 Thank you for the introduction. Thank you for the invitation to talk here. I'll, okay, the title's here. The subject of my talk, okay, my own interest to this subject grows, rises after the paper by Alek Lysavi, Kola Jorgov and Sasha Gamayun, who shows that tau function for Pendebeu equations, for example, for the most generic Pendebeu six equation, can be, has explicit expression in terms of conformal blocks for the Rasora algebra, or if you want negrasso partition function first due to theory, and this fact has main generalization. So this is why I came to this subject, and I was a beginner in this Pendebeu theory, and I stopped my talk with some facts about the Pendebeu theory, then. And the subject of my talk is a Q deformation of the conjecture of Gamayun Jorgov and Lysavi. The subject, the main result, Q deformation of conjectures, and other conjecture. Okay, we'll stay here. Okay, I start from the Pendebeu equations. It is known that there are six Pendebeu equations, but it is convenient to decompose one of them into three. So we have Pendebeu six, five, three versions, Pendebeu three equation, Pendebeu four, Pendebeu two, and Pendebeu five, one. And in this diagram, each arrow stands for degenerations, confluence for this equation. And it is easy to see the level of parameters. So the most generic one, Pendebeu six, has four parameters, Pendebeu five has three complex parameters, and so on. And the most degenerate ones, Pendebeu three, Pendebeu one has both of them have no parameters. For each Pendebeu equation, one can assign a certain rational surface. So this is a geometric approach to Pendebeu equation. Originally they were introduced as a second order differential equation with movable branching points. But there's another approach to them as a geometrically due to Akamoto for each Pendebeu equation we assign rational surfaces written here. And for such rational surface, we have so-called inaccessible divisor, in other terms, vertical leaves. And intersection form of the components of this inaccessible divisor is the same as intersection form on the scalar product on the affine root lattice. So to each Pendebeu equation, one can assign affine root lattice. And they are written here in the color magenta. The arrows here, in terms of root lattice, arrows are very simple, we just add one node to the corresponding root, not affine root lattice. And good news that today I will pick about very concrete example. One of the most degenerate, of two most degenerate Pendebeu equation, Pendebeu three, which assign to the root lattice D eight. Here I put it in the box. So this somehow general picture. And if you want, you can forgot it. Exceptional series, maybe we'll explain the equation, but after talking, maybe I'll say more, but probably the same answer. I don't know. Okay, so as I said, I will concentrate on this degenerate equation. So let's proceed. First of all, this is a second order on linear differential equation, which has no parameters. This is one of the conventional forms of it. Of course, it's not necessary to remember, but just to be explicit, I put it here. Another form, which is much more easy to remember, it is a Bollinia form of this equation. It is convenient to rewrite this equation as a system of two, as I said, total like Bollinia equations on two functions, two tau function tau, and another I do not by tau one. There are two equations. D2 is the usual second heriot operator with respect to logarithm. I admit formula how to write tau in terms of w, formula in opposite direction, is w of z is equal to this ratio. For this equation, we also have a symmetry. In the last six equations, the group of background transformations is very interesting. This is an extended, affine, real group of D4. But here, this group is finite, just group of 42. We have only one until your transformation, which I do not by pi, such that pi pi squared is equal to one. And this transformation acts in terms of total functions, very simple, man and maps tau to tau one and vice versa. And in terms of double, this is just the double goes to zero with w. And this is clear from this formula. Probably I also mentioned, so this equation has algebraic solution. So, generic solution of this equation cannot be expressed in terms of any sort of another elementary function. So, solutions of Pen-Levey equations are some sort of transcendental function, which are not equal to anything else. I mean, now we know that they are very related to conformal books, but they're not hypergeometric, not elliptic, and so on. And I think this equation don't have hypergeometric solution, this don't have elliptic solution, but have one, or even better to say two, algebraic solution, just w is equal to square root of z plus minus. And the corresponding tau function is exponent, up to some simple factor. Appearance of square roots has, of course, certain geometric origin. So, you see, there's no square roots in the original equation, but we have them in the solution. Probably I also commented, the solution is, okay, this is invariant of the Beckman transformation. And general, some kind of general. Okay, it was a very short course of Pen-Levey equation. Now, I come to the relation between these Pen-Levey equations and conformal books. So, this is a formula, which was conjectured by Comay-Unior-Guflin III in 2011. That's the tau function of this Pen-Levey III equation has this explicit formula. Now, what is written here? We have two variables, two parameters, s and sigma, which are integration constants in terms of the original Pen-Levey