin 4 are different from the first one just because there is no other fields just by the T-square regularized part of the stress energy tens so this theory contains more symmetry it is symmetrical with respect to this transformation and it corresponds to perturbation operator of this kind and there is the sort and probably the most interesting perturbation which is symmetric with respect to phi and with respect to this exchange and it is it corresponds to the sine Louisville model which perturb by this operator so there is in principle several perturbation and what I want to say now how the duality works in this integrable models so let's consider the first one probably the simplest one so it is the perturbed Lagrangian here we can use the first duality which to prepare this Lagrangian for the perturbation theory and there is the usual Coleyman Mandelstam duality because a and b b is small but a is not small the square sum of squares is equal or different is equal to one half so to prepare our Lagrangian we to do the first use duality of Mandelstam and Coleyman and to write this Lagrangian is the Lagrangian of Thuring model coupled with some exponential field and some other integrable with some field here the second exponent appears as the counterterm to work in the perturbation theory so we have a small coupling now we can use the perturbation theory when b is small and this theory works very well and we can calculate that the spectrum consists of two charged fermions and there is also some particle which must equal to m which disappears from the spectrum because it is not stable with respect to respect to decay but also what is interesting here we can there exist some charge generated by the current of spin 3 and which does not permit the reflection in this theory so this matrix is very trivial it is just the phase then we know that the particle which must to m disappears from spectrum so it should be at the appear in non-physical strip close to t equal pi and we can write the result in more or less general form the only we should do it is to compare to find the dependence of this delta on A so it is possible if we consider the asymptotic of the ground state energy in the external field and it is rather easy to do from Lagrangian and we should calculate this value from Lagrangian and from the scattering theory and see that they are the same and it is possible from Lagrangian this asymptotic has this form the all massive term we can take out so it will be massless tearing model interacting with and the energy of this model is well known it is this one from other point of view we can calculate it from the scattering theory from a symmetric approach and this we should here we should solve some integral equation some termodynamic better answer equation but in principle it is enough to know not a lot about all this kitchen so for example the same asymptotic can be calculated if we know the Fourier transform of our Kt equals 0 so comparing these two values we will find that this relation between delta and V and fix our scattering theory and then let us see what happens when we move from small b to big b and we will find that if we introduce parameter gamma which is inverse power of b its matrix again transform to this matrix which is proportional to the trivial one at gamma equals 0 it means that there is some other we coupling representation the dual representation for this theory to find the Lagrangian we can calculate this energy in this limit it is rather simple and from other point of view we will see that the theory now is completely bosonic it is other other we have some other also things which show us that the theory should be bosonic for example the behavior of this energy new threshold where a equal to m and we see that now our theory is described by the action which contains the free bosons and to support this we can calculate this minimum and compare with this matrix result and to see that they coincide and it means that we see that it we have some interesting transformation at small couplings we have charged fermions and in the weak couplings we have the charged fermions and the strong couplings we have the charged free bosons and in my point of view this is a little bit non-trivial but we understand that strongly interacting bosons can be considered like fermions but why strongly interacting fermions can be become them bosons it is a little bit it looks a little bit strange okay but another perturbation I will not talk a lot about it it also in this case we have S-metrics which is less trivial it should be the solution of some special equation S-metrics is restricted to be the solution of so-called Young-Baxter equation and the solution is the only known solution with the symmetry ua of sign Gordon model up to multiplication to some CDD fact and I will not discuss a lot of calculations I only will say results really it is the example where not minimal but some CDD factor appears not minimal solution of an equation and it is possible to to do again all possible calculation here and to find that Lagrangian for this we should solve some integral equation it can be solved but less trivial some algebraic function of this fact is given a parametric way and it exactly correspond to this Lagrangian the minimum of this Lagrangian because it is basonic one we can use the classical theory and the minimum gives us exactly this Lagrangian what is the interesting phenomenon which appears in this theory My opinion a little bit strange is the following not only fermion bason duality but also that probably I forgot to say before that if you will treat at the beginning fermion Lagrangian in terms of perturbation theory you will find that it has two particles to charge fermions and one boson which can be considered as the bound state but when we go to the strong coupling this bound state disappears at some point and from other point of view we can consider going from small gamma to big gamma here for small gamma this perturbation looks like attractive and there is no any bound state but for rather big gamma due to sigma model nature of this perturbation of this term we have the bound state appears even that the potential looks like attractive so now I want to say probably the most interesting dualities in this perturbed model the first is of them is known like the sausage model so this model has nevertheless this in for the perturbation theory in B this realization which gives us possibility to find this matrix and this matrix is now also restricted by Young-Baxter equation and it coincides with this matrix which we find with Sasha many years ago and we didn't know which model it correspond and called just the model factorized as matrix but in 15 years it found this model and married with it it was the sausage model which I will call later so again we can connect