 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, my own interest to this subject grows after the paper by Oleg Glysovy, Kolya Yorgov and Sasha Gamayul, who shows that tau function for Penderbae equations, for example, for the most generic Penderbae six equation can be, has explicit expression in terms of conformal blocks for the rasora algebra, or if you want negrasa 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 Penderbae theory and I stopped my talk with some facts about Penderbae theory. And the subject of my talk is a Q deformation of the conjecture of Gamayul Yorgov and Glysovy. The subject, the main result, Q deformation of conjectures, and other conjecture. Okay, we'll stay here. Okay, I start from the Penderbae equations. It is known that there are six Penderbae equations, but it is convenient to decompose one of them into three. So we have Penderbae six, five, three versions, Penderbae three equation, Penderbae four, Penderbae two and Penderbae 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, Penderbae six has four parameters, Penderbae five has three complex parameters and so on. And the most degenerate ones, Penderbae three, three, Penderbae one has both of them have no parameters. For each Penderbae equation, one can assign a certain rational surface. So this is a geometric approach to Penderbae equation. Originally they were introduced as a second order differential equation without movable branching points. But there's another approach to them as a geometrically due to a commota for each Penderbae 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 Penderbae 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 more degenerate Penderbae equation, Penderbae 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, I said, totally like Bollinia equations. On two functions, two tau function tau, and another I do not know by tau one. There's 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 squared is equal to one. And this transformation acts in terms of tau functions, very simple, manning maps tau to tau one and vice versa. And in terms of w, this is just, w goes to zero with w. And this is clear from this formula. Probably I also mentioned, so this equation has algebraic solution. So, genetic solution of this equation cannot be expressed in terms of any sort of another elementary function. Solutions of Penileva 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, 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 that the solution is a, okay, the reason here is invariant on the background transformation, general, some kind of general. Okay, it was a very short course of this equation. Now, I come to the relation between this kind of equations and conformal books. So, this formula, which was conjectured by Comay-Unior-Gofflin III in 2011, that's the tau function of this of this 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 Pindov equation. Recall that this equation has afforded two, so, a generic solution should depend on two complex parameters, s and sigma. s appears here, this formula has very clear predictive 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, satisfies this functional equation, and there's 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, about green part of this formula. This function f of delta z, either we take a limit of verisorac and 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 function, so more or less, this is the meaning of this formula. Okay, it was a conjecture, five years ago. 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 in the way, but correspond to either monodromy 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 this is actually conformal block. I don't have a short answer about motivation, you see. They have some motivation. Another comment is this function is the necrata partition function for pure SU2 theory due to a GT relation. So thanks, necrata partition function is given by explicit combinatorial formulas, summation of the partitions, and so on. So in this sense, this is 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 is Anton Shchetskin. Now I'm going to relate generic story about Penilever equation, which I mentioned before, back home transformation, algebraic solution to this formula. So as I said before, there's a back home transformation which acts on the solution of the Penilever 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 home 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 relation, 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, 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 in this case can be made rational. We can 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 there's not hypergeometric functions, I think not elliptic, so there are some transcendence, 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, okay, yes, depending on normalization, probably I put it here to conformal blocks, or just for good, I don't know what is that. They should be here, so we have some. So algebraic solution means that we have certain special relation between conformal blocks and an exponent, 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 deformity of different Penileva equations, which was introduced by Sakai. So if I understand correctly, history, so in 90s, there are many different equations, different analogues of Penileva equations, integrable different analogues were introduced by many people, Jimba Sakai, Grammatikas, Vramani, I think 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 analogues of the original Penileva property, that we study different, Penileva property means that we study differential equation without movable branching points. And Sakai introduced some sort of different analogues, different analogues of this property. So to go to the answer, then to each Sakai sign difference equation to the rational surface, which obtained by blow up of nine points in CP2. Here cheating a little bit, so these nine points don't mean that all are line CP2. Maybe we take one blow up and then take another blow up 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 reducible components of the anti-canonical device. And this latest is called surface latest as, and as before in Kamot 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 on all of R, so all possible R's, so corresponding all all possible rational surfaces, all possible different different equations. So you see after Q deformation, so before we have say eight, before Q deformation 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, okay, for this D41 we can set both differential or difference equation. So really we have more equations for Q deformation. And maybe I should comment that this fill A part of this diagram correspond to Q difference equation and this part correspond to D difference equations. Since I want to study Q deformation, I will leave in the first row. Difference is that you have 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 meromorphic solution on C star or on C, so this is some sort of difference. Yes, yeah, exactly. For each D I can assign differential equation, I can set 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 example, two different equations can have the same limit. So theoretically speaking, there are infinitely many ways to invent difference equation, which has the same limit. But we are looking for good ones which should satisfy some sort of difference when they were proper, it should be. If I understand correctly, the limit of this equation is not Pen-Levers, it should be something elementary. So differential