equation. Recall that this equation has afforded two. So, a generic solution should depend on two complex parameters, s and sigma. Let's look at the parameters. S appears here, this formula has clear pre-adjusted property with respect to s, with respect to sigma. If we shift sigma by sigma plus one, then tau function applies by the number. And as usual, tau function is defined up to constant multiple. So, constant with respect to the multiple. So, this factor is not essential. This is the power, and this function c is the explicit function given in this form. Given this form, one over product of two bar g functions. And bar g function, I recall, is satisfied with this functional equation. And this is a special solution of this functional equation. This is about the red part of the formula. Now, I don't know whether it's visible, but it's about green part of this formula. This function f of delta z is if we take a limit of the erosorocan formal block. Delta is the highest weight of the corresponding representation in which we calculate conformal block if you want conformal dimension. And the central charge is not written here, since during my talk, central charge will be equal to one. So, this explicit formula. And this formula relates one sort of, one sort of transcendental functions, namely tau function of many equations to another sort of transcendental functions, conformal blocks. And there, more or less, this space of functions are more or less equivalent. This is the meaning of this formula. Okay, it was a conjecture. Five years ago, the motivation, they have certain motivation from conformal field theory, certain explanation in terms of conformal field theory based on fermionic realization of the c equal ones can be represented in terms of fermions. And also, they represent not really there, but correspond to isomonodromy problem in terms of fermions. But this is some sort of speculation. There was some sort of speculation. Another motivation was that this series was not known, but first terms were known due to Jimbo. And then they calculated the next terms and realized that it's actually conformal blocks. So, I don't have a short answer about motivation, you see. It's just, they have some motivation. Another comment that this function is the necrata partition function for pure SU2 theory due to a GT relation. So, since necrata partition function is given by explicit combinatorial formulas, summation of the partitions, and so on. So, in this sense, this explicit combinatorial formula for tau function. Okay. This conjecture was proven two years ago, two independent papers, one of your presentation, and another is myself with Anton Shichkin. Now, I'm going to relate a genetic story about the Pilever equation, which I mentioned before, back-to-transformation algebraic solution to this formula. So, as I said before, there's a back-to-transformation which acts on the solution of the Pilever tau functions. And in terms of this formula, it's just a transformation which maps sigma to sigma plus one over two. I mentioned before that this back-to-transformation have order two. So, pi squared maps sigma to sigma plus one, and this is okay due to periodicity. So, this formula is consistent. Using this formula, we can rewrite Balini equation, totally like Balini relation. As a one equation on the function tau, and this equation will be differential, second order differential equation on that, and second order derivative equation on sigma. So, we have one equation on one function which is differential on that, and second order derivative on sigma. Algebraic solution, which corresponds to special parameters sigma and s, and using this formula, it's easy to find them given by this formula. It means that we have an identity. So, if you substitute corresponding algebraic tau function, we give the formula of, we give that this exponent is a linear combination of conformal blocks with certain coefficients which are, in this case, can be made rational. Factor out something, this is very simple, and get this. And this is interesting because this conformal blocks itself do not have a simple formula, or at least nobody knows it. And it looks like they're not hypergeometric functions, I think not elliptic, so they are something transcendent, and in this form, this identity is not trivial. Also, I mentioned that conformal dimensions which appeared here are probably now called as conformal dimension twist fields. I mentioned two papers about this twist field, but there are a lot of them in this formula. Yes, depending on normalization, probably I put it here to conformal blocks, or just for God's sake, I don't know what is that, but they should be here, so we have some. So, algebraic solution means that we have certain special relation between conformal blocks and exponents, and means that the corresponding fields are somehow special. In this case, they're known as twist fields, whatever. This twist is responsible to appearance of the square root of z here. Okay, so probably we are almost done. We are ready to go to the main subject of my talk is a Q deformation. Is there any question about what I have called? Good point to ask. Okay, now we are going to, now, Q deformation, and okay, we follow a so-called geometric approach to the deformation of different kind of equations, which was introduced by Sakai. So, if I understand correctly, history, so in 90s, there are many different equations, different analogs of Penilevaic equations, integrable different analogs were introduced by many people, Jimba Sakai, Grammatikas, Romani, I