 will talk about some results on minimal gravity. Это очень известный книжник полякова за молочкой. И я поговорю о каких-то континенциях, так сказать. Так что эта минимальная, так называемая минимальная гравития была изменивана и сочетана от демокнижника полякова и молочкова в 1987 году. И это очень важный случай, так называемый полякова за молочкой. И в этом случае это также изменивается гравития. В этом случае сочетание гравития является одним из минимальных моделей конформации полякова. И в этом проблеме, которая была открыта после первого книжника, была найдена сочетание функций гравития даже для геноз-зиро. Это трудно, потому что мы должны calculating какие-то интегральные сочетания с сочетанием гравития и обзорные какие-то специальные комбинации минимальных моделей, сочетания полякова и сочетания полякова. В таком случае только сочетание полякова. Это трудно, потому что мы должны интегрировать модельный сочетание гравития. Модельный сочетание гравития 8 лет назад, из-за того, как Олеша молочила на высокую эквейсию Льювилл Филдсери, в том числе, как мы работали с Олешей, мы обзорвали 4-пункта гравития сочетания гравития, которая была первая. Мы должны интегрировать модельный сочетание гравития. Но мы должны обзорвать и обзорить гравития сочетания гравития. И мы, вместо компута гравития, не гравития, а гравития-намперс, потому что это не зависит от позиции филды, а от интеграции. Мы можем компутать гравитию сочетания гравития. Это не зависит от позиции филды, а от интеграции гравития сочетания гравития. Это не зависит от позиции филды, потому что graduating constant lambda can be considered as coordinate on the space of perturbed, minimum gravity, level gravity. И еще. Это был в таком мутинге. Мы пытались компутить это cloudy by direct calculation of integral of motor space. Another discrete approach to two-dimensional gravity was invented by many people like Rizakov, McDowell, Kostov, Stohdacher, Bresnan, Douglas, Schenker, Gross. And in this approach, which is realized by integral over, through matrix models, we describe fluctuating two-dimensional surfaces like ensemble of graphs. And in continuous limit, we expect that these two approaches give the same result, не critical point. And in the end of the 80s, we discovered that the partition function in this matrix model approach satisfied to so-called string equations, also Cadeway equations. And the Cadeway times, Cadeway parameters of Cadeway approach T, MN, for Cadeway case, for Dickey generalization plays a role of perturbation parameters lambda in minimal gravity. And Douglas found that these dimensions, gravitational dimensions of these two sets of parameters in Liouville approach and in matrix model approach coincide. However, attempts to compare gravitational numbers itself, not only gravitational dimensions, showed that naive identification doesn't work. And this is the problem. In minimal gravity, we have a set of selection rules for correlation functions, for correlation numbers, which are inherited from selection rules conformal and fusion selection rules of minimum model of conformal field theory. Say one point function should be equal to zero, two point function should be diagonal, and then we have other selection rules, fusion. And so we have some restriction for if we compute correlation numbers of Liouville field theory, we have some restrictions for these correlators. So, how to reach? The problem is how to, is it possible to use Douglas approach? I mean, it's possible to conjecture that the correlation numbers and the generating function satisfy to Douglas-String equation, but correlation numbers are based restrictions from minimal, which followed from minimal model of conformal field theory. The idea to reach this fulfillment of the selection rules is the following. It was suggested, the solution was suggested many years ago by such as Malochikov, or Zyberkistadek. When we compute correlation numbers in Liouville field theory, we have to integrate of model space, and we met the problem of contact terms. And it is equivalent that the statement that there exists some non-trivial relation between natural, between KDW times, which are natural variables in KDW approach, and a constant Liouville field theory, Liouville minimal gravity, in which selection rules are fulfilled. And because of these contact terms, these variables, whose gravitational dimensions are equal, cannot coincide themselves, but has some additional terms. And appropriate choice of such change of these variables allowed this author, more Zyberkistadekher, to reach coincidences for one and two point correlation numbers in Liouville series of minimal gravity. But then, when after four point correlation function had been calculated, it became possible to make new explicit check against Douglas equation approach. And it was performed for the same series, Liouf series of minimal gravity in our work with Sasha Zomolochikov, and the full correspondence between Douglas equation, idea of such kind change of variables, get inside this was reached. The aim of the following work, which was done with Dubrovin and Muhometranov, was to generalize this result and check that it works also for other cases of models of minimal Liouville gravity. And analysis is based on the following assumption, that first assumption is that there exist in the space of perturbed minimal Liouville gravity, there is such special coordinates Tmn, which doesn't coincide with natural lambda mn, in which partition function of minimal Liouville gravity satisfies Douglas string equation, like it does in matrix model. Then the next assumption is the string equation define this coordinate, and this from partition function, or partition function generated function correlation numbers is nothing but logarithm of SATA tau