parameter lambda in S matrix with parameter B in in our scattering theory in Lagrangian and to see that everything is fine but if we consider this S matrix for small lambda when B is very big we will see that this S matrix tends to this matrix of O3s so it means that our theory is some you want symmetrical deformation of our sigma model and this deformation to say about this deformation we should say it is nice that Daniel said for me several words about this equation so-called Richie flow so we can consider this Richie flow with you want symmetry and today 2-dimensional target space and if we consider target space you want theory that there is really only one solution which is possessive property of attraction as it should and it has the parameter new and a new is related with effective curvature of our figure and its dynamics of its dynamic in renormalization group time it really it depends on one parameter you more or less which is this product and it looks in the beginning for small values of u it looks like sphere then it becomes a little bit like ellipse and then it becomes like sausage and really if we cut the sausage by two parts we will find that it is two written cigar which are glued together so this is from one point we have the solution of factorization equation from other the solution of renormalization group equation rich flow so we should now we want to prove that they correspond to the same model which is the dual to the field theory which we consider in the beginning goes to infinity so there are many of course possibility this exactly integrable kitchen knows many recipes and I will not talk about them but for example if we introduce the external field we will find that relation between parameter nu and parameter b or lambda in the theory of scattering theory and then we can find some termodynamic function like the energy in the one loop result one loop approximation how much time I have ok now we want to see what happens with our theory when we introduce to see how this renormalization group equation works we should introduce some scale and we consider our theory in the ultraviolet regime for example we can consider our theory at the circle and if we consider this scaling variable like nu multiplied by logarithm then in one loop all our results should be exact and if we consider now for example the central charge which is related with the ground state energy of our theory of the circle that it can be written that in the so called mini super space approximation which is the approximation which is exact in the this limit in the scaling limit we can obtain some differential equation of the time it is interesting that exactly the same equation appears later in the works of Grisha Perlman and the zero eigenvalue of this operator play the role of some entropy which has the non-decreasing property and but it is in some sense from the physics it was first it was written in our paper in 93 that its value is related with the central charge logic of theory should increase in when r goes to zero so when scale goes to ultraviolet and so this equation can be studied in this case in rather details Really everything can be expressed only in terms of parameters of parameter u and the equation which appears for example for the ground state energy will correspond to the values of m equal to zero and in this case this value of y zero of capital which is up to some constant multiplicative constant correspond to this central charge will coincide with some the most symmetric accessory coefficient about which Sasha talked yesterday it is when all fields have the same values of parameter alpha close to q over 2 so it can be calculated we can find it asymptotic for small u and for large u and just from this equation it in first it will be some Schrodinger equation in the potential in some in the small u it will be have the memory of sphere and all the values are very close to the spherical one but when we will go to ultraviolet here we will have the potential which will look like constant almost everywhere but at the end it will be some walls and it is easy to calculate this asymptotic if we know also the reflection coefficient in this quantum mechanical prob and compare it with the reflection coefficient in which can be calculated, was calculated exactly in Saint Louisville theory because this theory describe the ultraviolet property of this theory from the left and from the right due to character of the perturbation that we will derive the asymptotics when u is very big and now there is this we compare everything with like we can calculate with Lagrangian which was in the beginning of Lagrangian we can calculate all these values more exactly with arbitrary precision but there is possible also to calculate them from the equations so called thermodynamic beta ansatz equation which are couple system of integral equations it is not trivial but now it is possible completely exactly calculate more with arbitrary accuracy the value this function corresponding to ground state energy and it coincide with the corresponding limit which follows from the reflections amplitude and so here for the people who knows this TBA technique he has written the incidence diagram for this integral equation the first one correspond to the sausage model the second one correspond to some other model which I will talk soon and how much time I have 15 minutes, very well so now I want to talk about some other model which can be derived from the sausage model by some analytical continuation in the continuation in parameter u and parameter y in this case we will have the metric the sigma model with the metric which will be singular it will be singular and it is defined only for y more than minus u but all in any case geodesic instances on this surface are finite it means that we can again apply this equation this mini super space equation for analysis of this model but how will look the dynamic of these surfaces at the first case for the sausage model the surgery of two cigars here we have the surgery of two other figures it is the surgery of cigars with trumpets and this figure looks like this now we can consider u like in infrared in deep infrared from deep infrared to deep ultraviolet from u goes to minus minus infinity to going to the plus infinity and at the beginning when u is very negative our figure looks like bell then it becomes more and more long and so on unfortunately in the presence of the dam it is impossible to find some corresponding name for this model but so it has no some name but this dynamic is rather interesting and if we consider the surgery in this case it is possible to show that in the proper way the surgery is possible only when k number of our Lagrangian parameter of Lagrangian is equal to the integer so there is some kind of there is some kind of quantization