forgot several names, and then in 2001, Sakai invented certain unified geometric approach to all of them. It is some sort of clever different analog of the original Penilevaic property that we study. Penilevaic property means that we study differential equation without movable branching points. And Sakai introduced some sort of different analog, different analog of this property. To go to the answer, to each Sakai sign difference equation to the rational surface, which obtained by blow-up of nine points on CP2. Here, cheating a little bit, so this nine points doesn't mean that all are line CP2. Maybe we take one blow-up and then take another blow-up after pre, on the pre-image of the, on the additional exceptional divisor and so on. Morally, nine blow-ups of CP2. Then for each such surface, Sakai signed certain combinatorial data, latest, which is generated by irreducible components of the anti-canonical device. And this latest is called surface latest as, and as before in commota approach to differential equation, the intersection form on this latest coincide with the form on the certain affine root latest. And this is the table of all possible intersection forms of R, so all possible R's, so corresponding to all possible rational surfaces, all possible difference equations. So you see, after Q deformation, so before we have, say, eight, before Q different, we have eight equations. Now we have, I think, like twice more. So this part was the same, was for differential equation, but that means that it is the same for each of D, for example, for this D81, no, okay, for this D41, we can say it's both differential or difference equation. So we have, really, we have more equations before Q deformation. And maybe I should comment that this for A part of this diagram correspond to Q difference equation and this part correspond to D difference equations. Science 7, to study Q deformation, I will leave in the first row. D difference is a, you have a function on Z, and for Q difference Z belongs to C star, not equal to zero, and for D difference, Z belongs to C. So for example, it depends on what kind of singularities. So for example, you can look for the mermorphic solution in C star or in C, so there's some sort of difference. Yes, yeah, exactly. For each D, I can assign differential equation, I can assign difference equation. If I take any of A, say this, and take this limit, then in the limit, I can have differential equation assigned to this D. Different equations, A and D, for two different equations can have the same limit. So theoretically speaking, there are infinitely many ways to invent different equations which has the same limit, but we are looking for good ones which should satisfy some sort of difference penalty of property, it should be. If I understand correctly, the limit of this equation is not penalty, it should be something elemental. So differential penalty of equation lives here. So from A3, you can go to D4. This is called Q penalty of six equation which is invented by Jim Ben Sakai. Any more questions? Okay, there's another part of Sakai story which I want to mention to each surface. So one can assign so-called something's dual data. Which I denote the r-togonal to the togonal complement to the lightest r which I mentioned before. And here are the pictures of the corresponding r's. I don't have time to explain details of such notation, or such or even this notation, but just a paper, but I will sometimes use one of the table another, sometimes another. The other thing which I can say is that we have two lightest r and r-togonal. The rank of this group is 10, since we have nine blow-ups. So the sum of the ranks of r and r-togonal is also 10. Such lightest r-togonal is useful not to describe the group of discrete dynamics of the corresponding penalty of equations or whole group of discrete group which acts, which form our equation. And this group, roughly speaking, has a form. We take a while group of the corresponding affine root lattice, affine, yes, affine root system, and take also semi-direct product of the external automorphism of the corresponding affine root system. And this is the whole symmetry group. Therefore, the lightest r-togonal is called symmetry lattice. So r is called surface lightest and r-togonal is called symmetry lattice. Just, okay, and on the next slide, I put both lightest r and r-togonal, both tables. See here, for example, that here's rank two, here's rank eight, so sum of the rank is equal to 10, everywhere. And again, I will speak about very concrete equation. So differential equation lives here. It's d eight one. So the corresponding difference equation should be connected by arrow with this equation. So, and I want to study QD-form in scientist. I want QD-form conformal blocks. So my choice will be this, a seven one right, put it in box and draw the dimension. So as before, this is some whole sort of broad picture. And now in the next slide, I will say you in some details, what is the mean of this equation? What is the mean of this, at least of this notation? So you can forgot everything about this very large table. We have, so as I said, so a difference equation is assigned to a rational surface. In this case, we have several, in any case, we have not one surface, but certain module space of such surfaces. And this, we can choose certain coordinates of this space, which I forgot, I omit the exact definition, but I denote them by z and Q below. And on each surface, I have its own coordinates which I denote by F and G. Now, as I said before, group of disk