 Okay, so let me start the last lecture. So first of all, I wanted to remind you some key things, key features of a model space we are talking about. It's a model space PGS. And so it corresponds to the creative surfaces which you see here with red mark points, pinning given by the thread arrows and punctures here. And so I wanted to stress the following data. But first of all, we have discrete symmetries. And so this means that there is a group, discrete group, gamma GS, which is given by the automorphism group of G cross maping class group of S, semi-direct product with the product of the veil groups and product of the braid groups, braid groups over components and veil groups over the punctures. And it takes by cluster post-on transformations on the space. And the second feature of this space is that it has a center. And so the center is given via projection of the space to a torus. So the name for this torus we called AGS, but it is defined as a product over punctures of Z-cartan group and product over boundary components, again Z-cartan groups. But now cartan groups co-invariance by the action of W0, kind of reduced version of the cartan group. And so center, as I said, is given via this projection. And the last piece of data, and the one I'm going to talk right now, is that for each special point we have potentials. So who are those potentials? So special point S, then we have these potentials. So this is the functions WSI, which is just regular functions on the space, A1. And I belongs to the set of positive simple roots, meaning the set of vertices with the dingin diagram. And also there is a map, ROSA-BASS, this is the data which you see at the special point, potentials and projections to cartan. So as you see, there are many cartan's projections to cartan's floating around here. So basically every puncher gives projection to cartan, every special point gives projection to cartan, and using the special points we can combine those projections and get projections to cartan related to the boundary component. Okay, now the main question now, who are those potentials? So the potential functions, it's the following construction. So we use it with Shane in a paper about 2013 where these potentials played a role in statements like mirror symmetry, but now they play an entirely different role and will live on slightly different spaces. So it's very mysterious to me why these two subjects go together. But okay, let me give a definition. So you have three flags. You have B1 and B2, the usual flags, and A is a decorated flag. And so now we consider the model space of triples of these flags, denoted this way. So this is just A cross B cross B divided by the action of the group G. And so it's a subset because it consists of triples of this data, A, B1, and B2, such that the pairs A, B1, and A, B2 are generic. And so this is just space of triples. And now we want to define a function, omega i, which is going from this space to A1. So how do we define this function? So we notice that there exists a unique element of the maximal unipotent group, which comes into the definition of A, this decorated flag. I remind you that this A is defined as a pair, a unipotent group and additive character, which is not degenerate. So the model space of such data, maximal unipotent group and additive character to A1 is the principle of fine space for the adjoint group. Now I claim that there exists element of this particular unipotent group, which comes to the definition of this A, such that the first, so you have here two pairs, A, B1, and A, B2. And so this U sub A just moves them. This means that if you take A, B2, and this is U sub A multiplied by A, B1. This just means pairs, not modelization of the group. This parenthesis means to just consider pair. A, B2, it gives you some G to answer because they're in generic position. And therefore, there is this unit U sub A which moves them because A stays the same here. But now we got this U sub A, but also this little U sub A. Don't confuse with this data. But now the flag also gives you the character. And so you can just define this WI on this triple A, B1, and B2 to be the value of this character on this element we just defined. So you get a function. Okay, so that's kind of building block of the construction but not exactly what we are doing. Can you explain that in the context of SNR? Yeah, I mean I can explain this for PGL2. It's not SL, I emphasize, it's not SL, it's PG. And so it makes story different. What's the I? Simple root. I is a simple root. And how does the I enter the right hand side? It doesn't enter yet. So, sorry, actually you completely write that I should, wait a second. So, so far so far no, so sorry, what I said is not exactly right. So, sorry for a little confusion. So this psi, the definition didn't end yet. This psi is canonically written as a sum of psi alpha i's. Because for example, if you have apotrangular matrix which has values like A1, A2 and something else, it's a unimportant matrix for example. Then there are two characters taking this element and this element. And taking non-degenerate character means that we take linear combination of them. And so when you set character here, I have to specify. And so thank you, Joel, for the question. So I have to specify which of the components of this decomposition you take. So here I belong to the set I, okay? So what I said before is also good, but this is not what we need right now. There's also a good thing, some of them. But that's not what we need in this definition. Okay, before I go and ask the first question, so let me actually give you the general definition that shows how it works. And then we will proceed with the example. So now suppose that you have some boundary component. On the boundary component, we have one special point and maybe some other special points, maybe many of them. And so among those, there are the most left and most right to the one as we picked. But remember that when we pick a special point, we have two pinnings, that we have two decorated flags sitting here. So this flag can be called A minus, and this can be called A plus. And so they relate to this point as, because when you define the pinning, so you're supposed to put two flags here and give the data of the pinning. But we are not going to use any other decorated flags. We are going to use the flex B and B prime which you see to the first to the left and first to the right flex. Now we get the data because now we have the left track here, A. And then we have B and we have, my prime disappeared, and then we have B prime. And so now I can apply to this data our construction and get the function. So by definition, this Ws relate to the special point s and positive root alpha i is the value of this partial potential Wi, which we just defined on the flex A minus relate to s and then B and B prime. This is the definition. Now to François's question, so what is the example? So, for example, if you take PGL2, I stress not SL2, PGL2. Then we want to define a potential function on the triples. The first one is a representative for a decorated flag, which is a pair given by a vector and area form. And the second two are the flags, which in this case just lines. And so we need to define the number which corresponds to this data. Who can tell me what happens with the razor? It's fell, oh, that's not bad, okay, thank you. So, and so I remind you that here you have the action of GM by T and then here it X by T minus square. And so we have to take invariance with respect to this action. And so the cosets call this way. Now, what is the number? The number is omega of L1, L2 divided by this omega of VL1 multiplied by omega of VL2. So you notice that it does not depend on the choices of vectors here, not 0 vectors L1 and L2 because they enter numerator and denominator ones. But it does depend on the choice of omega and V. But it depends in such a way that on this orbits of the group GM is still constant. So it's a well-defined function. Okay, so we got potentials. Now, what do we do with potentials? So, some question? So we go to the next thing we do. We go to the topic which is called quantum potential functions and quantum groups. So I wanted to emphasize that at each puncture, and so you see this puncture on this picture, you have two kind of functions. You have the potentials WSI, but you also have projection to carton. This is data, the bottom data on the right. And so therefore, what you can do, so this is a map from PGS to A1. But we also have a map ROS of S from PGS to the carton group. And so we can form the function ROS up a star of alpha i. The alpha is a character representing the positive root alpha i. So we have two kind of functions. So we have R functions which are potentials and another R functions which are actually characters of the carton. And this is the data which we assign to a special point. And so now comes the theorem that given a puncture, sorry, given a special point S, we have the following. So that first of all, the functions WSI and ROS star of alpha i have condonical lifts to the quantum algebra. So it's very important these lifts are canonical. Cononical lifts to elements which we are going to denote WSI and also by the same ROS alpha i. And so they live in this algebra, q-deformed algebra of functions, Oq PGS and then WN invariance taken at all punctures. Now, this kind of space of functions is taking W invariance. Also, we agreed to call it Oq of log GS, just a definition. Okay, so first of all, we have these elements. Secondly, let me go here. These elements, what? N is the number of the punctures. Good question. So these elements, which I denote by star, give rise to injective map, let's call kappa sub S as a special point. From the quantum universal enveloping algebra of the Borel algebra, two Oq of this log GS. So let me stop here for a second. So you see you have, at every special point, we have now R plus R functions. And so it's a very natural question to ask what these functions generate if you take the Poisson brackets or even better can we leave them to quantum algebra functions and if we do what kind of relations they specify. So the part eight one tells you that you can leave them. This is the most important thing. And secondly then, after you prove that you can leave them, it's not that hard to prove that you have this injective embedding. They generate your Q of B. Now, how this embedding works? There are some generators of the quantum group, which are obtained by rescaling of the standard ones of the infotain jimba. And I will define them in a second rescaling. And so this EI go to WSI. And Ki go to rho S star of alpha I. And so again, it's quite natural from the picture because this Ki's, they give you the carton. And we do have projection