function for dispersion less generalized KDW hierarchy KDW, or different hierarchy, with some special initial conditions, which defined by Douglas string equation itself. Because Douglas string equation is not identical to KDW hierarchy, but it includes also some initial restriction for initial condition. And the next assumption that KDW times Tmn related with natural coordinates lambda mn on the space of perturbed minimal Liouville gravity is by some nonlinear transformation. And we established the form of this transformation from the requirements that correlation numbers should satisfy the selection rules, which are in minimal models of conformal fields theory. And it was very useful that we use some very convenient expression for partition function of minimal Liouville gravity. We use to get this explicit formula, we use some relation between Douglas string equation and the Frobenius structure, which is connected with this equation. Okay, let me put some details. Minimal Liouville gravity consist of Liouville field theory, phi, and some metasector. In our case, it will be one of the models of KUPE, a minimal model of conformal field theory. In this model, we have the set of primary fields, which is called phi mn, which is enumerated by dots of cut table. And we consider this series Liang, the next one. In this series, we cut table consist of one or two rows, and we can choose this field due to, like, the set of primary fields. And we have the following restriction for operator product expansion, fusion rules for operator product expansion, which is written here. And we will use this restriction from projective invariance we have for the genus zero, also that one point correlation factor is zero, two point is diagonal, if dimension of the field in metasector, the net coincide. And for high correlation numbers we use fusion rules for product expansion, and together with these two rules result we get some restriction, which is written here, such kind. Okay, now, the gravitational sector in polycoff approach is defined through path integral or two-dimensionary matrix. And in conformal gauge we have, it's described by Liouville field with this action. And central charge, you know, it is conformal field series. Also Liouville field series with central charge is connected with coupling constant B, like it's written in these relations. And to define Liouville gravity we have to suggest that the central charge is metasector in Liouville field, the sum is equal to 26. And from here we get this value for coupling constant. Okay, observables in minimally Liouville gravity are created by a step writer. And the simplest observables looks like it's written here. They are integral of our surface. And OMN is a product of primary field of metasector and exponential field Liouville field sector. And it is one, should be one-one form. And from here we get this value for so-called gravitational dimensions. It depends on the field phi. Observables, OMN and from the bottle, Pico. These operators have the gravitational dimensions in terms of cosmological constant mu, like this. And they satisfy the same selection use like fields phiMN from metasector. It is definition of correlation number. And this is the generating function. Since to compute correlation numbers we should integrate over two-dimensional source, we get the problem of the following problem. When two or more points in this integral are inside, this contribution of such situation are not controlled by conformal field theory. And we can add delta-like contact terms. And this ambiguity is equivalent to possibility to add to correlation number. Some in other correlation number of different order. These contact terms also equivalent on the language of coupling constant to adding to such kind of change of variables. These terms respond to some delta-contact delta-like term. And so any addition of contact terms and such change of variables. We can restrict this change from dimension consideration. We can impose restriction that only such terms in this sum we can have, which has the same gravitational relations as the lambdaMN. It means that we can restrict it imposing this relation for terms of only such constant C are not equal. Zero if this relation is fulfilled. But however, it's not exclude such change because in minimal model we have because the gravitational dimension are some rational numbers. We have a lot of relation of such kind. So there exist different system of coordinates lambda and general minimal gravity and dogles. And variables in dogles approach do not coincide. And we can use this freedom to try to fulfill certain rules of minimal level gravity. So we will do in the following way. First we will estimate of free energy or partition function, generating function or correlation of numbers from dogles equation. And then we will choose such change of variables that selection rules would be satisfied. There exist explicit formula for 3-point and 4-point correlation numbers in minimal level gravity and in 3-point function. It's useful to normalize this 3-point function in the following way, such to do it not depending of the normalization. And it gives us possibility to compare the accelerators in minimal level gravity and in another approach comparing such accelerators which do not depend of the normalization. So for 3-point and normalize 3-point function we have this formula and for 4-point function we have this formula which was calculated using his high equation of motion in Liouville field theory. Explicit formula. And we will try to find these contact terms in such a way to fulfill selection rules first. But then we will compare also this 3-point and 4-point function where we can do it because in one approach from dogles string equation we can get any acceleration