transformation is a semi-direct product of the while group of the symmetry lattice, of the root system of symmetry lattice and external automorphism, which is now is the de-hedral group, group of symmetry of square. I also put here presentation of this group using generators and relations as zero and one are reflections of while group. So they have relations, they both are square equal to one and no other. And P1 and P2 are generators of de-hedral group. P2 is a rotation of 90 degrees and P1 is a reflection. So this satisfies this equation. And here is the action of the de-hedral group of the while group is written. So actually, external automorphism of this group, of this root lattice, of this root system is the two, which we can permute to root, to simple roots. And P1, here should be this one. Okay, P1 should, I think, permute them. Oh, no, no, no, no, no, no, no, no, no, no, no, no, no. P2 permutes two roots as permute to seven corresponding reflection as zero and this one. And P2 squared and P1 commutes with this affine. And the action of W is given by the following formulas. This formulas come from geometry of this rational surface, but I don't have time to explain what is the definition such surface and how to use its geometry in order to write these formulas, just to give an answer. All this is done by Sakaya. This is how a formula looks like. The most non-trivial parts of the formulas are certain rational functions. For one can check that transformation defined by this formula satisfies these relations. So for such formulas more or less clear, but for such rational functions, this looks for me non-trivial, but it is okay. So the small group acts both on coordinates on the modular space of surface and coordinates on the surface, so on the total space of point of bundle. Acceleration surface, it's a blow-up of CP2 in nine points. And but such combinatorial data, these nine points are not generic, so some of them lie on the premise of another one and so on. And the sort of the generation is encoded, some, there is some combinatorial description of the sort of the generation which correspond to this surface. So actually we have only four points on CP2 and so on, then three points in the premature one of them and so on and so on. This lattice, so this core parameter Z in Q, parameterize the generation. So combinatorial data is the same. Maybe I'm answering the wrong question now. Okay, thank you. Okay, let's proceed further. I also introduced element T in this group. This element of infant order, translation is W. And in order to run, I used standard notation for different equations, I denote by X upper, by T of X and X down, T inverse of X, then you see T shift Z to QZ and T inverse shift Z to Q inverse Z. These formulas are obtained from this formulas. It is impossible to check this on the different slides, but it remains to believe. And for function F and G, we have certain rational function and using this formula, we have this, we can write the one equation, the function G. And I will say next slide that, but I can say that even now that this equation is actually Q deformation of the original second order differential Pendevere equation. So if in the limit Q goes to one, this in the leading order, this equation goes to Pendevere three, three equation. Of course, this limit should be arranged. So Q goes to one, it goes with G and Z. So Z goes to zero in certain speed, G also goes to zero. There is this certain regime under reach, this equation goes to this Pendevere equation. In another regime, it can go to something elementary and so on. Okay. Now I want to write formula for tau function, in order to Q deform original conjecture, I want to write formula for tau function. And the first question is, what is the definition of the tau function? And for differential equation, Akamoto gave the original definition using Hamiltonian formalism and so on. And for difference equation, at least I understand situation is not so simple. For example, for our concrete equation, we haven't find in the literature the definition, so actually we invented definition of the tau function, but following the Tudor approach. And I mentioned there are, probably there is no convenient definition of the tau function for such equation, but there are several approaches. So I mentioned last paper by Naomi, but there are several papers of Naomi and his co-authors. And I also want to mention paper by Rink and Borodin, who defined that tau function, not for Pendevere equation, but for linear difference equations. And differential Pendevere equations arises as another problem for linear differential equations. So there should exist a relation between Q deformed Pendevere equation and Q difference problem. So problems that is a relation between their definition and definition, which we need, but this is not done. So this I just mentioned, but actually we follow somehow Tudor approach, but it is not necessary to know what is his approach. I put it as a theorem here. So we can introduce for tau function, because there are some reason why for tau functions looks to be good choice, and define action of the transformation P1, P2, S1, S0 on them by the following formulas. And also then find action of the T. So these four columns are already given below, given above in the previous slides. And these formulas are our definition, and one can check that all relations are satisfied. All relations of the group are satisfied. Moreover, one can check that if F and G are expressed in terms of tau function by this formula, which are very similar to formula