to carton assigned to this special point. This is our data that we're talking about here. And so this carton proceeds to the quantum group. And the potentials give you these EIs. And now the third part of the theorem is that for any boundary component of pi with two special points, exactly two, you can still ask the question, what does this UB of Q generate? And the answer is that the corresponding map gives rise to a map of the UQ of G, again, injective to OQ of log Gs. And one little technical condition that, not that technical, you have to consider some kind of subspace of the space where this auto monodromy, we were talking about the previous lecture, is actually identity. So this auto monodromy, remember that we have for each boundary component, we have projection to this HP. And so I called the auto monodromy. And so we just equated to one. OK? All right. Now, who are those generators? So here, this EI is just Q minus half Q minus Q inverse EI. And this FI is Q to 1 half to Q inverse minus Q, the usual FI. And KI is the usual KI. And so these are the usual Greenfield-Jumbo generators. OK, so this is the connection of the potential to the quantum group. So why we force to such a rescaling? This implies that there exists a unique anti-involution star from UQ of G to itself for which all these new generators are stable. If you don't do rescaling, that, of course, will be violated. OK, so now what is the key example of this station? The most interesting example, the most interesting special case where all these features are already present. The most interesting means the simplest one. The simplest one is when you have disk with two boundary points. It's a key example. So let's consider disk. And so it has two special points and a puncture. So this is the two special points. And so this is the puncture, P. And in this case, we still have the additional data which now you draw as follows. So there are two decorated flags on the left and on the right here and here. And so we call them, if this is point S1, we call them A1 minus and A1 plus. And this is A2 minus and A2 plus. And they are related by, so there is a pinning. This means that the h distance between them is 1. I'm sorry, I'm behind. So this v star of the i is equal to v i. The star is the identity on all the generators. On the generators, yes. Yes, yes. So, I just put plus the star of q. Also, oh, star of q is q inverse. Sorry, I didn't hear the letter, probably. So you're asking about k. So this is the standard involution, which offers a type which acts on quantum and quantum Poisson variety. And so in order to do something, you want to have self-adjoint operators. And just remind you that we have the generators plus quantum serrations and plus kind of carton relation, which I'm not going to remind you. And I'm not going to use them. So in this case, this data we were talking about before, a special artist follows. So we have this outer monodrami, which is a map mu from this model space Pg related. So I will use this kind of sign for this picture. Two carton group. And this mu is given by the product of the projection to carton at one point. And then you apply the standard involution on the carton to projection to the other point. That's what it is. And now we wanted to cut out a little bit this model spaces. Because as we said, as you see from the left-hand side, part two, that the center of this cluster Poisson variety, in this case, will be carton at the puncture and carton from outer monodrami. And you will see why, but we don't want to have any kind of contribution from the outer monodrami, so we put it equal to 1. That's very important. And so we reserve a name for the model space we get. So it says that R reduced model space is just given by the conditions that outer monodrami is 1. And then just to complete notations, we have this L model space, which is the one obtained from RG by forgetting the flag at the puncture. So this may be a little complicated, but what I'm saying is that we consider this basic model space, which is nothing else, but I'm not talking about this. But what it means is that you consider in drum version model space of connections on CP1, which has one regular point and one irregular is this kind of irregularity, which corresponds to this picture, this kind of generic type of irregularity. And then the model space of stock's data is given by this model space. So we consider this model space. And so it actually was this key idea which we had with Volodypok. And long ago before, very long ago, that the key idea was that if you have correct model space, this one, then this is quantum group. But to implement this, you need to have really correct definitions of this model space, for example, to make sure that you know what corresponds to the boundary, because otherwise you get a wrong space. Now, how this idea implemented in the theorem? First of all, the classical level. So again, whatever I'm talking about today is joint work with Linkvishan, which is a piton archive somewhere in April. And so the claim is that this model space, lg0 dot, is actually equals coincides with g star, which is a Dreamfield dual force only group. And so I'm going to give a proof of the theorem, which is actually construction of all the data, because that's where you see how it will work in the quantum station. So I'm sorry, I didn't understand the definition of lg0. OK, so once again, I'll say. So you have this model space, model space of stock's data. The first thing you do, you notice, as I said, that it has too much of the center, and you kill the outer center. Then its dimension, model space, will be exactly the dimension of the group g. It still has some data at the puncture, which was this flag at the puncture. And so you forget this data. So now you're talking about plain model spaces of g local systems with spinning on the boundary sides. And the spinning satisfies the conditions that the auto monodrama is one. And so I claim that this is the guy to consider. And first of all, this guy is a group. So this is a group. And this is just a model space. And so the first thing you need to do, you need to define the product map. The product map on this model space. And so you do this by using gluing construction map, which we talked about in the first lecture, plus this kind of encircling. I'll tell you this in a second. So what do we do? So we draw a picture that we have these two disks with a puncture. And then we have the pinnings. So first of all, if you just consider the product of these guys, then by doing the gluing map, you end it up on the disk with two punctures and two pinnings remaining. Because you just glue. So the gluing map allows you to glue the pinnings. These two pinnings, you glue them, they disappear. And you glue uniquely. Now, what is the point here? The point is that this map, of course, exists if you don't put condition out to monodromes 1. But if you don't put this condition, then this map is going to be vibration. If you do put this condition, then this map is essentially isomorphism. At least it's finitely many different from being finite cover, but on real positive points will be just isomorphism. And after that, you just encircle these two points. And so you go to this picture where you have still two pinnings. And you have now one puncture. And so this is the product map. OK, that's number one. So and this map is a Poisson map. So it's a product map, which is Poisson map for the Poisson structure we consider. You can still note that if you consider this slightly bigger space, it's maybe a little bit too many notations, but just let me spell this correctly. So if you don't impose condition out to monodromes 1, what you get is precisely b minus cross b plus. If you impose condition out to monodromes 1, then you cut it down as you shoot to get the dual Poisson group. Now the next question is the unit. So what is the unit? And so this is, by definition, the trivial G local system plus standard pinning at both sides. Standard means one of the pinnings, but the same in both sides of the picture. So you draw it like that. And so you have two pinnings here, but they're the same because the local system is trivial. So it makes sense to say that it's the same because you're kind of moving this pinning through its trivial local system so you can do this. Otherwise the puncture does not allow to do this. Then the inverse. So the inverse is rotation by 180 degrees. And very easy to see that this is indeed the inverse. And then the next one, what is the geometric R matrix? Again, I'm not talking about R matrices at all, but the geometric R matrix is the element of cluster model group, which does the following thing. That you started, you now have to go on the product of these two spaces, which according to the bottom line is just the same as you have these two punctures here. And then what you can do, you can just interchange these two punctures. If it's 1 and 2, you get 2 and 1 just by this move this way and this move this way. So this is this geometric R matrix map. And so notice that I didn't. So that's it about the model space of this L, G, or dot. So you clearly see it's a group. And it says the morphic is a Poisson group to G star. And so the conclusion for this discussion is that by geometry plus cluster Poisson geometry, cluster Poisson properties of everything we're talking about, cluster Poisson nature of our model spaces, PGS. So we get to the conclusion that if you take just this OQ of this LGS that gives coordinate free kind of new way to think about a new geometric definition of quantum group. So once again, so when I say this, I don't talk about EI, Fi, Ki. This is kind of relating to the previous language for the quantum group. You can just say that you consider this model space. You don't really bother by all this EI, Fi, the relations, whatever. You just take this algebra, and then this algebra immediately has all the properties. Because by the product map, you immediately see that this OQ of L, G, sorry, is a Hopf algebra. This is immediate because the product map is obtained is a cluster Poisson map, basically. So the gluing map, the most interesting one, is a cluster Poisson. It preserves the cluster Poisson structure. And then there is a forgetful map. You just forget some of the data. And so it's the forms of G star. And so that's what it should be. And now if you happen to want it to relate to the classical language, then you just notice that this Greenfield Jimbo generators, they're given basically by potentials. Because I emphasize again that on this model space, you clearly see that there are some functions. You clearly see that there are these potentials, and the Carton group there. And so they are naturally quantized. And so you get the generators. But