of the coupling constant and now we can find again to see this equation it will change now a little bit but it can this again values can be expressed in some retrieval way at least the ground state can be expressed from the previous one and when we have the point when u y is close to minus u when our figure is looks like the bell it means that imaginary period goes to infinity and in this limit it is possible to solve this equation exactly and to find that this corresponding conformal field theory we can obtain all spectrum of this conformal field theory and central charge from this equation and the calculation show that the central charge and corresponding spectrum of dimension correspond exactly to so called z n para fermionic conformal field theories these models describe the critical behavior of z n analogous of isent model z n isent model and here is almost exactly the spectrum of this model of course we have really in for exact value it will have here n plus 2 but we our approximation works in the case when this is rather big so then if we consider now the perturbation theory near this exact solution then perturbation will be in terms of models of elliptic function and really in this parameter which in the quantum case in exact case it will have also some correction but this what is interesting that this perturbation theory is convergent from in all region from s equal to 0 to s equal 1 which correspond to the ultraviolet region it convergent but for ground state is very very slowly convergent so in now in the infrared режим we have the conformal field theory what will be in the ultraviolet regime in this case we will have the potential of this type so now our potential will be we look like the wall from one side and from other side it will look like attractive potential and in this case we have some also new phenomenon there appear some bound states in this equation it means that in quantum case there are some models of these surfaces related with the this attractive part of the potential which which appears for m more than 0 but it is written the asymptotics when u goes to infinity it is for ground state it is the same as for the sausage case but for non-ground state they are different a little bit different and here behavior of the level when m is order of 10 when we have many different levels here we move from infrared from conformal field theory to the other conformal field theory corresponding to the here is something like Bell ZMPara here we have attractive some attractive part but really we have the theory on the long cylinder with c equal to and we see that the levels all levels have the same monotonicity properties all they decrease or as it do they should because it's minus with minus it is the central more or less related with central charge and here we have all levels discrete levels here they collect to the to the continuum spectrum like it should be and very long cylinder so at the end I want to say that now if we calculate the right and left arm matrix they will be different and really if this arm matrix can be derived if we add from other side the perturbation of the following side here is again sign Gordon perturbation but with the dual field field so altogether this potential will look like this for this model with singular matrix and but these two terms are mutually local they are local with one respect other only if i2 is integer over k equal 4 is integer so we have another this coupling constant quantization I can say several words more about some perturbation theory near this infrared point but where it's possible to calculate more or less everything but I don't think it is very important for people not working with integrable models so then probably some concluding remarks that the quantum field theory which can be considered the perturbation integrable perturbation can be massive and massless and they have some rather interesting duality properties like fermion duality and duality with sigma models All this picture can be generalized really sign Louisville model can be related to the Sainton model which also has some interesting properties of duality and the sausage model also can be generalized to probably the most general solution and exact solution of this equation equation of freedom equation in with three dimensional target space and it also this theory is also dual to two parametric family of integrable theories which contain almost all known almost all known integrable quantum field theories Thank you I have a question I have a question about this sigma model with the metric whose name Shell knows your name if you could possibly terms to the transparency where it is known if you know what I am talking about the long one yeah here Shell knows your name you say that the surgery by the way I somehow dislike surgery in this particular case It is mathematician introduce this word Can you explain is it possible to explain in simple words why in this case the surgery demands new to be The problem is that you should match two way function when you are very very long one it is the way function of trumpet and another is the way function of cigar and you this surgery can be done without with some anesthesia really only when this value k is proportional to the integer it related that the number of for example winding and momentum number they should be how was commercial and this appears only for integer values so that's what I thought it's not possible to understand without going to the anesthesia yeah but if you take some if we consider Lagrangian formulation which correspond to this our matrix which gives exactly the matrix which describes the all ultraviolet properties of this theory the right one and it will be not sausage but some different from it when we have the duality between winding and momentum then you will find that this theory also is well defined only for this values No, I understand if you try to do duality with that you necessary come back come out to the conclusion that k must be k must be integer but from the point of view just of the matrix it's something right but from other point of view the theories flows in the infrared to the paraphirmionic theory and this also if n is not integer the spectrum will not be finite it will have infinite number of primary fields it will not be the rational field theory so the theory is really good when this parameter is integer Other questions? What is known about the conformal blocks of sign Liouville? What is known it is known something in some sense it can be related with the express through the blocks of Liouville theory some of them but Exactly something like A, G, T relation it is not known at the moment but it still rather nice problem because it is the simplest model probably after Liouville theory so sometimes it will be solved I am sure there are no other questions