 So, this is the second lecture, and I want to start with a little reminding of what we are going to do. So, the main goal was to study a modular space, the character variety, a modular space of local systems on surfaces, but in order to achieve this, we proceed to a different space which has much more structures, which allow, in particular, to go back, but also allow to treat the modular space of local systems the way we want. And so, let me just remind you, actually define this modular space. We denoted P, it depends on the group G, and on the created surface S. So, first of all, here is the reminder of what is the decorated surface. So, it's the surface, first of all. It has holes, and it also has punctures, so it's arbitrary surface with punctures, and it also has holes, and by default, each hole have at least one special point. So, these red points are special points. Everything modular, isotope. Now, we defined last time two important varieties which we need. So, first of all, there was a flag variety B, and secondly, it was the principle of find space A, which I remind you is just modular space of pairs U and Psi, where U is the maximum unipotent, and Psi is the character of U, which is a non-degenerate character. And so, this space is isomorphic to G mod U, where U is the maximum unipotent subgroup. Now, there is a very important for us map H. It goes from the configuration space of pairs of a fine flex, which is by definition is just A cross A divided by the diagonal action of G to Cartan group. So, we say that it assigns to a pair of a final decorated flex, this H invariant, which lives here, and its definition is clear from the Brody composition, because if you write A cross A divided by G as just a double cosets using the fact that A is G mod U, then this is just G mod U mod U, and by Brody composition, this is just U times some element of the Cartan group times, and that's a little non-trivial fact, it's a lift, a natural lift of the well group element to the group G times U. And so, this is this H. Now, it's important to me that it's a completely canonical map, and I remind you that the G is a joint group, and so, if G are joined, means has trivial center, then this H is symmetric, H of A1, A2 equals H of A2, A1. All right. So, if, do you mean, is it a barential map, only to find where the, this W we have here should be, you only want to consider W is W not? I did not say how I lift the element of the well group to the group G, so I need to talk about data, fruit system and so on, so find other to do this. Better be which, but, I mean, if we already have different W appearing. Yes. So, I use this one. So, and it's symmetric. So, you will see example for, for PGL2, well, you will see how all this works, okay. So, let me give a definition, which is a kind of a non-standard definition. The word is standard, the definition is non-standard. That says that a pinning is, the following data is a generic pair of decorated flags A1 and A2. Generic means that this pair and that the composition corresponds to a maximal length element W0 in the well group. So, it's a generic pair, such that the H distance between two elements, these two decorated flags, is one. As I said, usually pinning is defined in a different way and this becomes a property, but I prefer to have it this way. And then we'll say that this is, so, remind you that we have a natural projection, configuration of two decorated flags goes to configurations of two flags. This is canonical projection pi and we say in this case that, that, so, this is H bundle and we say a pinning A1, A2 over pair of flags B1, B2, where B1, B2 is the pair of flags which obtained from A1 and A2 by this projection. So, that's the language we use. Okay, now we can go to the key definition and I'll give you an example later on, right after. So, key definition that the moduli space we're looking for, the moduli space PGS, parameterizes triples. So, these triples are denoted by L, beta and P pinning, where, first of all, L is a G local system on the underlying topological surface S. So, B, beta, sorry, is called a framing and so this is a flat section of the associate local system of flags. So, this associate local system of flags, we don't know that LB, you can just say that this is L cross B over G or this is the same thing as L divided by action of somebody else's group. So, it's a flat section of this guy near each mark point. I remind you that the mark points are this blue and red points, the punctures and special points on the boundary. And so, what this definition says is that we have to basically take a flag near each of them and if the punctures is just a flag which is invariant under the monodrama around the flag. Now, the third part of this definition tells you what is P. Oh, sorry, I forgot to tell you a little condition. So, this will be fine as is, but actually I want to put one more condition such that for any boundary interval, what is boundary interval? Boundary interval is a piece of the boundary which you see between two special points. So, you have a data like that and then you have a surface here. So, by the, the data of the framing means that you have a flag here and a flag here near each of the two special points. And this is our boundary interval, I. And so, the condition is that the pair B1, B2 is generic. What does it mean that a pair of flags in different points is generic? This means that they're connected by a unique interval and so we can use the local system data to pull them to the same point and there they become generic. So, it doesn't matter to which point to move them. Now, finally, the third condition is a, so, so far this definition is exactly the one which we used with Volodyfok for a long time. But now comes a little new thing. So, we need this pinning A1, A2 over this pair B1, B2, which you see on the right, for each boundary interval. Okay. So, now how the story looks like on the picture. So, we have flags here, but also in addition to those flags, we have those pinnings now. So, it looks something like that. And we also have pinnings here. And now, when I say we have a pinning, this means, let me just give one example, that we have here pair of degraded flags over this pair. And we also have a pair of degraded flags over this pair considered on this segment. So, this means that near each special point we actually have nearby two degraded flags. So, this is the data. Okay. So, let me actually make a cartoon of this data in the simplest possible non-trivial example. So, let's suppose that S is just a disc which has a single puncture and two special points. So, it has a puncture here. And it has two special points. And so, if you want to make all the data, first of all, you have to say this is okay, this is point S2, this is point S1, this is puncture P. Now, you want to draw this in a bigger format. And so, okay, this is still a puncture P, but still two points. But now, the data we have is the following one. So, here we have flag B1. Here we have flag B2. This is the decorations sitting over those points. But also, nearby, we have two flags. So, let's call them A1 minus, and this is A1 plus. And here we also have another two degraded flags. So, this is A2 minus, and this is A2 plus. And here we also have invariant