you may not want even to consider them if you want it. So then you get just some algebra which has all the properties of the quantum group. Do you remember that the JLM, this theorem was also put by Boltz? By who? Boltz, Philip Boltz. No, no, no. Wait a second. So I'll. Not one. Well, I would first of all say that this model space, I believe, wasn't considered. He did not consider it, but he considered a different form. OK, OK, OK. But I'll tell you, let me finish with a kind of punch line we'll get. So this L space, is it? How close is it to the space of two pinnings, modular action of the group G? I mean, you have a local system. So it gives you the local system as the main player. So there's also the local space? The extra pinnings give you just two cartons. It's a little bit. And the main body of this space, it comes from the fact that it's a G local system. Yes, with this kind of additional two red points, you have flexes in those points. It's a loss of data. So the local system itself is, again, a Carton. Carton model W. So this is the two flexes, which you see. They add you lots of dimensions here. And the final two cartons you see from the pinnings. So the main point in all this discussion is the following. So the main point is that this part one, which tells you that these elements have canonical lifts. This is a very non-travel statement. So now let me comment on that. So what part one means is that, first of all, for each river, C, which defines the cluster Poisson data, there exist some elements WSI and this rho S up a star of alpha i, which depend very much on this choice of this square C. So they define specifically for the square C. And in this square C, they are Laurent polynomials. They belong to OQ of this torus TC. Secondly, most importantly, that for any cluster. Where is the choice of the question? OK. So this both down to the question. So the good question. So this theorem, part one of the theorem, in the cryptic form, contains a lot of very non-travel information. And I'm trying to explain what this information is. So it says that there exist functions which belong to this OQ log Gs. What does it mean? What is OQ of log Gs? By definition, it's defined as follows. For every quiver, you have to consider, so what does it mean to define a function in quantum function log Gs? This means that for every quiver, you have to define Laurent polynomial with coefficients as the QQ inverse. And then this part one. And then part B is that for any cluster Poisson transformation, which takes your quiver C1 to C2, you can define, you can see the following. That first of all, in this situation, you have W C1, just emphasizes dependence. And you have W C2, which depend on this S and I. And so they must go to each other under this isomorphism. So you have the isomorphism between the fraction fields of OQ of T C1 and fraction fields of OQ of T C2. And so this isomorphism, so you have some element W C1 sitting here, W C2 sitting here. And they're transformed by this isomorphism to each other. So you cannot just say that you have a Laurent polynomial one coordinate system. Then it's good for nothing. Then also you do the same for this cartons. So they corresponds to each other. And so only proving that you have such a package would mean that you constructed a lift. And as I said, it's a non-trivial, quite non-trivial theorem that the lift exists. And actually deforms the original functions. What do we have to do with the way S is cut? No, S, from the very beginning, you see theorem given a special point S. So S is given to you. No, the surface. Oh, the surface S. So in this result, it has nothing to do with S. So S, the surface S in this setup could be anything, any surface. You just have any kind of surface here and whatever. And just happen to have one special point S. It doesn't matter what happens outside. That's also a point. Quiver. So I didn't talk about these lectures how you define the cluster process on structure. And so in cluster process on structure, you define actually some infinite collection of quivers to start with, which are related by cluster transformations. And then they generate even bigger collection of quivers, infinite collection of quivers. And so you have to start with the quiver which corresponds to the cluster process on algebra, cluster process on structure, on this model space, and all its descendants. And you're saying I shouldn't think of this quiver as cutting surface into S. I would say that, yes, this quiver can be obtained by a certain construction, which it takes about half an hour at least to explain. So I'm not doing this right now, huh? I hope you explained it before that. I did not explain this before, yes. I explained this for Ersel, too, but not in any other setup. Yes. You have a construction of potential function for all quivers? Yes. So we have a constant. For every quiver, so we take one quiver and then we lift the potential function in a certain way. And then we prove that this lift does not depend on any choices. Because when you try the whole problem of defining a lift, it's problematic because you can consider like q minus q inverse to some element x. So when q goes to 1, it dies. And so you can always add something like that. And so when you say that you found a lift, it's not too much. So you have to find a lift, which is the lift, which means that if you transform this lift by cluster transformations to all other coordinate system, you'll find the lift which you're supposed to assign to them in that coordinate system. So you find a compatible system of these lifts. And so the construction, of course, initial construction depends on a choice of many things like triangulation of the surface, some data which I'll talk about a little later on, and so on so far. And so you have to ensure that for some reason, it extends. And particularly, you have to ensure that if you take your lift, then it will remain Laurent polynomial after any cluster transformation. This is very non-trivial condition because cluster transformations have denominators like 1 divided by x plus y, something like that. And these denominators seem to destroy the Laurent polynomial in nature of these elements. The statement is that yes, you can lift in such a way that it's not get destroyed. So my question is, you take a group G on the surface as a triangulation. No, no, no, there's no triangulation. So originally, there's a group G on the surface S. Right. OK, so then you get these quivers. Then somehow I get quivers. I didn't discuss yet how. But you don't get all quivers this way. You don't get all quivers this way. You get a tiny part. All quivers. It's OK. So it is so canonical that I mean it is actually so the main part of the proof is to show that this lift is unique in a sense that there exists always some system of coordinates where the sleeve is given by monomial. And therefore, it's unique because the monomial lifts unique way because supposed to be starting variant. So this function of w, of course, they lift elements which are starting variant. And so the key point is that in some situations it's going to be monomial, so you don't have a choice. But then you have to prove that in all other situations it's kind of compatible because you can have two different situations when you have no choice. And so why is it compatible by class transformation? That's a question. The answer is yes. Yes, yes, yes. I mean if you don't have this answer, then this would be completely meaningless because all the discrete groups I'm talking about, they're all acting by class transformations. And if you don't know that you can do class transformation preserve your things, then it's completely useless. So you cannot do a thing with this. OK? So once again, the key point that you have a huge discrete group of symmetries. And you must preserve this discrete group of symmetries because otherwise it's not a reasonable object. All right? So now as I said, so this was our kind of suggestion with Volodya. But it was waiting till, first of all, you need to construct a cluster for some structure. That's what we did now. But for PGLM, we did this before. And so Alexander Shapiro, a few years ago, and Bruce Schroeder, they actually find a lift. So they defined Kappa for PGLM using exactly these ideas. But there was no spins and so on. So they just used one cluster coordinate system using so-called special cluster coordinate system, which we defined by Volodya Fouk. And when we defined the cluster action. So the difference between GLM and other groups is that for PGLM, if you choose a triangulation, if you choose a triangle, there is a preferred cluster coordinate system, which has lots of symmetries, which are absent in all other cases. OK, so now as an application, so the main application of this is that you can quantize the whole story. And so let's erase this. So why you can quantize? Because there is this machine of cluster quantization, which the first part of previous lecture was devoted. And you can just apply this machine in this particular case. So as an application, we can define the principle series star representation of the quantum model double, which is I remind you. This guy is uq of g, uq check of g check. Now, why this works, of course, because actually, we're talking about here. So what you really do, you construct, you do this for h for Pg or dot, which we talked about before. And so here we know what to do, because it has cluster portion variety structure with all the symmetries. And so we already have it. Then if you map one to the other, you get this principle series. But my point here is that it's actually an interesting question. So what do we mean by star representation of a quantum group, which is infinite dimensional? Because if you look at the classical representation theory, when we talk about finite dimensional representation, there are no questions, neither for usual group nor for quantum groups. But if you talk about the dimensional representation of the classical groups, like SL2R and so on, so you really wanted to consider group representations, but not the algebra representations. Because if you consider the algebra representations, there are way too many of them. And there are much, much, much more of them than the group representations. And you won't get only those algebra representations which corresponds to group. And then there is a very non-trivial theorem, which was developed over many years, which says that the adequate notion is the notion of Herschandra models. So you indeed consider representations of the Lie algebra, but you insist that when it's restricted to maximal compact subgroup, it actually comes from a presentation of a group. And the spectra is finite dimensional force, the representation of this maximal