Pen-Levers equation lives here. So from A3 we can go to D4. This is called Q Pen-Levers 6 equation, which is invented by Jim Ben-Zakaya. Any more questions? Okay, there's another part of Sakaya story which I want to mention. To each surface, one can assign so-called some sense dual data, which I denote the R-Togonal to the Togonal complement to the lattice R, which I mentioned before. And here is a picture of the corresponding R's. I don't have time to explain details of such notation or such or even this notation, but it's just a paper, but I will sometimes use one of the table and sometimes another. The only thing which I can say is that we have two lattices, R and R-Togonal, and 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 lattice R-Togonal is useful not to describe the group of discrete dynamics of the corresponding plane-level 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 lattice R-Togonal is called symmetry lattice. So R is called surface lattice and R-Togonal is called symmetry lattice. Just, okay, and on the next slide, I put both lattices R and R-Togonal, both tables. You 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-8-1. So the corresponding difference equation should be connected by arrow with this equation. So, and I want to study Q-deformed in scientist. I want to Q-deformed conformal blocks. So my choice will be this, A-7-1 write for the box and draw the notation. So as before, this is some 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 forget all the reason about this very large table. But we have, so as I said, so difference equation is assigned to a rational surface. In this case, we have several, in any case, we have not one surface, but certain model space of such surfaces. And this, we can choose certain coordinates of this space, which I forgot, I omit the exact definition, but I don't know 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 symmetries of square. I also put here presentation of this group using generators and relations, as 0 and S1 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, this root system, is the two, which we can permute to root, to simple roots. And P1, here should be this one. P1 should, I think, permute them. Oh, no, no, no, no. P2 permutes to roots as permute to the corresponding reflection as 0 and S1. And P2 squared and P1 commutes with this affine. And the action of W is given by the following formulas. This, these formulas come from geometry of this rational surface, but I don't have time to explain what is the definition of such surface and how to use its geometry not to write these formulas, just to give an answer. And all this 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, it looks for me non-trivial, but it is okay. So this trial group acts both on coordinates on the modular space of surface and coordinates on the surface, so on the total space of the point of the bundle. Accelerational surface, it's a blow-up of CP2 in nine points. And, but such a 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, we have some, there is some combinatorial description of the sort of the generation which corresponds to this surface. So actually we have only four points on CP2 and so on, then three points in the premise of one of them and so on and so on. This lattice, so this coordinate prime regime Q, prime tries, this is 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 an infant order, translation W. And in order to, I use 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 sort of national function and using this formula, we have this, we can write one equation, function G. And I will say, next slide, but I can say it even now, that this equation is actually Q deformation of the original second order differential Pendevere equation. So in the limit Q goes to one, 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, so there's this sort of regime under which 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 conjection. 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 as 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 the definition of the tau function but following the Tudda 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 about this. There are several papers of Naomi and his co-authors. And I also want to mention paper by Rinke and Borodin who defined the tau function not for the Pen-Lever equation, but for linear difference equations. And differential Pen-Lever equations arises as another problem for linear differential equation. So there should exist relation between Q deform Pen-Lever equation and Q difference problem. So problems there is a relation between their definition and definition which we need, but this is not done. So this I just mentioned, actually we follow somehow Tudda 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. Is there 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. Those will 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 look very similar to formula which expressed differential function. Solution of the differential Pen-Lever 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 before. The second order differential equation on G. In terms of tau function, this equation reduces to system of two, Bellinia again, like Toda equations on tau functions, tau, use two tau function, tau one, tau three. And last comment that we have made several checks that on the, and that is followed. So on the level tau function, all what I can get action of W is a Laurent, even no series, Laurent polynomial as here. So for G, it's not true. Here we have 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 the limit Q goes to one and get differential in the equation. And for tau function, our goal will be the same as what was informed by gamma union of Lissa v to introduce only one function tau. Depend on two parameters, U and S. Here I use U, but U is, I can say that U is a Q in power of two sigma and sigma and S are direct analogs of sigma and S 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 Balini relation. This Balini relation is equivalent to Balini relation on the previous slide, it works slowly, maybe. So this is my problem. To find function tau, which will satisfy this Balini 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 this sense, I will solve difference in the equation. Here, it's a type, here should be the 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 questions? 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 QD form conformal blocks. So I wrote this tau function in terms of the QD form conformal blocks, I will write it. But before I need to recall what is QD form conformal blocks. First, just in words, this square of the vector vector of the VeroStora algebra. So it has the same representational definition as usual conformal block. But for actual calculation, we use another definition or statement. This function is a Nekrasov and Stanton partition function for pure SU2 gauge theory, five dimension. So therefore, this function has an explicit combinatorial form. Summation of a pair of partitions, and here is the Nekrasov type product. So the function defined as a power series in Z. Here I used two parameters, U1 and U2, but actually this function depends only 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 VeroStora case, it will correspond 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. In the representational definition, we do not use additional Heisenberg, we use just VeroStora. But if we take a limit, this is the same. For Heisenberg, we're taking the ones. Maybe this is not important. But, so this two is essential. So U1 is not, this is not fair. Okay, I would like to put as a lemma that this series is a series converge. As a series, as a series on Z. 