flag. So, this is some other invariant flag B. That's all the data. But when we say pinning can emphasize that we mean we have a pair of degraded flags assigned to this boundary interval and to this boundary interval. Okay? All right. So, that's the main definition. Now, what's good about this? So, before we go to discussion, what's good about this, let's talk about pinnings. So, that's a claim example. So, let's suppose that the group G is a group PGLM. Then I claim that a pinning is the same thing as a projective basis in a vector space, in m dimensional vector space. Which means that we're talking about a basis, E1 and so on, Em, a basis. Modular equivalence relations that basis E1 and so on, Em. By definition, it's equivalent to basis lambda E1 and so on, lambda Em. And so, on the picture, this is just for PGL2, for example, this is just two lines. I mean, it produced you two lines. But actually, you have to... Okay. So, you have to... Now, you can say this in a different way. So, this is two lines, but you also have to pick pair of factors in these two lines and consider them up to a scalar. So, this looks a little inconvenient. And so, you can say that this is the same thing as a collection of m lines. Let me just put this way. This is a line plus one generic lines, which I call the line P, the pinning line. And the lines L1 and so on, Lm, which is the one which correspond to original basis, so to speak. And so, on this picture, this is a line L1, this is a line L2, and this is a line P. And you see how it works. So, the line P defines your isomorphism between these two lines. And so, it exactly defines your pair of a basis up to a scalar. So, you can think about pinning this way. Now, why these two definitions are equivalent? So, let's call this definition star, and this is definition two stars. So, the first definition is equivalent to the second for the following observation that you can always pick m plus one vectors in such a way that the sum is zero. So, in this case, this means that we put here E1, E2, and this is E3, such as the sum is zero. This is, here we're going from two to one. So, if you have data two, then we can pick this guys and then take out of this the basis E1 and so on Em, the first m of them. And why is the verse it's also clear that if you have a collection of vectors, m of them, just their sum produces a line. Em plus one equals P. Yes. Yes. So, P is spent by Em plus one. Okay? Now, let's do even more specific example. Let's see what we're going on for PGL2. And so, before we discuss PGL2, it's actually a good question. So, what is the principle of fine space for PGL2? Let's first of all consider the case of GL2. And so, the principle of fine space for GL2 is given by pairs of vector and two form where V is a vector in two dimensional space minus zero and omega is an area form, also known as zero and V2 is a two dimensional space. So, this is the principle of fine space for PGL2, just vector in the form. Now, there's an action of the group GM, the multiplicative group which acts by taking V and omega to TV and T minus two omega. And so, if it takes a quotient, that's a PGL2. So, it's a quotient of a GL2 model of this action. So, we denote the elements, the cosets by notation VW in this kind of parenthesis. Okay. Now, the question is where is the map? I promised you that there is a map, H. Is this APGL2 just the same as V2 minus zero? No. So, if you consider, in a sense, we don't know what V2 is because we mod out by the diagonal group. So, for example, we mod out by element minus one, minus one. So, as soon as you take a vector, you're bound to consider negative of that vector. So, let me explain, so, what Joel, so, Joel asked a subtle question. And so, what he asked, he says, let me just, now you see the point and you see the point of that discussion. Okay. So, what Joel says, he says that what is principle of find space for SL2 and this is, of course, V2 minus zero. Now, how to define this function H here? It just takes two vectors and, okay, I said V2 minus zero. Maybe we fix here some volume form because we want to talk about the group which preserves the volume form. Still, you can take this volume form and evaluate on vectors V1, V2. That's all good. That's right, but it's actually non-symmetric. So, if you change the sign, you get minus of that. This definition does not have a property of symmetry. If you want to get the definition of PGL2, you're kind of trying to say that you want to take V2 model of plus minus one. But the way to do it is this. Then you don't get confused. For example, how you define this map H? So, if you look at this definition, so what would be your map H? Joel, I'll tell you. So, if take V1 and W1, V2 and W2, then you map them to the following number. So, this is the only number you can imagine. Take V1 of pair of vectors W1 of V1, V2, multiplied by W2 of V1, V2. And so, this map is obviously symmetric and obviously descends to the quotient-based action GM by exactly this kind of action of GM I discussed. So, as you see, there's a little subtle thing here. And from this picture, you see that it does not depend on anything. It's absolutely canonical map. In spite of the fact that we have to use some choice of a lift of the very group element, but nevertheless, in the end of the day, we get absolute canonical map. And symmetric. So, as I said above, here you explain why you get two more, you explain why the star and star are double-star with the same, but why are they equivalent to the original definition? Why they're equivalent to original definitions? This is a good exercise, of course, to ask the audience. But basically, you take, you cook up. So, you take E1, E2, and so on AM, and you make a flag out of this. E1, E1 plus E2, and so on. And you make the volume form. E1, W1, W2, and so on, and so on. So, this is how it goes. Now, the original definition. So, you take E1, then you take flag. You have, okay, you can put it E1, W2, and so on. So, what you need to do, you need to produce two flags, first of all. And these two flags, like E1, E1, E2, and so on. And you also can read the backwards, like AM, AM minus 1, AM, and so on. But you also need to produce some volume form. And the volume form is dual to just which product of those guys. And this is a usual flag. To get the fine flags, you just put the wedge between them. Okay? And so I claim this is the azimuth. Okay. So, all right. Now, why do we want to have this model space? So, the key point is that there is a gluing map. This is absolutely key property of these model spaces. Let me actually go first here. This model space has several crucial properties. The first of them, of course, is this gluing map, which I'm going to find. And the second one, as you'll see, is that this model space allows you to localize the very notion of a local system. Okay. So, the gluing map. Suppose that we're given two boundary intervals. Let's call them I1 and I2 on S. And so it looks like this. So, we have a surface and we have either different surface or maybe the same surface. So, I set on the same S, but this S could be disjoint. Then you just have two different surfaces and two boundary maps. Or it could be the same surface and you just glue two boundary intervals together. So, we have