compact subgroup. So the theorem is that if you start this representation of a group, then you get Herschandra models. And actually, this is basically the same thing, which is it's a very non-trivial theorem due to many people. So the last proof is due to Bernstein and Kroth's like 204, I forgot. But it's a quite non-trivial statement. It's very unclear that you can reduce analysis to algebra. I mean, group representations to Lie algebra representations. Now you turn to the quantum groups. And that situation is much worse, because quantum maximal compact group cannot be quantized. So there is no Herschandra models except for SL2. So this approach cannot work. Then you ask, OK, so if you just consider a representation of a quantum group as a map from this algebra to some space of functions, like our S space, OK, you've got something. But that's not the thing yet, because it's not the thing even for Lie algebra representations. And so the suggestion which we have here is that actually when you talk about this model of spaces, it's not just a group, but you have many more structure even basically by this huge discrete group of isomorphisms, of discrete aftomorphisms. And why you need it? For example, because if you have some quantum group representation, and let's suppose you change it in a certain reasonable way. For example, there is the section of the braid group which was defined by Jan and Lustig in around 1990s, which says that instead of talking about these EIFI generators, you can rotate it. So Jan was talking about this quantum wave group action, which kind of rotates them. And Lustig was talking just about braid group action acting by aftomorphisms of the quantum algebra. And so if you construct any representation of this guy, you're supposed to manage to show that if you add the Lustig, let's say, braid group action, you get the same mean equivalent representation. If you don't get it, then this is not the good object to consider. And so I'm going to show that this kind of approach, it gives you all these properties. So this principle series of presentations comes with additional rigidity, which in particular proves the statement which I just mentioned. But let me recall the classical setup that there is this Gelfand Neymar principle series. And again, it's not just a bunch of representations. So this story works for K, which is R or C or QP. It doesn't matter. And so the point is that if you take group G cross H, it acts on the principal affine space. And therefore, you can take the L2 of the principal affine space in your favorite field. And it's decomposed into integral of some representations. Let's call them R, which are parametrized by unitary characters of H of your K. And so there are some decompositions. So this alpha is a unitary character. But it's more than that. So the number one statement that you have is construction. Then the second statement is the construction of Gelfand, the Yangi Gelfand, Sergei Gelfand. And Grive, it's 73, which tells you that there exists unitary equivalences, I sub w, which takes this principal series representation at alpha to principal series representation of w alpha. For any element of the well group. OK. And so for example, if you take G to be SL2, then A is just two dimensional space minus the origin. And then this intertwiner is basically Fourier transform. So it takes function of f of x1, x2 to integral of f of x1, x2 is a character like exponent of 2 pi i in the classical NPR situation. And x1, y2 minus x2, y1, dx1, dx2. So it's a Fourier transform, but shifted by using the symplectic structure. Somehow twisted a little bit. So then the square is 1. And there is more. So you can consider like principal affine space, takes all differential operators here, and then consider stabilizer by carton. And this is basically universal algebra of G and so on so far. So relation with the differential operator, some principal affine space, which is the backbone later on localization and so on so far. So I claim that this picture gives you all this kind of package immediately and without basically any extra work. So how this construction goes. So I remind you that what I said that you have this A H of G and it embeds to A H of this modular space. And then there is this triple of spaces, the Schwarz space, Hilbert space, and the space of distributions. And all action is here. So this algebra, this algebra is there for that one, it takes on that space to get a star representation, as we discussed in the beginning. So you have this plus Regi define properties. So what are the properties? Let's first of all, just as before that, first of all, if you consider, let me talk about Hilbert space. The Hilbert space related to this modular space for the disk with puncher and two special points. So it is decomposed as integral of some representations, H lambda d lambda, where lambda belongs now to real positive points of the carton group. So please keep this in your mind. Not all real points, but only real positive points. Secondly, the real group W acts by cluster Poisson transformations and commute with auto monodromy. It's a little technical, but I add this. Therefore, it acts on this restricted model space, RGO dot. And therefore, the main package plus this properties immediately implies that we have this unitary intertwiners. So what they do, so this is the analog of Gelfand and Greif intertwiners. And so what they do, they provide maps, IW from this H lambda to H of W of lambda for every W in the real group. And so this shows that although we get a sequence of familiar representation parameterized by the points of the positive points of the carton group, those which lie on W orbits are equivalent. Not just they are equivalent, they are explicitly equivalent. The next thing is that you have this braid group story. And I again stress here that the very fact that you have such intertwiners, it's a quite non-trivial statement even to understand. So it's an intertwiner which acts on Hilbert space, but it commutes with everything. It commutes. It takes the Schwartz space to the Schwartz space. It takes the action of the algebra on the Schwartz space to the action of the algebra on the Schwartz space. I mean, it's identical isomorphism. And they all commute. OK, now the third story is this braid group action. So it's a theorem that Z-map we're talking about from UQ of G to OQ of L, G, or dot, which is conjectural isomorphism. But this is not known. And actually we don't really care about it at the moment because the story always lives here. So this map, you see that here you have this braid group action. And here referred to Leustich's definition. And so if you take this braid group action here, so on the other hand, here you have the action of the braid group BG. Why? Because whenever, remind you that whenever you have a boundary component, it's not on this list, but I just said this last lecture, whenever you have a boundary component, you always have a braid group action on this boundary component if you have even number of special points. You have two, so you have the braid group action. And so the statement is that this braid group action maps precisely to our braid group action. But now our braid group action lifts to unitary intertwiners because in our situation, you have this group BG acting by unit transformations on each of the spaces H lambda. And this action, so let's call this section isobeta for any beta for the braid group. So the section, when you apply by element of your algebra to some vector, you get a rotated element by the braid group action multiplied by i beta of s. So this intertwines the braid group action. The very fact that it lifts to unitary intertwiners, again it's a corollary of the fact that this BG action is claustrophoson, which is quite a non-trivial statement on its own, especially when you're talking about the boundary. But as soon as you know it, so it's automatically quantized and automatically gives you the section. And the last thing, and then I will make a break. So the last thing is that you have the center. So you have the center of the universal envelope in algebra, which embeds to the center of this OQ of p, g, or dot. And so here we have this row, the preimage from the puncture called mu. And so they basically match. So here you have to take double invariance of this action. So this basically corresponds to that. So that's it for the first part of this lecture. So we'll continue after break, maybe start like in 7 minutes at 3.40. And so what I'm going to do now, so now I introduced this representation of the quantum group. They, by the geometry, they form something like a braided monoidal category just because of the geometry involved. And I wanted to give two different realizations of this tensor category, kind of conjectural, but related to different worlds. And so that will be the subject of the second part. So the next topic is quantization of modally space local systems. Gs, now notice that my surface is now this way. But so let's move it for now the most general. But S, TQFT. So remind you that we have this modally space local systems where there's no conditions, punctures, no action of W. The Braille groups and punctures were forgotten. And so what we had that this log Gs modally space, it produces a quantized algebra, OQ of log Gs, which is defined as OQ of PGS. That's where this modally space PGS is really needed. And then taking W and invariance. So this is the definition because there is no other way how to define the quantization of local space of modals, local systems, which has all the properties. You have to go up. You have to go to this big modally space. Then we do the steps. So we consider the modular double, language model double as usual. So we take OQ, which we just defined on the left, and tensor this with OQ check of log G check S. So this is the algebra. Star algebra we're talking about. And we have this gamma Gs equivalent quantization, this gamma Gs equivalent cluster Poisson structure. And therefore, on PGS, again, I stress here that all the work is done on this modally space PGS. But in the end of the day, we are interested in the smaller modally space. And we also have a quantization of this. And so all this put in together implies the main statement, which was the goal from the first lecture, the theorem, that if the plan constant has absolute value 1 or is positive, then this language model double of the modally space log Gs has a gamma Gs equivalent, equivalent the following. Equivariant star algebra structure. Secondly, it has a series of star representations. In the triple. So this triple is the spaces, the Schwarz space, Hilbert space, and the dual to Schwarz space. So we have representations. And as you remember from the first lecture, representation