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. Last I mentioned this function is a topological string partition function for local p1 times p1 geometry. So this function up to simple factor, infinite program symbol squared is a topological string partition function. So this function F, which has three interpretation if you want, Q VeroStora conformal blocks, necroson 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? F is the same as before. Q deformant conformal block, necroson partition function of pure SU2 theory. The main, the new ingredient here is this function C. Also with this factor, 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 Z in some power. In order to state main conjecture that this tau function is solution of the Balinia different equations. We need to impose 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 state that R is equal to one no 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 as possess symmetry so maybe convenient but this 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 suppose 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 in 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. So it's 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 on the previous slide satisfies this Boolean equation. As I said before this Boolean equation is actually equivalent to the corresponding Q difference in the equation. This main question. Q difference in the equation of the surface type A7 would 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 and this relation is conjecture. No, no, this conjecture is. What is proven that this relation is equivalent to this but it is not victimized. So if you introduce times it's basically a singular vector for Q-virus or it's essentially like Q-virus or constraints is it related to this state or isn't it related? So it has actually proven. No, no, I know the paper but I don't know if it's this relation. I don't know, I'm not sure this is related. My understanding, I don't know. I don't know if this is related. Okay, I just had two comments 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 say three or maybe five summits on the left side on the right side, but in conformal limit and limit where Q goes to one we have checked all terms. And everything's okay. Continuous limit is true. So I think this conjecture is true. Let me briefly mention two other parts of the of 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'll have also analog of this different of this background transformation. This analog permutes tau one and tau three and this claim which is the second which is not so clear it is a question on algebraic solutions. I recall that differential pin lever three equation has one two algebraic solution plus minus square root of Z. The same holds here and this is even simpler. You can start 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 pagandar product. In terms of conformal blocks this formula is equivalent to the relation that this double pagandar product is equal to the sum of Q deform conformal blocks. This is a certain Q rational coefficients and here I have power Z. So probably I forgot it in the first slide. Let me make one more comment and about this pagandar product and then I go to the conclusion. That I said that algebraic solution for pin lever d8 is a exponent. Tau for correspondant function is exponent and this should be compared of the Nekrasov partition function for pure U1 C which is also exponent. Here algebraic solution is a double pagandar product and this also should be compared with the Nekrasov could be compared with Nekrasov partition function for pure U1 theory five dimension which is also one infinite product. So the fact that exponent the Q deformation of the exponent is 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 phenomenon here. We have exponent of Z and here is exponent of square root of Z and in terms of CFT 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 conjecture. So if I consider myself as a mathematician so some of my questions to prove. If at least try. If I, okay, if I forgot about this question then the next question is to generalize. And now I recall this big table here called symmetry table of the symmetry table for symmetry lettuces and I recall that we live here now. Naive conjecture is that for this line, for this point of the equation are related to five dimensional and cross-partition function with n fundamental multiplies. So here I discussed case where n is equal to zero. And naive conjecture that for any n at least here is this holds. So n is not less or equal to seven. This, for example, it is known that if n is less or equal than four then we can take to goes to one limit of this equation and get that there should be relation between corresponding differential and the equations and corresponding four dimension across a partition function. And this relation is actually known due to Gamayoun-Yorkov-Nissavi for this case, we know that there exists a limit. Another important or even no important relation is that 20 years ago, Cybert argued 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 is 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 two function of the corresponding Q-Panelier equations can be given in terms of the cross-partition function of this gauge series. And of course, I don't know here we have actual group of symmetries is a affine real group. So where, how affine part comes. So probably this affine part comes due to the fact that we have not only one across the partition function, there's a whole series, but this is a question, maybe more interesting than proof, but this one. So it is. Okay, second part of the discussion, and that I, from the, how 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 background transformation of the 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 will look double, looks to be new phenomena, which arise after Q deformation. And it would be interesting to understand this in terms of our formula, of the tau function. So actually this symmetry is used in this, such simple case, to that transformation z to the inverse and Q to the Q inverse. And second transformation transparent, more or less, conformal blocks has such symmetry and also p-cum- functions has such symmetry and so on. But to z 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 clear into certain symmetry of conformal blocks. And, okay, two comments, but first about this symmetry. So I recall that this, this Q-deformed conformal blocks is more or less equivalent to strain partition function for local p-1 times p-1 geometry. And for such geometries, there exists fiber-based duality, which intersects to interchange two factors of p-1. And in terms of this function, this duality has the form. But this identity is a formal identity, the formal power series in variables u and u-z. And if absolute value of Q is not equal to one, the 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 formal power series, they are equal, but this not, but do not 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 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 Giulio Alessandro and Dalba, who constructed Q differ, also who also constructed Q deformation of the formula for the level three power function using this paper mentioned in the previous talk and many others. 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 more in the next talk.