these boundary intervals. And then, of course, we can glue them on the surface. So, we get now the glued surface, which looks like that. That's a trivial operation. Now, the claim is that we can do this with these model spaces. This is the property which the usual model spaces of local system do not have and model spaces, which we considered before, also doesn't have. But this one does. Claim. There exists canonical dominant. Dominant means onto. The risky onto. Gluing map. So, that's called gamma. So, related to the gluing of these two boundary intervals, which takes model space related to the original surface and goes to model space of whom. So, I glue here the surface and I get a new surface, which I wanted to note by S. So, if S prime, sorry. If this was S, this is S prime. And so, we go to this PGS prime. And it has a following property that, first of all, it's a principle H bundle over the image. It's not map onto. It's only the risky onto. And also, the image dense, as I said. It's the dominant map. Okay. So, how the construction goes? The main observation for this construction is that if you consider... So, construction. If you consider the collection of all pinnings, then this is a principle homogeneous G set. And so, if you have any two pinnings, there is only one way to match them. There is a unique element of the group which matches them. And also, if G is adjoined, I remind you this, then if you have a pinning P, which is again by far a decorated flex A182, then you can create the reverse pinning given by A2 and 1. And as we discussed for a joint group, this is still a pinning. For SL2 or not a joint group, this is not going to be a pinning, but in this case it is. And so, now we do the gluing map. So, we still have pair surfaces to glue. So, let's draw them again. But now we have the pinnings here. And so, this is my pinning P. And this is the pinning Q. And so, this is my surfaces. So, we go from here, first of all, to the same kind of picture, but where we reversed one of the pinnings. Because originally, okay, so, we just make them going sort of parallel. So, it's P and Q prime, Q star. And then we just glue them. We reverse Q. And then, most importantly, we glue. And so, we get a local system over this guy. So, how we get this local system? So, we impose conditions that P equals Q star. And then, if you had a local system sitting here, and a local system sitting here near neighborhoods of this I1 and I2, there is one and only one way to identify these two local systems here. Because of the pinnings. So, the pinnings allow you to glue. And after you glue, you inherit all the data you had. And in particular, you are going to inherit here some flags. Because when you glue, you take this flag, for example, B1 prime, B2 prime, and B2 prime, B2 double prime. When you glue, I identify them. So, you have a well-defined flag on the glued surface. Okay. So, you can glue. Now, the immediate corollary of this is that the model space PGS is rational. Because remember, there were two main problems we had to fight when we wanted to quantize a local system on surfaces. And the first, the major was the one that it's not a rational space in general. And so, we cannot hope at all to introduce coordinates. But this one is rational. And it's obviously rational. So, the proof goes as follows. If you consider the model space on the triangle PGT, then it's easy to check. It's birational to configuration space of three flags and product in a natural way of three carton groups. And so, now, when you do this process of gluing, so, if you had your original surface, you cut it on triangles like that. And you keep going. And then, you have this pinning data everywhere. But what happens when you glue? One can show that when you glue, you just multiply elements of the carton group which you assigned by this isomorphism to the triangle. So, you just multiply them, and so you don't lose the rationality, so to speak. And then you glue them all together and you end up with something like that which still has pinnings outside. So, you can still keep gluing this. But inside, all pinnings are already gone. So, you get something like that. And so, this is, of course, a vibration of the fiber, in this case, H cubed. But also, because of the composition, it's evidently produced a rational structure. So, it's enough to have a rational structure here which is very easy to do in order to inherit the rational structure of the whole thing. So, in a sense, if you, for example, coordinateize somehow this model space, you get some coordinates on the model space. Almost for free, so to speak. So, I want to emphasize that in this definition, it comes almost for free. So, it's just built this way. But now, let's see what it gives us for the simplest case which is PGL2. So, example. So, let's say that G is a group PGL2. And so, as was evident from this example, the kind of the smallest building block for those model spaces is the one which is related to the triangle. And so, let's just draw a cartoon for the triangle. So, PGT over the triangle for the case of PGL2. Parameterize the following data. So, first of all, we have, so, first of all, we have local system, but there is none because the triangle is trivial. Secondly, at the vertices, we have flags which are just points of the projective space. And finally, as you can see from that discussion on the blackboard on the right, the pinning in this case is just an extra line. So, this means that you have some extra lines here. And so, all together, let's call them P, Q, and R, all together, you end up with six points on projective line. Again, they are very, they have different nature because those points, the white ones, they are just flags and these points, they are pinning. They are objects of different nature, but for PGL2, they look on the same foot. And so, this is birationally, as I just said, configuration space of six points on projective line. But then this implies that we have canonical coordinates on this model of space PGT. Why? Because look, so, if you look at this decomposition, which I started from, then in this case, this is just a point, nothing is going on. But H and H and H are three copies of GM. And so, this gives you three coordinates. But more specifically, to not define them, I want to introduce, first of all, cross-racial of four points. And so, how we define the cross-racial? Actually, cross-racial is a little confusing notion because there are several ways you can define it. And here, only one is a good one. So, we consider the lift. So, we have these points, which live on projective line. And we leave them to the vectors, which live in V2. And then we take, remember that you can take a volume form, omega. And so, we can take volumes of those pairs of vectors and divide it this way. Volume of x2, x3 times volume of x4, x1. Actually, x1, x4. And now you notice, I said, tilde. Now you notice that this cross-racial does not depend neither choice of omega, nor choice of these vectors because we have each one vector downstairs, one upstairs. So, that's a well-defined invariant. And also, you notice that it's normalized by the cross-racial of infinity, minus 1, 0, and x to be x. All right. So, now we can define the canonical coordinates. They are related