means that this algebra x here, and there forms a dual space. But it does not act here. And so the mapping of this gamma Gs, it acts everywhere. And this construction is equivariant. So these two actions respect each other. And there is a spectral decomposition. So the spectral decomposition in general, you decompose this Hilbert space, AGS, as integral of some Hilbert space, which depend on some lambda. And this lambda is element, you take this torus, which describe you as the center of the Schwarz space, and take its positive points, d lambda. And this lambda actually belongs to this guy divided by w to n. Because different lambda, which differ by the action of w, they give you equivalent representations. And so you need one to write down spectral decomposition. OK, so this is somehow the main statement. But then there are some related questions. So the question is, what are you going to do if you cut your surface in pieces? So the question is how to relate these Hs for different S. And here I mean the following. So you can take some space, some surface, by the way. And you can cut it by a loop, gamma. And so you get one or two surfaces, depending what was the original surface. So you get a new surface. And so if this one was S, this one will be called S prime. And so if your quantization data here was parameterized by this lambda, the one I was talking about here. So here you have extra parameter, let's call chi minus and chi plus. And actually, if you cut it, you'll have condition there related. So you basically have one parameter. But this is the extra parameter of the center. So you have to understand that you have here lambda, kappa inverse, and kappa plus. And this belongs to this original HGS real points multiplied by the product of two cartons at real points. And not just two cartons, but actually this is cartons divided by w here as well. Because that's how this picture works. And so you need to relate this bigger Hilbert space to this collection of smaller vector spaces. And that's what we had a conjecture a long time ago, how they are related, should be related. So you call this model of final conjecture. But in order to state this conjecture, you have to notice that we have this map. You have a restriction map. You can restrict gamma as a loop. You can restrict your G local system to the scattered surface. And so this way you get a map from PG S to PG S prime. And therefore, you have the dual map on the level of algebras, which is a map from this H algebra related to the pair G and S to the algebra related to the pair G S prime and here G and S. So you have such a map of algebras. And so the conjecture, so this is an old conjecture. It's Volodyph Fock and myself. And this is 2007. And so it's kind of one of the key things. You need to know it's modular final conjecture. And says that the relation should be very natural. This H, G, S, and lambda data is the original data of your irreducible representation, is given by integral of the data related to this G S prime, same lambda, but also this chi and chi inverse, d chi. And so this chi belongs to the positive points of carton group divided by w. And this is not just, so as stated, this is a statement about Hilbert spaces. But it also has variant. You can look at the S spaces as well. And then actually, that's where you actually formulate the statement, because this should be an isomorphism over A H G S prime models. I remind you that this algebra does not act here. So you have to have this decomposition. But then on the corresponding Schwarz spaces, this algebra acts here naturally and here through the restriction, dole to the restriction map. And so it has to be compatible, of course. OK, so that was the conjecture. So it is known in some cases. So why do you put chi inverse in the picture? It depends. So it depends on the question, what do you mean by monodrame? Monodrame means element of the carton group. Carton group means the semi-simple part of the monodrame, the actual monodrame. But monodrame is taken with respect to some element. And this element depends on the orientation of the surface. And this orientation is inverse. This loop has one orientation on the left and one on the right. OK, so the good news that this is known. So it's known for PGL to actually before we were doing this. So it's by York Teshner. And it's also recently approved for PGLM like last year by the same Alexander Shapiro and Gus Schöder. But for other groups, it was hard to state this. Another thing which I forgot to say, and I beg your pardon. So when I was talking about the main theorem about cluster portion structure, there was a works of In Lea, a series of works, who established a cluster for some structure on somewhat different model spaces. This was the previous one we considered, XGS for classical G and also for G2. So it's lots of interesting construction. But his constructions are case by case. And actually, you need to do construction for simple lace group, but because then you can do folding. So it's the main thing is a group D, D series. Then you can get G2 and B groups on A we already had. But still, it's case by case. But he found some interesting particular cluster corded systems and then proved what needs to be proved. OK, so we have lots of things here. But now let's go to the main points. So the point is that we now relate this to this Louisville Todes theory. So when you say Todes refers to works by Fagan and Edward Franco in the 90s, like early 90s. And so this works as follows. So we have two kind of pictures of the story. So we have the run picture and beta picture. And so they're the same. So the run picture. This is a story related to Louisville Todes. But it starts as follows. So we have sigma, which is now genus G Riemann surface with n-punchers. And this is the first time in the lectures when we have surface with a complex structure. So before, it was completely topological. Now over this, this sigma, they form the model space mgn. And we need some bundle over this model space. Let's call it lgn. And so this bundle, we can do it like C start to n plus 1 bundle. And so this bundle is, first of all, a determinant bundle on the curve sigma given by determinant of pioschomology. So then it's a product of a bundle which you get from the punctures. So you have puncture P i. And you can take the contention bundle to sigma at this point. And when sigma varies, you get another line bundle. And each of them are punctured. So here it's minus 0. So this is C star bundles, all of them. Now, when you look at this C star n plus 1 bundle, you say that you can take the fundamental group of this. And so this is a fundamental group of this bundle. And it, of course, fits to exact sequence gamma s, gamma s hat and gamma s. And so this is the variant of the making class group which you want to consider. You can also say in an equivalent way that this guy called lgn is a quotient of some kind of bigger variant of the tecumular space by the bigger variant of the making class group. So basically take universal cover of this and then there's group acting there. OK, so these are the notations. The first round of notations. Now, the second round is that I have to rely, yes. Oh, yes, of course. Thank you. The second round of notation is that I need to invoke w-algebras. And I'm not going to talk about this simply since I don't have any time left for this topic. But I just need to say that there exists. So if you start with g hat, which is g hat smoothie, lii algebra, then you can do the following that you can define this w-algebra by something called the influence circle of reduction applied to this guy. It's also if you can denote this something like that. It's a semi-infinite cohomology. And the most important thing is that if you take the level kappa representations of g hat, then you can move them by this ds construction to by taking actually semi-infinite cohomology with respect to currents with an important group of this v psi. I forgot about psi here. Once again, I do not explain anything here. I just say that there exists w-algebras and kind of briefly indicate what is the worth related to this. And so this goes to representations of w-g. All right. So why do I need to say this? Because inside of this w-g, you have a Virasora algebra. And now you can ask a well-defined question. You can ask what happens with level kappa representations when you restrict them to Virasora. And then there is a well-known formula, which tells you that the central charge of the Virasora on this ds of v is given by the following formula, which is very important for whatever is going to happen. This c sub g equals rank of g multiplied by 1 plus dual coaxial number multiplied by dual coaxial number plus 1 and certain q squared. Who is this q squared? So q squared is even more important guy. So q squared is h plus h inverse plus 2. And the q itself can be written using our beta, which we used all the time. It's beta plus beta inverse. And I assume here that the real part of beta is non-negative. Now the condition that q squared is non-negative, this condition just means, if you decipher this, that this h has absolute value 1 or h is non-negative. So this is the conditions which we used everywhere. And so in this language, it transforms to the condition that this q squared is a positive number. All right. Then one more piece of bookkeeping. When you talk about Katsmudi algebras, you don't talk about Kappa level. You shift it by dual coaxial number. And then this is actually our h. So this is the dictionary how you go between the world of Katsmudi literature and this setup. Now, Fagan and Frankel, who defined W algebras in general, I mean, they were defined by physicists by the Molochukov in the middle of 5th or 3th and 4th elements. So Fagan and Frankel general case gives a mathematical definition, not these formulas. So they say that there is something which we can call a solitary series of representations, the lambda of W algebras. And so for Virasura, you just take textbook by Viktor Kats and Reynand Skovsky. And in one of the chapters 2, I believe, or 3, you find this representation written up for Virasura algebra. Now, who are the parameters? That's what really important for me. So the parameters, it's element of the dual carton and non-negative q. OK, so then these parameters alpha and q, they allow you to produce lambda, which is q times rho for g plus i alpha. And so I am actually going to switch to this alpha, which is just plain cartons, like shift in the usual representation theory by rho. OK, and so then the statement is that this model's v lambda is equivalent to v omega x on lambda, where this section is in standard ways. So it takes on alpha rather than on lambda, actually. OK, so this is the data. So if you