which expressed differential function, solution of the differential Pendevere equation in terms of tau function. So if F and G expressed in this formula, then the action on the tau function is consistent with the action on double on F and G, which I mentioned above before. The second order differential equation on G, in terms of tau function, this equation reduces to system of two, Bellinia again, like Tudor equations on tau functions, tau, we use two tau functions, tau one, tau three, and last comment that I have made several checks that and that is also on the level tau function, all what I can get action of W is a Laurent, even no serious Laurent polynomial as here. So for G, it is not true. Here we have a real, we have rational function, G minus Z is denominator G minus Z. But if I will act on tau functions, I will get only Laurent polynomials. And probably this fact could be explained using some relation to cluster algebras, but I don't know. But this looks like one more hint that our definition of the tau function is good one. Now, I consider all my function which I introduce, F, G, and tau as a function on the variable Z and also Q. So I write action of T as a shift on Z, so H up is H of Q, Z and so on. Then equation on function G has this form and here I can take a limit few goes to one and get differential in the equation. And for tau function, our goal will be the same as what's more informal by gamma union, you're going to introduce only one function tau. Dependent two parameters, U and Z, here is U, but U is a, one can say that U is a Q in power two sigma. So in sigma and Z are direct analogs, so sigma and Z in continuous formula. And for this function, we set tau one given by this formula, tau three given by this formula. Assume periodicity and we want to have such boolean relation. This boolean relation is equivalent to boolean relation on the previous slide, work slowly maybe. So this is my problem. To find function tau, which will satisfy this boolean relation, maybe also periodicity, also periodicity, then I will find a solution, I will find function G by formula, which was on previous slide. So in the center, you will solve difference in the equation. So here, that's the type, here should be inverse. Thank you. Probably this T looks different from this T and this is also type. And this T also looks different. This is also type. There are several types. More question. Now we almost come to the main conclusion, main conjecture. So main conjecture will be on the next, main definition will be on the next slide, but before in order to write it, I need to recall what is Q deform conformal blocks. So I wrote this tau function in terms of the Q deform conformal blocks. I will write it. But before I need to recall what is Q deform conformal blocks. First, just in words, this square, we take the vector of the, of the rostro algebra. It has the same representational definition as usual conformal block. But for actual calculation, we use another definition of statement. This function is a necrase of instanton partition function for pure SU2 gauge theory, five dimension. So therefore, this function has a explicit combinatorial form. Summation of a pair of partitions. And here is this necrase of type product. So the function defined as a power series in Z. Here I use two parameters, U1 and U2, but actually this function depends on the ratio of them, U1 over U2. So everywhere I will write just U. Also, during the, also today, my talk Q1 inverse will equal to Q2 will equal to Q. For the rostro case, it corresponds to the C equal one in terms of topological vertexes. Vertex, it means that our topological vertex is not refined, so that's very good. And this formula have maybe better say U2. When we take a limit, we don't have this U1 counterpart. To be honest, I not really understand the difference in this notation for SU2 and U2, maybe you will explain. For in the representational definition, we do not use additional Heisenberg, we use just the rostro. But for when we take a limit, this is the same for Heisenberg we take in the ones. Maybe this is not important. But, so this two is essential. So U1 is not, this is not important. Okay, I would like to put as a lemma that this series is, is a series converge. As a series, as a series on the, on the, and converge for if absolute of value of Q is not equal to one. And U is also not Q in power n, so for such U we have poles. And last I mentioned this, this is function is a topological string partition function for local p1 times p1 geometry. So this function up to simple factor in infinite program symbol square is a topological string partition function. So this function f, which has three interpretation if you want Q over a sort of conformal blocks n cross n string partition function and topological string partition function is main ingredient of the conjunction. Definition, the tau function will be given by the following form. What we have here, the f is the same as before. Q deformant conformal block or n cross partition function of pure SU2 theory. The main, the new ingredient here is this function C. Also with this factor but doesn't, and this function C, if you remember in conformal formula we have Z in some power divided by the products of Barnes G function. This is some sort of analog of Barnes G function and this is some sort of analog of the Z in some power. In order to stay, to state main conjecture that this tau function, the solution of the Balinia difference equations. We need to impose a following conditions function C, system of three