to the sides of this triangle. And so, the coordinate xr is just the cross-racial of the point x1, x2, x3, r. The r is the one which comes from the pinning. And similarly, cyclically shifting this formula, you get the other coordinates, x2, x3, x1, p. And xq is the cross-racial of x3, x1, x2, and q. Okay. So, they are assigned to the sides of the triangle. So, you clearly see it here that, for example, r is assigned to this side and so on. So, what you do, you take the cross-racial of these four points. All right. So, that's what we do in the case when we have a triangle, but the rest is already basically forced on us. I would say almost forced. I mean, it is forced on us. We just have to calculate. That's why I say almost. So, gluing triangles. We have the following picture. So, we have to take two triangles like that. And we have to glue them according to the corresponding sides. So, let's say that we have here two sides of the spinnings p and q. And let's say that the original data was points x1, x2, x3, y1, y2, y3. Yeah. Let me make this y2, y3. Why? Because they follow, well, that doesn't actually matter that much. So, we wanted to glue them. And so, when we glue, we have to impose conditions that p equals q star. So, we have to do a non-trivial operation which you have to perform to actually glue them. And if you do it, you get some number which assigned to this edge. So, here, first of all, we get four points, x1, x2, x3 are inherited. But this one, so you identify x3 and y3 and x1 and y1. And then, you use the pinnings to make this identification unique. This means you have a projective line here with two points and projective line here with two points. If you just want to identify this point with this and this with this, there is c star ambiguity. But because of the pinnings, there is a unique way to do this. This means that, for example, this point goes somewhere to the point x4 on the projective line we are talking about. Now, it's just one projective line, so to speak, which we think about assigned here. And we postulate. So, we just said that the new coordinate assigned to the edge is just a product of the coordinates which we assigned to this edge, right edge, and this right edge. So, we just multiply them. So, okay, I said it's postulate, but it's basically, once again, we get some product and now there is a little statement about this product. And this statement, it's possible to define this a priori. It's a little claim that this xe, which we defined as a product of the coordinates which were assigned to the red signs corresponding to pinnings, it's actually, again, a cross-ratio of now x1, x2, x3, and x4. Those four points. And so, that's it. So, now we have defined a collection of functions on the model space for PGL2 related to arbitrary decorated surface because now we can glue all the triangles together. So, we take product of coordinates on the sides which we glue and all together we get some kind of system of functions which correspond to the edges. And so, the main claim here is that this is a rational coordinate system on this space which is basically obvious, again, this is obvious from what I said before because from the gluing construction, it is obvious that this collection of functions, the collection of coordinates, you don't have to prove this. This is kind of given to you by the construction. So, we proved the following theorem that given an ideal triangulation tau of a decorated surface S, the functions we defined, the functions x sub e assigned to the edges of this triangulation, provide a vibrational isomorphism. So, it's a vibrational isomorphism from the model space PGL2S to the model space GM, counted as many times as you have edges of this triangulation. By the way, I'm not sure I said before what is ideal triangulation. Ideal triangulation is the triangulation of your surface which has vertices at the marked points. So, if the collection of marked points is empty, as I said from the very beginning that it cannot be empty by default. Yes, ideal triangulation means a triangulation of these vertices at the marked points. And marked points are both punctures and special points. All right, so now we have... If your surface has some boundary, including boundary... Yes, yes, yes, including boundary. For example, the main example is a triangle. So, where is this triangle on the blackboard? Yeah, here. So, this triangle, this model space is three-dimensional because it's basically six points on projective line. And these are the coordinates as you can easily see. Remember that these points has to be generic. So, this is precisely the coordinates and actually unique coordinates on this set. But now, the beautiful thing about this is that as soon as you understood what you do here, the rest is forced on you by the construction. Yeah? In particular, it becomes the vertex being function. For example, the surface can be just surface with one puncture. So, for example, you can have a puncture torus, which on topological picture looks like that, and you present this like this. So, this ideal triangulation has two triangles. You, of course, glue the sides of the triangle. And so, you have an ideal triangulation built from two triangles. Is the fact that it's a puncture manifest itself in any of the data? Yes, because the model space which I consider, this is actually a beautiful question from Sergei. So, the model space assigned from here. Let's look. This is the model space of, first of all, PGL-2 local systems on the puncture torus. It's a self-redimensional. But we add some little extra data. We choose, for each puncture, there is only one puncture. So, we choose invariant flag. In this case, it's just an invariant line, invariant under the monodron. That's it. Now, in order to introduce coordinates, we cut the surface on two triangles. And so, when we cut it out, we just get two triangles. And in order to handle this, we insist that we put the spinnings here. So, we insist to consider a much bigger, at first glance, model space. Then it has coordinates. Now, after you glue them, you glue all three pairs of sides here. After you glue all these right sides, they're gone. You don't see them in the answer. But you inherited the coordinates from them. And so, what happened here is that you basically had PGL-2 local system on puncture torus. What you added to this picture was just a little data. You just added an invariant under the monodrome eigenline. So, it's just basically two-to-one data. You can choose either this eigenline or that eigenline. But after you did this, you can localize this way. And this data, it's a data on a topologically trivial space on the triangle. And so, in a sense, this tells you how you reconstruct, how you localize the topological notion of a local system on topological trivial objects on triangles. And in order to do this, as usual, when we try to localize topological objects, so, we have to introduce some extra data. And so, in this case, the extra data lives, of course, on the 2D part of our story, which is the original local system. But it also lives on 0D, zero-dimensional strata. This is our flex. And it also lives on one-dimensional strata. This is spinning. So, these three data, like L, better P in the definition of this modular