function principle but we don't know how to get up to now 5-point and more point function in direct approach. And doing this we will use selection rules. In dogles approach partition function of q p minimal gravity is described as follows. We take two differential operator p and q. p is differential operator of q small order u alpha of x is some coefficients of this differential operator which depends of variable x and also of times. And p is the second operator which is defined as a positive part of the differential operator which is defined in the brackets. And t k alpha e are cadaver times. It means that if we fulfill this is dogles equation and it follows also that if we define from this equation the coefficients u one say and u two and others they will depend of the times. They will obey evolution equation but this equation contains additional information about initial condition for our coefficient u alpha. And instead of using this equation because we are interested in genus zero we can take quasi classical limit and consider instead of commutator plus on bracket pq which you find like this. And instead of operator q and p we can consider the symbol. So we have plus on bracket of two function of variable p. p is symbol differential operator d. We have this polynomial. It's already polynomial of variable p. p is not the same number which is in definition of minimal real gravity. And we have to solve this equation. Plus on bracket pq is equal to one. This equation is equivalent to the following action. So the same list action principle of the list action which can be written like this. S is nothing but zero in four in point when p goes to infinity of this polynomial. We have this Laurent polynomial. And this denotation c alpha is the residue of this degree of polynomial not integer degree. Now we want to ask to find the terms which we should include in this definition. To do this we should do some dimensional analysis. I will skip it. In result we get the following identification of the dimensions. It's written above. Our times tk alpha is... should has... It's better. We will nominate... We will nominate this like tmn. Should has... Let me skip these details. This is the result. If only the finite number terms which is written here we have this degrees of our polynomial. Then variables tmn will have the same dimensions as the dimensions of the variable variables. Now these variables and lambda variables connected... We will suggest that they connected by such kind relation Each term in the right hand side has the same dimension like this one. So there exist some... Not so many, but there exist some terms. It is polynomial. In fact it is polynomial of lambda. The right hand side. And now... Now we will try to satisfy the selection rules. And for this it's useful to... It's convenient to use the following expression for generating function. In Douglas approach generating functions connected with solution u1 star solution of string equation like this. But it's possible to write explicit formula which is useful for the following analysis. To do this we use this connection between string equation and rabbin structure connected with it. Let the... Let i is algebra of polynomials model Q prime. Q is our polynomial which is connected with given... With given series of models. So we have... In this approach naturally we have series of minimal models instead of pq. We use the following limit lecturer. Here k is integer part of this ratio. p0 is... Reminder. Reminder, yes. p0. So p0 is... And all this series connected with polynomials q of sorry p, not this one. Which is... Let me... This is definition of our algebra. Our algebra of polynomials model Q prime is derivative. So we can define also... Define also linear form The space of such polynomials which is written here. Here is residual. And we can introduce some basis in our algebra. And c, alphabet, gamma is structure constant of our algebra. Which is connected with this bilinear form. G, big gamma. And this tensor c, alphabet, gamma, like this. And then we can... Then the statement is the following. That the solution of this equation. This equation, this relation. We can solve, so to say. This relation between partition function f and... Solution of string equation. And write this formula. Here we have this tensor. Which is obtained. We get got from structure constant. Of our algebra. We can compute structure constant. We get this tensor. And S is nothing but action. Which I wrote before. Let me show where is action. This one. So we have this formula for partition function. For generating function. First of all, it should... We have this integral over d u alpha. And this integral should not depend of the path. It means that this one form, differential one form should be closed. And also we should check that this Douglas relation of this form. Such defined by this formula. Partition function and solution of our equation fulfilled. We need to have this to check that this one form is closed. We need this two equations. It's nothing but associativity. Requirement associativity of our algebra. And also this recursive relation. Which for residues of different order. Which can be checked that they fulfilled. And so we have explicit expression for partition function. And we make some admissible change of variables. From kodewa to lambda. And we'll find simultaneously the form of this substitution. And correlation number step by step. 1.2.3. Consider a simple example. This case is kodewa, in fact. Case or liang series of minimal level gravities. And in this case we have our second order. We have a very simple form for our generating function. SU is such polynomials of the only variables U. It's written here. And we have, I use here somehow the numeration of variables T