conditions. This condition can be viewed as a second order difference equations as a function two variables, U and Z. Here is a second order difference operators with respect to U, here with respect to Z and here with respect to both of them. Function R is not essential, I can say that R is equal to one. All one can say that R satisfy homogeneous version of this equation where right side is equal to one. This is not necessary for Balinia equation but it is sometimes convenient to impose that functions U and R satisfy symmetry with respect to U to U inverse. I recall that conformal block satisfies symmetry and also this products satisfies this symmetry. So maybe convenient but it is not necessary for Balinia equations. This is the definition. So the most material part here is of course function C. For example, after such formulas, you can ask whether this function exists and the answer is positive. This function exists and there are a lot of such functions. Here I put two possible solutions. One of them is actually more or less the same, this was before Q deformation. Something like Z in some power but now this power which is called sigma before, now this is a logarithm Q over logarithm Q. Logarithm U over logarithm Q. Therefore, we have some such strange formula. And advantage of this formula is that in this case function will be miramorphic in Z, miramorphic in U. This function of course not. I don't know whether it is a good question what is the best answer for function C. This function is defined up to the multiplication periodic function and both U and Z or even of this multiplication by solution of the homogeneous equation which were on the previous slide. I don't know. At least such function exists. I have two rather random examples. I also noted from the definitions clear that our function satisfy periodicity condition with respect to U, this is in here. And if C and R are a universal variant then the corresponding function will be also universal variant. Now conjecture that function tau defined previous slide satisfies this Boolean equation. As I said before, this Boolean equation is actually equivalent to the corresponding Q difference pendulum equation. This main question. Q difference pendulum equation of the surface type A7 and seem to be right. One can rewrite this equation as a Boolean equation conformal block. I put it as a theorem but actually this was a long but straightforward calculation. I don't know proof of this. So this conjecture is equivalent. But anyways, you can say that this, okay, this relation is conjecture. No, no, this conjectures. What is proven that this relation is equivalent to this, but it is not big difference. So if you introduce times, it's basically singular vector for q-varus or, it's actually like q-varus or constraints is applied to this state but not applied. Because actually it's wrong. Hmm. No, no, I didn't know the paper but I don't know it was this relation. You did it for all the U.N. years when you were... I don't know. I'm not sure that this is related. My understanding, I don't know. I don't know if this is related. Okay. I just had to comment that first of all we, of course, we have checked this relation up to certain power of z. And second, we checked this equation, conformal limit of this equation and this additional check because up to certain power of z, we use only first three or maybe five summons on the left side and on the right side. But in conformal limit, limit where q goes to one, we have checked all terms. And everything's okay. Conform, continuous limit is true. So I think this conjecture is true. Let me briefly mention two other parts of the the differential pin lever equation and the liquid deformation. First, I told you that there is a background transformation for differential pin lever three equation and here we have also analog of this difference of this background transformation. This analog permutes tau one and tau three and this is clear. Which is the second, which is not so clear, it is a question on algebraic solutions. I recall that differential pin lever three the equation has one, two algebraic solution plus minus square root of z. The same holds here and this even simpler. You can, during my talk, substitute this g of z in this equation and get everything's okay. This equation is satisfied. I don't know whether this classification of algebraic solution of this equation, but anyway, naïve analog is a solution. And therefore, we can substitute this solution to the conjecture and get the following results. So if tau function is U inverse invariant, then for special values of U and S, S is before is plus minus one and U should be square root of Q here, this is essentially reduces to this double-paragymn product. In terms of conformal blocks, this formula is equivalent to the relations. This double-paragymn product is equal to the sum of Q-deformed conformal blocks. This is a certain curational coefficients and here I have power z. So probably I forgot it in the first slide. Let me make one more comment about this program of product and then I go to the conclusion that I said that algebraic solution for pain-leveur d8 is a exponent, tau for conformal function is exponent. And this should be compared of the necrosis partition function for pure U1 theory, which is also exponent. Here algebraic solution is a double-paragymn product and this also should be compared with necrosis, could be compared with necrosis partition function for pure U1 theory, five-dimensional, which is also one infinite product. So the fact that the exponent, the