space, they tells you about the localization data on all 2D, 1D, and zero-D dimensional boundaries of this surface. And so, that's how you localize topological notion. I emphasize again that after you localize on the triangles, you don't see local system at all, but you can recover it when you glue back. And so, that's the point about this modular space, that this modular space which allows you to localize local systems. Before we introduce the data related to the sites, we were not able to do this. Okay. But as you remember, we wanted to have not just a modular space with coordinates. We already have this. But we wanted to have a modular space with a Poisson structure. So, where is the Poisson structure? So, following these principles, in order to introduce a Poisson structure, you have to do it just once. You have to introduce this on the triangle. So, we define a Poisson structure on this elementary modular space, PGT. By the formulas, so using that blackboard notation that xpxq equals just xpxq and this forces us because we want to have cyclic symmetry to have a similar definitions for the other Poisson brackets and xrxp is xr times xp. So, there's nothing to check here. It's just a Poisson structure. And then we come to the second theorem. If that one was theorem A, this is theorem B, is that there exists a unique Poisson structure on this modular space PGL2 S such that for any ideal triangulation tau of S the gluing map which is a map from the product of PGL2 T, the modular space assigned to the triangles down to the modular space we are looking for PGL2 S that this gluing map gamma is a Poisson map. Now, if you think about this theorem, it's actually what you need to do is two things. First of all, this theorem is going to tell you what are the Poisson brackets between the coordinates which we just introduced before. We are going to calculate them in a second. And secondly, the theorem tells you that if you change the ideal triangulation, the Poisson structure is going to stay. So, let's start addressing this. So, let's do it here. Poisson structure is supposed to be anti-symmetric. It's a Poisson structure. Then what is the right hand side? Xp times Xq. It's a very good question. You see that the orientation of the triangle is crucially involved in this definition. So, we use the orientation to tell you who is the first and who is the second in this Poisson bracket. If you change them to the opposite, you're going to pick up, of course, minus sign. So, this is a product. It's a function, so functions can be multiplied. It's a magic thing. Once again. So, the order of p and q... Oh, I see. It would be... A star or something? This one is both changing the edge p to p star or something like that. No, no, no, no. It will be just a minus sign. So, in this definition, Xq Xp is going to be minus Xp times Xq. Or Xq times Xp. That's symmetric. So, this Poisson bracket is Q symmetric. But in this formula, p is before q. And p is before q because that's how they look on the picture. So, on the picture, it seems that we're talking about counterclockwise orientation of the... We're going about surface-having counterclockwise orientation. In the pictures, you have to tell yourself which orientation you take clockwise or counterclockwise. You say you have orientation of the surface, but how you depict on the pictures. You have to decide. And then concrete formulas, concrete pictures will depend on this. But the definition is okay. Okay, let me do it here. So, we define a Q symmetric function epsilon ij of pairs of edges of tau using the following recipe. So, we say that this epsilon ij the Poisson tensor is given by the sum over all possible triangles with the following properties. So, first of all, what we sum is such gadget which I'm going to define in a second. It has value 0 minus 1 or plus 1. And t is just some triangulation of your triangulation. And now the key thing is how you define this gadget. I and j are edges. Yes, that's a good point. I, j are edges of tau. And so by definition the symbol i t j is going to be this way i t j. It can have three possibilities. Plus 1 minus 1 is 0. Plus 1 if this triangle t is such that i and j are its sides and they go this way. This gaze means counterclockwise. So, if they go the opposite way you put the minus sign and otherwise you put 0. So, this implies that my epsilon i j could be any number between 0 and plus minus 1 and plus minus 2 or plus minus 1, plus minus 2 or 0. So, it could have five values. For example, if you take a puncture torus then you have just three sides 1, 2 and 3 and so if you calculate epsilon 1, 2 for example it is by cyclic symmetry is going to be the same as epsilon 2, 3 and as epsilon 3, 1 and this is 2. It's not 1. So, why 2? Because when you do the calculation you have to think about this pair also this pair. So, somehow the contribution between the Poisson bracket between 1 and 2 comes from this angle and this angle. So, the two contributions is plus 2. Okay, now of course the claim is as you can imagine the claim is that if you calculate the Poisson brackets by the rule I just explained the rule means there are two rules here. So, the main rule is that you have Poisson brackets on the triangle and then you glue all the triangles together. So, you multiply coordinates on the edges and so you get some coordinates on the product which turns out to be cross ratios and then to calculate the Poisson bracket between those cross ratios you basically have to write them down as a product of this original coordinates, initial coordinates and use the Poisson brackets between initial coordinates assuming that the Poisson brackets are 0. So, if you calculate it this way you get the following answer. This is what we expected. This is epsilon ij xi xj. So, this claim tells us that this epsilon tensor is the Poisson tensor for this Poisson structure. Okay, so this is calculation for one particular coordinate system. So, the question is what do we do if we have different coordinate system? In this case we of course need to change the triangulation and it's known that any two triangulations can be changed by a sequence of elementary transformations called flips essentially any two. So, there is some kind of supply of triangulations which you use here and any two of them related by flips. But before we do this, as I said we have a flip of triangulation so and the rest of the triangulation changes. So, we wanted to ask the question how the epsilon function changes under a flip, an edge k. And the answer is the following. So, if you do the flip, you get a new Poisson tensor and this new Poisson tensor press via the old ones using the following formula. It's minus epsilon ij if the edge where you made a flip called k here is one of the i or j. And if not it is epsilon ij plus one half of the following products. We take epsilon ik times epsilon kj plus epsilon ik times epsilon kj. So, why would I write one half of this? Because I want to put absolute values. So, that's how it changes. And okay. So, this is a result of calculations that whatever situation you had you have the following formula for the epsilon function. And then the next question is how the coordinates change under a flip at k. And the answer is the following. Sasha, do I