of times. And then we should to get generating function for correlation numbers. Because we insert this resonance transformation. After this we have our function. Generations function like some polynomials of variables lambda. Serious in fact variables lambda. And s is polynomial of lambda also. Derivative of action of these variables. And then we, all these polynomials looks like this. Like polynomials of variables U, which has some dimension. Cosmological constant mu, who also has dimension. And some numbers B, C and so on. Which are parameters of our resonance transformation. And we should find them from our selection rules. We have some selection rules, which is written here. It's some convenient switch of two dimensionless variables. And we have such formula for our partition function. And the upper limit X star is solution of string equation. It means that it's zero of our function Y U. And from dimensional analysis we can understood that our function Y0 and Yk. It has some polynomials of some order. And then they go polynomials of, which includes degrees of X with step 2. Therefore, it's convenient to use such change of variables X. We get another set of variables, which now are function of Y. New variable Y. And we get for one point in two point relation numbers, which we get above from our definition of partition function. We get the following two explicit formula. And it is one point function and it should be equal zero. When K is not zero. And this is two point function. It should be zero when K1 and K2 are not equal. And from dimensional analysis we know the order of these polynomials and this one. And I will skip. It's very simple. We arrived to this folding statement looking to this requirement, which falls for our selection rules. One point function should be equal zero. We arrived to the statement that Yk is nothing but Jacobian polynomials of such four. The result is something in this table. We get the Jacobian polynomials of this order, like this one. With how it is called. Нет, это пункция, которая... Нет, нет, нет, я имею в виду. We define Jacobian polynomials as... Как? Kernel. Remind denotation. So it should be zero if n is not equal m. So we arrived to this statement. If we can do it, it is done in general. So in the way which can be used in general case also for other series. But namely for this series, we could use another way of doing. And we arrived, it was done in our work with Sasha. And we arrived to Legendre polynomials. But Legendre polynomials connected with Jacobian polynomials like this in this case. Then we do it the same for 3-point function. And get other resonance terms. We get formulas like this for other terms, resonance terms in our action. And again we can use in such a way to delete all unwanted 3-point correlation functions. And then after this we can compare the result for non-venition correlation function. We get this formula for 3-point function, for normalized 3-point function. In this case we have this formula for... In minimal gravity approach. And we can get formula for 3-point... We get the same formula for in this approach, in Douglas approach. Then we do it the same for 4-point function. And we get some very simple explicit formula, which is written here for 4-point function. And again it coincides with the result of direct correlation. And at last we can formulate the general result for this series. In terms of the former language Legendre polynomials. It is a conjectural statement. Partition function for this series is equal to this integral. X star is a solution of this root of this polynomial. And Y is this series. Here we have Legendre polynomials of this order. Differentiated n minus 1 times. And it gives the answer for... It would be interested to prove this formula. It's very simple. I don't know how to do it. And I have to end the finish. The same can be partly repeated for the next series. We have due to connection with Froberian structure. This formula for generating function. And again we arrive to the following results. To compute 1 and 2-point unwanted correlators. We can choose in a special way the resonant change for this order. It's done. It's performed in terms of again of Jacobian polynomials. In the case of Liang case a series it was in terms of these polynomials. In the next case this polynomial. And for 3-point and 4-point we can... In the cases when 3-point correlators are not equal. When 0 we can result which coincide with honest direct computation. But there is some open problem I will skip. Thank you very much. I have some naive question. Long ago there was the following picture which we had of this minimum gravity. We said that in conformal fields here we have primary operators. We have secondary operators. And we interpret secondary operators as deformation. As halomorphic deformations of primary operators. And when we coupled to gravity these deformations become gauge artifacts. So they are not there. And we are left with algebra of primary operators. And it should be equivalent to some kind of topological field theory. Or matrix theory and so on. And so the picture was extremely simple or maybe simplistic. So what's the modern view of this? It doesn't make any sense this picture. Нас это случилось в таком направлении. По-моему, по-моему. Я думаю, что это должно быть. Потому что мы не понимаем такие простые слова. Я старался несколько раз. И я всегда загадывался. И я делал эту идею. И что-то другое. Спасибо. У меня есть вопросы? Я имею в виду вопросы. Это возможно, чтобы стать пилимом? Пилим? Пилим, да? Наверное, да. Наверное, да. Наверное, да. Также, я... Я имею в виду... Мы считаем здесь только... Очень простые. Очень простые. Какомологи, если интересно. Леонсу Керману. Да, вот что-то по-моему. Болезненно. Мои вопросы слезте, Саша.