have complex Riemann surface, you want it to put to the puncture these oscillatory representations and take the covariance. Before I proceed there, let me just say once again, what do you get for SL2? So if g is SL2, then we get this way as oscillatory representations. We are models, 1 half q plus i alpha. And the central charge is going to be bigger than 1. And h is the eigenvalue of the L0. It's going to be bigger or equal to c minus 21 over 4. And this is a very well-known. Again, if you take, for example, textbook of Victor cards, you will see that the unitary representations are given by some domain where this is c line and h line and they are all unitary. And we consider only some kind of wedge here. And we talk about a little piece of unitary representations here. H, H, H, there is no h bar. So if you want to be confused, then you notice that you have h bar and h. If you're not yet confused, you notice that you have dual coaxial number, h check. If you still manage to guess through this, you have the algebra of the cartangrof. It's, again, h. And we can keep going. So sorry, this is the standard of notations. It's h. It's not Planck constant. So then there's the RAM data. So this is the RAM data on tau gn hat. So we assign to Riemann surface v-sponsures p1 so on pn, coin variance of oscillatory representations v lambda k sitting at the puncture pk. And this means that we take the tensor product and then, in the case of vessel 2, take coin variance with respect to the Lie algebra of vector fields, which are meromorphic and holomorphic everywhere except those functions, which they could be meromorphic. And so this coin variance with respect to sigma, so it's a vector bundle on mgn. And the main player is this w algebra. And so inside of this guy, we have a line which depends on sigma and the data. Let me call the data here alpha. So alpha produces this lambdas, but alpha is just what we see here, alpha. And so this is just the product of highest weight vectors. So this is the highest weight line. Now, if you take this coin variance, so coin variance, for each Riemann surface, coin variance give you some infinite dimensional vector space. When this Riemann surface varies, you get some kind of bundle of vector spaces. And actually, you can argue that they should live on the extended model space rather than the original model space. And so what you get, you get infinite dimensional vector bundle with flat connection on, you can say, this mgn, you can call it lg n star. It's a bundle over mgn. OK, so we want to have a name of this. So we say this is v lambda 1 tensor and so on, tensor v lambda n wg. Now, the key point is that this vector bundle has a flat connection, but this flat connection is, so you cannot integrate this over a finite moment of time. So it is flat, but one cannot produce operator of parallel transport for this connection. Because you try to integrate it and it goes to infinity. So that's a typical situation, actually a typical connection in infinite dimensional vector space will be just like that. It exists only infinitesimally. So on the other hand, we have the data which we produced. OK, so this was the RAM story. So we have this beta data. And this is data on this kind of extended techmalar space tau hat. So this comes from this, you know, all this work, so from work with Volodyphok and then with Shen. So it comes from quantization of log gs. And I emphasize that all kind of recent results is our work with link with Shen, so it's a recent thing. And so what it gives you, it gives you, first of all, gamma s hat equivariant local system, systems, because you have this s gs and the citizens side of Hilbert space gs, and since it's about generalized function gs. And as I was stressed many times, so you have gamma s equivariant representation of this algebra ah of gs acting on s spaces. OK, so now what are the parameters? That's the key point. So the parameters, the parameters are lambda which sits in this carton group, taking positive points of this carton group. That's where parameters of our presentation sits. Now what are the parameters here? So here parameters, so you take some alpha k which sits in h of g hat. And now I want to move to the Langland's dual guy. So I wanted to put here check and put everywhere check. So we take Langland's dual on the DRAM side and the usual group on the usual side. It's actually, as you'll see, it's not that important because that story is Langland's self-dual. But in any case, so this produced parameters which is lambda k which is q rho g hat plus this i lk alpha k. And so the first question is can you actually relate them? Because this is, so it sits here and the parameters here sits there. So a priori they live in the groups which are a little bit different because this is Lie algebra and this is a carton group. And you have, this group is actually a product of carton groups over the punctures. So roughly speaking, they are the same. But in order to say that they are really the same, you have to use identification. So you have to identify the dual to g hat is canonically carton group of r plus. And so why this is true? Because by the exponential map, this is actually Lie algebra of g. But then the dual Lie algebra to g hat is carton logically the same as this. So that's how you get this main identification. So now after this, the parameters match. So the parameters of oscillatory representations are now exactly the same as parameters of quantization. If you consider the length of the dual here and the original group g there. And so now I can state a conjecture which relates them. So first of all, so what you wanted to say and what we, as you will see, cannot say, we wanted to say that the Ram equals Betty. We wanted to say that the space of coin variance on the drum side equals to the space of the quantization space we have from our side. And we cannot say this because this actually cannot be true. So because if you consider the space of coin variance, it's kind of very discrete space. It has no topology. It has descending filtration. And you cannot say that you can hope to identify it with a Hilbert space or Schwarz space because they are topological spaces. They cannot be isomorphic. So what can you say? The conjecture is that there exists a gamma s equivalent pairing of vector spaces, vector bundles with flat connections on tau gn, which is continuous on the Schwarz space. So what is this pairing? So on one hand side, we have the space of coin variance. And so this is something which we attach to G check. On the other hand, we have the vector space we produced in which we kind of wanted first to be the same, but it's not quite the same. So it's supposed to be a pairing between this and the Schwarz space, two complex numbers. And so this pairing should be continuous on s and non-degenerate. So now how the data match? Because here we have some parameters alpha. And here we have the parameter lambda. And of course, they match as they're supposed to. So lambda k is this q rho g check plus i lambda alpha k here. So this lambda is from there. This alpha is from there. And they identified using the isomorphism. And so OK, so the statement in the version 1 is that there exists a pairing between these two bundles with connection. I have to give you a comment here because this vector bundle has a connection which is highly non-integrable. I mean, you cannot have a parallel transport here. And it's just connection of mgn. So this connection is coming from representation of the mapping class group. And so it's by definition integrable. And it's basically representation of the mapping class group. Again, they're very different in nature. But even before we go there, you can say that equivalently, you can say that there exists a map of gamma as head-activarian bundles with flat connections. It just dualizes in this picture. So if this we call pairing C alpha, and this we can call the same way. So C alpha is a map from the space of coin variance WG hat to the dual to the Schwarz space, to the space of distributions. So here I use, of course, the fact that this pairing is supposed to be continuous here. And so the tautologically, it defines this map. But here it's an abstract vector space. No topology, it's just a pairing. But on the other hand, here there's a vector, highest weight vector. And so it maps, therefore, somewhere. So it maps to some kind of vector which lives here. And I say that this is a conformal block, by definition. This we define conformal block for this tautous theory as the image of this map, the image of highest weight vector. So now the key point is that this way, we can say that you just have a map from the tecumular space to the space of distributions. Get a map from this extended tecumular space to the space of distributions. And this map is gamma s-equivariant. Now the equivalence of this map really deserves a comment. Because if you think about this pairing, then the left guy, this really sits on the model space mgn, all bundles on this. So that's where the RAM story sits. And the beta story, it sits on the tecumular space. And so this space comes from non-integrable connection. And so the fact that this pairing compatible with this connection gives some infinite condition on this map, but not more. But the question is, OK, so how can we say that this map has monodromy? And this map looks like it doesn't have monodromy. So this is a model space. If you take some point sigma, if you take a loop in this space mgn, then you come back, you get exactly the same space of the covariance which you had originally. There is no, we cannot talk about monodromy here. And so when you come back, you get the same vector space. And here, it's supposed to be paired with something which lives in a bigger space. And so when we come back, we live in the dark mirror space. But it looks to me, it looks like if you come back by this loop, we come back to exactly the same space of covariance on the left. And so on the right, if you have any kind of natural correspondence, like natural pairing between the deram and beta side, then on the deram side, you get exactly the same guy you started from. So you must get the same guy you start from on the beta side. But I claim that you get not this guy. You get this on the beta side. You get the action of the element of the tehnular group which corresponds to this loop. This seems a contradiction. So how this actual contradiction results? So what happens is the following. That in order to define this map, so you have to take a surface and you have it triangulated. You have to put some data on the surface. Then this surface at the moment doesn't have any complex structure it produces this vector space here. Then when you move around this loop, so you of course take