understand correctly that this form kind of from the i could be an internal edge or a boundary? Yes, yes, yes, yes, yes. There's no difference between in this formula internal or external edges. Yes. Which is kind of clear from the main formula for the triangle. So, if you look at the triangle and look how we introduced Poisson brackets, we exactly obeyed this rule. So, how we did the Poisson structure, and this is the last thing I'm going to tell you before the break. So, if you start with original triangulation and had some coordinates here like x, a, b, c and d, then you're going to end up with a flip triangulation. And the coordinates here will be x inverse and then goes some formulas which look a little strange if not ugly. b times 1 plus x inverse inverse and d times 1 plus x inverse inverse and here's a times 1 plus x. So, this is the result of straightforward calculation and you can check the Poisson bracket is preserved. So, this means that the theorem is proved. So, you recalculate the coordinates, they satisfy Poisson bracket and the only thing which somehow desires to be understood better is the following. So, that up to this point everything looks actually extremely nice and kind of canonical. So, with this idea of introducing the boundary data so the whole definition of coordinates in the Poisson bracket become completely straightforward and kind of forced to us by the original step. But this is a little strange. So, I mean this formulas do not look nice. So, we are going to see after the break that actually they do look nice but you have to do it in a more sophisticated way. So, you have to make this you have to go to the quantum world then the formulas become very nice and then when you go down to the classical world and do the calculation then you get what you get. So, actually the whole story is nice and both formulas on the right-hand side of the blackboard actually consequences of some more fundamental and nicer formulas but we are going to see this after the break. So, the break is up to 1540 so let's say like 7 minutes. So, let's continue. So, we have this coordinate change given on the blackboard and there are some comments about these formulas. The first comment that this formulas star they involve the addition multiplication division but not subtraction and this means that they make sense for any semi-field k. I remind you that semi-field is a set where you have this three operations plus multiply and divide and so this implies let me go to different part of the blackboard this implies that for any semi-field k we can define a set p pgl2s with values in this semi-field. So, this is something we cannot do in usual algebraic geometry but we can do in this situation because what we do is just declare that in one coordinate system a point of your guy is just any collection of elements of k defined to the edges of triangulation and then you just recalculate how this coordinates change by using those formulas and because they on the nose satisfy the pentagon relation which is obvious from the construction you don't have to check anything the corresponding point now you know how the corresponding coordinates how the coordinates of this perspective point will look in any coordinate system so you define a point as well as in this semi-field at this moment we don't say anything else but just we can define this set but also the second comment is that the action of the mapping class group is explicit in this coordinates which is obvious and this implies that this mapping class group acts on the set of k points is equal to s k points that we have the action of the mapping class group now let me explain actually what this means and how this works so if you start with some triangulation of a surface if you apply the element of the mapping class group then we get some hugely distorted triangulation of the surface and so and going on but so these two triangulations can be by a flips so we can write a sequence of flips which relates this one to this one and so what happens you notice that coordinates of their if you're talking about usual local system for example coordinates of the pullback local system with respect to coordinate system obtained by pulling back the triangulation so there by definition the same thing as coordinates of l with respect to tau this is topology because when you pull the when you move the triangulation and move the local system nothing happens however what you really want to do you want to express the coordinates of the moved local systems in the coordinate system related to tau and this way you get some complicated subtraction free expression because it involves some sequence of flips and that's how the mapping class group acts on local system and therefore on any k-valid points of this model space okay alright the next question is what actually we get from this Joel asked me a question in there during the break so what's going on for general G and for general G the point is that the same thing is going on the same format and therefore it's a good idea to see what actually we get so we get some explicit collection of coordinate systems there are infinitely many of them the mapping class group acts on them that's why infinitely many of them model is action of the mapping class group actually we still have finitely many of them but little number is infinite we have formulas which allows to recalculate what's going on and the new thing which happened here is that these formulas make sense for any semi-field so let's play with this what kind of semi-fields we have the first example of course the semi-field of positive numbers this is topological example you add them, multiply, divide but should not subtract the second is the semi-field of tropical numbers which is defined as a set as a set R with it 3 operation plus multiply and divide given as max plus and subtract and then you can restrict to the subsets you can restrict to the subset of tropical rational points and tropical integer points and obviously those operations preserve the subsets so we have more semi-fields but there's one more semi-field R bigger than 0 of epsilon which is just given as Laurent series like that so you have an epsilon to n plus an plus 1 epsilon n plus 1 plus and so on up to infinity and you insist that this coefficient and only this coefficient is bigger than 0 then again you obviously can multiply and divide them and you also can add them up you will never get negative sign when you add them up so it's semi-field again and now this semi-field is intuitively clear this is also intuitively clear and those are explained through this one because there is a map which takes Laurent series with a positive leading coefficient just to tropical integers and what it does it takes the Laurent series like that just to number n actually if I take maximum it's minus n I could take minimum here still will be semi-field but I prefer to take maximum then I have to put minus n the point is that when you add two power series and you look at the leading coefficient it can get smaller I mean so if I don't put here minus if I just put n then it can get smaller and so if you change the sign you have to put maximum now this explains where does this semi-field come from they are just images homomorphic