Q-deformation of the exponent as such infinite product is known. Any infinite product can be viewed as a deformation of the exponent, but I think this relation means that such infinite product is natural here. Also we have the same phenomena here, we have exponent of z here is exponent of square root of z and in terms of c of t it relates to the fact that we actually consider twist fields we need to take. And here we also have the same square roots of z. Okay, I have three more slides and I don't know five more minutes. Okay, first comment is that the main statement is a conjecture, so if I consider myself as a mathematician, so some of my questions to prove. If at least try. If I forgot about those questions then the next question is to generalize. And now I recall this big table here, or it's called symmetry table of the symmetry table for symmetry lettuces and I recall that we leave here now. The main conjecture is that for this line, for this point of equation are related to five-dimensional and cross-partition function with n fundamental multiplets. So here I discussed case where n is equal to zero and the main conjecture that for any n at least from here is this holds. So n is not less or equal to seven. This for example, it is known that if n is less than or equal to four, then we can take q goes to one limit of this equation and get that there should be a relation between corresponding differential and the equations and corresponding four-dimensionally cross-partition function. And this relation is actually known due to Gamayoun Yorkov and Lisa V. So for this case we know that there exists a limit. Another important or even no important relation is that 20 years ago cyber-harguers that this gauge series, so SU2 here should be SU2 somewhere. SU2 here, SU2 gauge series or U2 I don't know with n fundamental multiple has global symmetry E n plus one. And, but I don't know whether exist any, I don't think that this coincidence, but I don't know any explicit relation between this fact that this gauge series has such global symmetry. And our desire that our hope that the corresponding, that q tau function of the corresponding q-pollineve equations can be given in terms of the cross-partition function of this gauge series. And of course I don't know where here actual group of symmetries is a fine whale group. So where, how a fine part comes? So probably this a fine part comes due to the fact that we have not only one across the partition function, there's a whole series, but this equation may be more interesting than proof. Okay, second part of the discussion and that I, from the, if you look at the original of the formula for this group, which you see that I use only part of this group, only shift t and also p2 squared, which is, which goes to the back from transformation of the different corresponding differential equation. So this part is a, has clear meaning in the limit. But actual discrete group is larger. And so this actual group looks to be new phenomena which arise after q deformation. And it would be interesting to understand this in terms of our formula, of tau function. So actually this symmetry is used in this, such simple case to see that transformation to see to the inverse and q to the q inverse. And second transformation transparent, more or less, conformal blocks has such symmetry and also p-cummer functions has such symmetry and so on. But to see, goes to the inverse, I don't know whether it's possible to say something about our formula for tau function and say some correct statement which involves our formula for tau function and also such symmetry. Naive idea that this symmetry should means, should be equivalent to certain symmetry of conformal blocks. And okay, two comments. But first about this symmetry. So I recall that this q to the form of conformal blocks is more or less equivalent to string partition function for local p1 times p1 geometry. And for such geometry there exists fiber-based duality which intersects to interchange two factors of p1. And in terms of this function, this duality has the form. But this identity is a formula. Identity is the form of power series in variables u and uz. And if absolute value of q is not equal to one, this series do not converge. So I don't know how to use it. So if I define this conformal blocks as I did before, as a power series on z, then as I stated before, the series converge, everything's okay. So for absolute value of q not equal to one, left side is defined, right side is defined, but they are not equal. And as a form of power series they are equal, but this not, but don't converge. So I don't know whether any relation between fiber-based duality and this question, z goes to z inverse. And probably this question is more or less, is very related to the question on the previous slide, what is the relation between the symmetries of this affine group symmetry and symmetry from cyber paper. Okay, and my last remark will be more or less the same as on the previous talk. I mentioned another paper by Julio Alessandro and Alba, who constructed Q differ, also, who also constructed Q deformation of the formula for believe this three tau function using this paper mentioned in the previous talk and many other, I just wrote here two sentences. And maybe Alba will explain more. It's interesting to note they work in different regions so in their case absolute value of Q is equal to one. And if I know correctly, this approach are not completely, our approach are not somehow related. But I don't know details and maybe Alba will say one in the next talk.