your triangulation with yourself. And so when you come back, you actually get a different triangulation on the same surface. So it's already looks somehow different. So if this was triangulation tau, it was triangulation gao tau. The gamma corresponds to this loop. So it's a loop gamma here. But your construction, on the other hand here, you start with a space of coin variance. And you get the same space of coin variance when you come back. But the point is that the construction here was producing isomorphism or map to the vector space related to quantization of this guy. And after the move, it's quantization of this guy. This is different in original data. And if you wanted to have two Hilbert spaces here, like h gamma, h tau, and your cluster, and h gamma of tau. And so these spaces are different. And actually between them, there is an intertwiner, i sub gamma, which we constructed in the way. And so naturally, when you come back, you come back to the vector which sits in this vector space. If you still wanted to observe it in this vector space, you have to apply this centered vinyl. That's how you get the action of the mapping class group. And so once again, I stress that it's a kind of interesting and kind of new, at least for me, mechanism how you get away from the vector bundle, infinite dimension with the vector bundle this connection, which is flat, but not integrable. But nevertheless, it kind of catches monodrama on the right-hand side. You catch it because your construction depends on the data. And this data moves continuously. So it's a monodrama of the data defining the quantization. It's not a monodrama of anything else. And so the data has monodrama, and therefore, your map is twisted by that transformation. OK. What? Which properties does the map? It's a certain map, and it's a gamic variant, but it's compatible with flat connections. So the compatibility with flat connections gives some kind of constraint on the original pairing. That's all I can say about this map. It's not too much, yes. But at least I can say where this map goes. I can at least say for SL3, for example, what is the space of conformable blocks? Because usually, if you take the lengths of the conformable block, you get the correlation function. And so even three-point function for SL3 is not something which was defined or believed by at least by some people to exist. OK, so let me finish this story with drawing some picture which relates all this and explaining class ingredients in just a few minutes. So how the story looks like. So here we have coin variants. We lambda 1, tensor and so on, tensor, will end the N with respect to WG check. On the other hand, we have this quantization space. And remember, it's not a single space. It's rather a small space, log GS, which sits inside of the Hilbert space of log GS. And also here, there's a space of distributions. And so what is conjecture number one is that these two spaces, not that they are equal, but there is a canonical pairing. All right. On the other hand, there is a third player here because by using this modular font or conjecture, as I will explain in a second, you can say that there is a third guy here, which is space of invariance of this modular double of quantum group in tensor product of the principal series representations of this modular double. And so you can schematically write it down as it likes this representations. And then you take here a H of G invariance. So this is the third guy. So since this is conjecturally pairing, and in the case of SL2 or PGL2, there is a work of York Techner, some of its co-authors, which actually gives you a good evidence that such pairing should exist. It's stated in different, he was doing somewhat in a different language, but you can say that his works is a good evidence that this picture is correct. But then there is a smaller font or conjecture, which is known again for PGL2. It's basically this set of works. And so you can go this way. And so you come out with their ideas that that's supposed to be this link. And so this link is kind of reminiscent of a continuous analog of Kasdan Lyustik's works from 1990s, where you have not representation of W algebra, but integral representations of Katsmudi algebras. And here not infinite dimensional representation of quantum groups, but finite dimensional representation of quantum groups. But now this I explained. And so the only question which remains to be seen in the last moment is why this is the same thing. And I claim that this is clear as usually from the geometric picture. The geometric picture tells you immediately this is true, because how actually, maybe I do it here. So here we're kind of talking about something which does not exist mathematically, but could be called like a continuous tensor monoidal category with the options given by the principal series of presentations. I emphasize that every moment you try to say that there is a tensor category with a continuous spectrum. So you're in trouble, and you need to work hard enough to explain what you do. And you can't apply any technique which you know from this usual tensor category business. So you should use this as an analogy, but then the analogy is kind of complete. So, and you don't think in categories you think about homes. In this case, for example, first of all, you never think about homes. You think about space of invariance. Then you can develop such a formalism, but still you cannot use the main benefit of categories that like you take tensor product of two objects, you decompose this and so on. So you have to rebuild your language. But model of this is a continuous analog. OK. And now how we actually defined tensor product on representations of quantum groups. So we were using these pictures. So we take this guy with a puncture, multiplied by this guy with a puncture, with a pinning, sorry, I said puncture, I mean pinning. This is a product. This is a puncture. And so there are these special points here. And we considered this outer monodrama equal to e-spaces. And then this is essentially isomorphism. And from this you see that representation of these guys is going to form tensor category because now you can, so if you just take representation of this and representation of this, then that tensor product is by definition this H which relates to this guy and so on and so forth. And then you have a map by encircling. And so you have whole palatable structure. So now you see that you have kind of tensor product here and so on and so forth. Why I'm saying this? Because if you accept this and it's done so, so there is this kind of geometric way of thinking, that's what I call quantum geometry of surfaces. Then calculation of something like invariance is a geometric procedure. Because all you need to know, you need to know this model of counter conjecture. And so that's again, you don't need to use quantum groups. You just need to use whenever you have any device which contains this model space, then you have such a category. It doesn't matter that it's related to quantum groups in this setup. You can maybe imagine that you can imagine some different way to assign algebra of answers representation to this model space. It will have the same properties for formal reasons. And then you say, okay, you take tensor product. What is the tensor product? So this is the Hilbert space which sits here with two special points. But then what does it mean that you encircle them? This means that you actually producing, so you encircle them means the following that you actually producing picture like that. And that's how the picture looks like. That's what it means that you had the surface. But then you can adjust kind of identical thing. But then you can cut it here. And then you see that you have this space. This is a space of invariance, therefore of triple product. Because you take tensor product of two representations. I mean, it's very easy to see from this picture's space of invariance. But that's exactly what our quantization picture. So here, the surfaces we're talking about, so there it's a Riemann surfaces, but it's a Riemann surface related to surfaces like that. Here it's topological surfaces. So it's like that. But then by this kind of, by modular functor conjecture as a main input and by this easy geometric thing, you see that this is the same as the space of invariance. So I just explained the picture, which shows this. And so this shows that if you have, if you have this modular functor conjecture, then this space, indeed, the space of invariance. And then you say that, okay, that these two spaces in a pairing, this means that these two spaces have canonical pairings, so on, so forth. So all in all, in the end of the day, you get the following picture. So what you're really talking about there, you have this correlation function in Todor theory. And physicists say that it's supposed to be given as integral of e to some action and some e to alpha phi one, e to alpha n phi n, phi alpha n d phi, which we don't know what it means, but it produces some correlation function, some function of the punctures, because you have your Riemann surface there with punctures p1 and so on, p2 and so on, pn. And there is a complex structure here. Now we have two sides. So we have the drum side, and we have the betty side. And this is the two guys. So on the drum side, you have the following structure. You have the space of coin variance, W, G hat, plus filtration. Why filtration? Because you have the highest weight vector here, and then you have all this tensor product of highest weight vector, then you have all the descendants. It's usually filtration. So you can think, okay, this is a vector space with this filtration here. On the betty side, you have again a different vector space. And so I can just call it H, but you can call it H betty now. And so this is like H the drum. And so now when we say that there's a pairing, so you can interpret this. Imagine it as a comparison isomorphism. So between H the drum and H betty. Again, it's a little more subtle because it's not a Hilbert space here. This is not a Hilbert space either. It's a discrete space. They are not isomorphic, but they are pairing, and there is no isomorphism here whatsoever, just a pairing which is not degenerate. But it looks like you're talking about the Raman betty realization of some motif. And so then you have this filtration, which is like Hodge filtration. And so it all looks like in attempt to make sense of this integral, which doesn't make sense, in a kind of Hodge theoretic way. And so it seems that this suggests that there should be some kind of chapter of algebraic geometry, which talks about this infinite dimensional quantum motifs. Quantum, because we have a plant constant everywhere. And this is an