images of this one and so basically this is a major one and this is somehow the descendants of this semi-field so all semi-fields are natural but then there is a question so what do we get so the question is who are those sets gl2 s of k so can we interpret them in a meaningful way and of course the first question is what we get when the semi-field is a semi-field of positive numbers and in this case the answer the theorem that if you take this model space p, pgl2 s with coefficients in r plus then what we get is the extended and I will define this in a second the tecumular space tau s now I have to define the right-hand side as some little enlargement of the tecumular space so it works as follows so I hope you remember the definition of the usual tecumular space which was given as a previous lecture and then to get the point of the extended one we just need to get a little bit of data so definition the extended tecumular space tau s parameterize pairs which consist of a point of the usual tecumular space plus an eigenvalue monodrame let's call the second value lambda p monodrame around each puncture p so basically we add for every point we add two or possibly one if it's uniport monodrame bits of data but now what's nice about the extended tecumular space is that's a claim it's actually a consequence of that theorem if you want it on this extended tecumular space is different morphic to r to the number of edges of any particular triangulation and in particular it's different morphic to r to n remember that the usual tecumular space which sits here is a manifold with corners so it looks like that and if you go to this one the corners are gone you just have plain rn so why is this a consequence of the previous theorem because basically by the definition I gave you there are positive points going to be isomorphic to r plus raised to power n when the number of edges because you just take one coordinate system and prescribe positive number to the edges that's it you can say but this doesn't yet prove the theorem because it just says that the left-hand side is r to n but how you identify the left-hand side and the right-hand side so let me just give you I'm not going to prove the whole theory but I just going to give you a map from the extended tecumular space to the set of positive points of this modular space and so first of all you of course pick at an ideal triangulation tau and then you see the following picture that if you start with your ideal triangulation then it looks something like that so you have some points let's say p1 p2 p3 p4 and you can consider the quadrangle related for example to one of the chosen diagonals E now you're going back to the universal cover this is the universal cover so it's an upper half plane and when you go to the universal cover it remembers that you have what it looks like punctures here but actually if you go to the universal cover remember that the tecumular space looks as follows you have this next here minimal geodesics and they were developed to geodesics like that and this foregone will be lifted to the ideal foregone which looks like that and so we have this point this and this like that that's how you leave this foregone but notice one thing that when you leave this foregone I choose this red point here but I may also choose this one so that depends which eigenvalue I take because these two points this one and this one on the projective real line they adjust this projective real line at the same time as a kind of accident pgl2 accident here this is also rp1 which is a flag variety for pgl2 this is a kind of classical accident here and so now we treat this line not as a boundary of the Shevsky hyperbolic plane but as a flag variety and these two points you can think about them as just eigen lines of the monodrama but remember that we have to choose one and so we choose one everywhere and then if you do it then this picture leaves this way so now after it leaves this way obviously so we get here some points let me call them p1 tilde p2 tilde p3 tilde and p4 tilde and if you want to leave this edge the edge going to be leave this way but the key point is is the following remark which is a separate remark that for any points on a circle let's call them x1 x3 x4 the cross ratio of these points is positive if and only if cyclic order of the points agrees with the cyclic order on s1 and so here by construction you have a lift of the ideal foregone so the cyclic order of the points is going to follow the order on the line and so this implies this remark implies that the corresponding coordinate xE is bigger than zero that's all we need so we produce a map which takes a point of the extended technical space and creates a bunch of numbers which it turns out to be by this little common positive numbers related to the edges of the map from here to here because this set is understood at the moment as a collection of positive numbers assigned to one particular triangulation so that's it we get the map then it's easy to prove that this map is one to one I'm not going to do this so the conclusion is that this structure is such that r plus points give a technical space how about the tropical points so now the natural class of points and so I'll give you just these two examples because the whole goal of these lectures is not education and technical theory but applications to representation theory and these two examples have crucial importance for representation theory as you'll see a little later on so that's why we're talking for so long about them so especially this one so if you take the tropical set to be the because this is sorry the semi-field to be the semi-field of rational tropical numbers then what we get is another theorem that if you take this model space p pgl2s and evaluate this at the q-tropical points then the set you get the kind of notation of this guy is this one but this is by definition the set of rational x or sometimes we say unbounded laminations on s now as before I need to define the very notion of a rational lamination or unbounded lamination but then that's what we get is a result that after I define this notion is defined intelligently geometrically the result is that this geometric definition of the set delivers you the tropical q-tropical point of this model space now what's the definition is this definition the very notion of laminations is due to Thurston and there's some variance on this and so definition is little long so a rational x lamination so I prefer to give you a picture of this definition first and then I'll just explain what you see on this picture so the picture is that you start with a surface which looks like that and it may happen that this surface has some boundary component and on the boundary component you necessarily have some number of points at least one these are those points and then lamination is just a collection of non-intersective curves but this curves could be loops so they can go for example this way and they can have multiplices which are rational numbers so it could be for example three tenths here or it can have loop like that and it could be like five here the numbers are positive and you can have for example an arc which is going from cusp to here or it can go from cusp to here or it can go between the