example. Okay, so I did not have time to tell you how you define this cluster Poisson structure. Very sorry, it's about half an hour to tell you. It's not difficult at all, but it takes a little time. So you can find this in this chapter 5, I believe. Now, Pepe is Link with Shen. But other than that, so that's the picture which we have. And so I wanted to stop here, and thank you those who survived for surviving this lectures. Yes. Question about the first part of the lecture. You mentioned that you had a new geometric definition of quantum group. I mean, we just say that quantum group is, or Q, is the quantum space of regular functions on this model space, which we call L, G, or dot. But not all quantum groups, that particular one. What do you mean? No, I'm talking about classical quantum group which corresponds to Dinkin diagram. Not a fine quantum group, not young and not anything else. Yes, it's a classical quantum, it's a quantum group which corresponds, find it, quantum group which corresponds to finite dimensional semisimply algebra, which was the input G in this talk. The point was that you define usually quantum group and you completely break your symmetry. You say that you have generators EI and Fi, then you go ahead, you use serrations. And basically what you're saying, you're saying that you quantize like your GLN, but in order to quantize GLN, the first thing you did, you choose a basis in the vector space. This is EI, Fi is just equivalent to using a basis. And that's a little strange because you talk about symmetries, even quantum symmetries, so called quantum groups, not quantum universal elements in the algebras. So you want to have symmetries and to define the symmetries, first of all, you break them. So that's a little strange situation. And so in this approach, the symmetry is never broken because you have this model space in which you make no choices whatsoever. So it's a model space related to this punctured disk with two special points on the boundary, some other data, there was no choices. Then it turns out that due to specific nature of the space, this canonical generative still sits there, but they are not maybe that canonical because there is still action of this braid group on the whole object. And it's actually by permuting them. So none is better than the others from this sense. But that's how the standard approach to quantum group appears. But what I'm saying is that in a sense, I don't pay attention to this. So I'm saying that this space is a quantum group. And if you want to do infinite dimensional story with a quantum group, so you look at the space, you work with the space and you better not work with E, I, F, I and K at all. And so all notions, okay, you have R matrix, you have some formulas, you have some universal matrix, all very good. But in this approach, you don't need any of these formulas. You don't need any formulas at all. You just need one input, you need to have a cluster post on structure on the space, and it has to be equivalent with respect to large discrete group of symmetries. And so just that, and property of the spaces, like gluing map and the fact this cluster post on map, this kind of funtorial construction, this recovers all the properties which you need, which you can try to recalculate. But I actually, not that, I don't quite understand why it's important now to calculate them. And the other thing which is on one hand side, on the other hand, if you look at this construction, so if you really want to get back to quantum group and using the indigenous EIFIs, then you need to introduce these potentials, okay. Now the very strange thing about this is these potentials were originally invented using a different setup. And so actually this is something I forgot to say, maybe I use one minute to say this. So it's really important I forget to say this. So there is this dual model of spaces. There's a space which we can search in this lectures, PGS. And there is another model of space which we defined with Volodypok and they didn't consider in this lectures, AGS. And if you take the functions on this AGS, this is precisely upper, this is where upper cluster algebra lives, defined by Bernstein-Felmini-Zelivinsky, upper cluster algebra, not cluster algebra, it's this guy. And so then there was a conge... So when we define this cluster Poisson variety and this cluster Poisson variety, that's a dual space. Dual in what sense? So it's the main point is that there should be some duality which is kind of mirror duality, mirror like for example, homological mirror symmetry and so on, which relates this and G check. Okay, so that's... So and I claim that we proved now that there is a cluster structure here and cluster Poisson structure here and they indeed dual to each other. And before we get some kind of portion space which is called XGS and if surface has boundary, this is not a correct space because this dimension is strictly less than dimension of this AGS. And so this not even, it can be a candidate for the dual cluster Poisson space. The point is that whenever you have one, you have the duals the other, but they have at least to be the same dimension. Okay, if your surface has punctures, if you want to run this duality for, I'm about to finish. So maybe put it here. So you have this PGS and AGS. But it's also very interesting model space when you consider this local systems which you forget about the action of W at the punctures. So forget flags at punctures. Then here, this was our proposal with Linkvishan that you should consider this guy with potential. And so this is supposed to be again homological mirror symmetry duality but this is treated as Landau-Ginsburg model. So this is the same construction. Exactly same construction, a different setup. So it is very interesting question. So why is the same construction appears twice in such a distant subjects as Landau-Ginsburg model, homological mirror symmetry and construction of the quantum group? So there should be some, there should be something there and I don't know. Okay. I have a question which is confusing me since the beginning. When you can do this principle series of conditions, why do you always take a tensor product of two different like dual guy? Oh, because I'm forced to do this. So I'm forced to do this, this is a very good question. So I'm forced to do this by construction. So if I don't do this, it is still there. So it's a property of the construction, not something which you impose. So you start with this OQ of PGS and it acts in some let's say Hilbert space and there's Schwarz space, you construct some representation, okay? And then you consider take centralizers of this action. And the claim is that this is OQ of log G hat S. So it's a check, what? So if you have this construction here, you recover this guy. If you start with this construction, you're supposed to take different model space, not PGS, but PG hat S. And this is a joint group, by the way, so it's supposed to be a joint. So you run a different construction which produces a different space, but turns out to be the same space and the centralizer is the one you had before. So it's not that it's my desire to take this language dual, so it just happened to be there. And this comes, this idea that it should be there, it comes from the, if you do quantization of cluster Poisson varieties, it's exactly the situation, but the dual cluster Poisson variety, language dual cluster Poisson variety enters naturally to the construction. And then you just need to check that cluster dual Poisson variety structure is precisely the cluster Poisson variety structure which you have here. That's a kind of lig group statement, but it's a part of the general setup. And in the situation where you have a quantum group, is it something? Same, same. Same thing. What's the classical analog of that? I don't know what the classical analog actually, probably there's no classical analog, but so also I consider here the quantization when H is bigger than zero and absolute value of H is equal to one. In this case we have star, in this case we have star. What this star does is the usual star which takes star of XI equals XI. And this guy takes of generator XI to YI, the generator on the dual guy. So if this leaves here, then this leaves there. So this star structure is not a star structure on OQ. So you do not get star algebra structure on the quantum group. It's simply not there. It's only on the model of double. And if you don't use it, then if you look at the formula how it relates central charge to Planck constant. So this will give you a regime where C is between one and 25 and this is C like bigger than 25. They continuously go one to the other, but you don't want to miss this region because I said C is bigger than bigger equal to one, but there's two regions and they're served by entirely different constructions, so to speak. And one of those constructions does not exist on quantum group, for example. It exists only in the model of double. I actually was missing all the time through. Any more questions? I don't know. Yeah. Is there a difference in nature between H bar of minus one or H bar positive? What do you mean difference? I mean, do you see? No, they're very parallel. So I wouldn't say they're the same. They're different numbers, but... So when we were talking about last lecture quantizing the central twine operators, they are naturally within the setup. So that's all I can say. That's not, I don't know. So for H bar is modulus equals one, the Q coming from this H bar has modulus just arbitrary, right? Yeah, it's exponent of i pi h. So, and the star doesn't take... So the star here, we talked about the first lecture, but the star, this star, the computer star, it takes Q to Q check. So it changes, the original one takes Q to Q check inverse. It was taking Q, the real one takes Q to Q inverse, and this one takes Q to Q check inverse. But on the level of quantum torus, this is a very straightforward statement, but it's a nice fact that it's actually, it's glued together in cluster, right? You can glue them together, that's what I'm saying. It's a priori, it's unclear why you should be able to glue them, so you can glue them together, so story. After that, the story goes. So the story, if you just consider one quantum torus, there's not that much, I would say, intellectual value there. So you, the whole story, this business becomes interesting when you have different cluster Poisson coordinate systems and when you consider intertwiners. This is kind of the main output. It's like the usual story, you want to go to numbers. That's how you get to numbers. If you just consider quantum torus algebra, you cannot get a single number out of this. Okay, there seems to be no more questions, so thank you again.