cusps like this or any other way you can imagine here but they do not intersect that's the main point so the point is that we may or may not have some red curves which go to every cusp or to every boundary segment so we may not have any of them but if you have them they come with weights for example like three here and one tenth here or whatever and the main point is that they do not intersect and also if you see two loops which are isotopic to each other you can just join them and add up the weights so that's the definition in words so if you want me to write them down as a long set of words this is a fallen one so this is homotopy classes of a collection of a finite number of non-self intersecting and pairwise non-intersecting this is the conditions they do not intersect curves on S these multiplicities in the set of positive rational numbers positive rational numbers and this curves can be either closed or ending at either closed or ending at casps or boundary intervals that's a long long long story and in the end of the day I just need no condition which is important condition that besides all this I have to choose some orientations of the punctures which support the curve plus orientations plus or minus of the punctures supporting the curves okay so on the picture I did not put yet this data just waiting till we get here so now I can put this data so I have to put somewhere I put plus somewhere I put minus for example plus minus minus and here I put nothing so this puncture there's nothing because there's no curves there okay so that's the notion of the lamination and then the claim is that the set of these laminations equipped with positive weights is nothing else but the set of qt tropical points of the same order space so as again I'm going to give you a construction one way so I'm going to tell you how you produce numbers if you have this lamination and till I'm raising I'm just going to say that this will play a crucial role in implications to representation theory because this gives you the simplest example of this canonical basis kind of new kind of canonical basis in algebraic geometry in particular in kind of related representation theory which go beyond Leustic's canonical basis so this this gives the simplest possible example of this and the kind of the first example of this kind of basis and so it turns out that this basis will be parameterized not by rational but by integral laminations of this type integral means that the weights supposed to will be integers not rational numbers so that's why I care so much about them this is the set which will appear as a set which parameterized some linear basis on some other dual model space which we'll talk about little later on but now let me give you co-ordinateization so this is a map from the set of laminations on s q to p pgl2 s with values in this tropical semi-field so as usual we pick an ideal triangulation tau and then what we do is the following so let's suppose that we have some puncture and then we have some collection of curves which ended this puncture so like this one and this one and so first of all this puncture has attached to this sign plus or minus remember that and depending on the sign we will rotate infinitely many times this curves to one way or the other way if plus following the orientation if minus following the opposite orientation of the surface and so this will give us the following picture so let's say we have here plus then we kind of rotated infinitely many times like this but we also have to do the same with the other curve and so with the other curve let me do it in slightly different color so it goes this way where is the triangulation picture triangulation at this moment didn't appear yet it's not important wait a second so this is just a pre-procedure so we rotated this that way this is the first step the second step is that if we as Joel said have a triangulation then we are going to assign the number to this triangulation and then we count so what we count we count the curves which come this way or the other way and we do not count the curves which go this way this curves we don't count only the curves which intersect the diagonal and then we count this one with the sign plus and this one with the sign minus you can ask me what does this one means so that's a good exercise for you so this picture together with the orientation of the surface tells uniquely what the word this one means so you can distinguish between red and kind of little orange here and so one is with the plus the other with the minus it does not depend you look from here from there and after that you just take the sum of all the weights and then for each edge E take the sum of the weights if they are positive and subtract the sum of the weights if they are kind of the negative sign but always originally are positive numbers you remember that but when you count them you count them with plus and minus and so in the end of the day the numbers you get you get arbitrary numbers positive and negative and so this is the statement is that this may be the isomorphism in particular you can check that if you do the flip and this is kind of fun statement here the main point is that under flip if you have happen to have such picture you have X0 here and X1 X2 X3 X4 and you go to this one minus X0 and here you get something like X3 plus maximum of X0 0 here you get X2 minus maximum of X0 0 with a minus and the same formulas here and here that's how the coordinates we should just introduce work under the flip and then if you compare them with those formulas you clearly see that these formulas are obtained by tropicalization of those ones and so that's exactly the claim that we get not just assignment which assigns to elimination collection of rational numbers but assignment which assigns to elimination a point of a variety with this positive structure with values in tropical points this just means that under the flips they put the same way as the tropicalized coordinates there okay so that's it about the elementary geometry and of classical geometry of this model spaces yes so how does the picture 1, picture 2 relate to each other you have a curve that's got located there you see when you have a triangulation you have to count how many times this curves intersect the triangulation so anyway how do you just think about this so you take them with a plus and minus and so eventually they will cancel out if you take so I'm saying that the numbers here will be so should be alright so I beg you I don't I want to argue why so but you'll get a finite number let's not try to to dig into this one if I will have to have correction I'll do it next time but alright so I didn't want to speak that much about this particular subject so it's important for us for the for the following reasons so now we are going to say that we have a similar structure when we have arbitrary group G and so the output of this that if we do have such a similar structure with some kind of strange coordinates change and so on which will be explained in a minute then we do have the corresponding notions and we do have the notion of the technical space just take the positive points and the space of laminations just take the tropical points and then we will play with the spaces so now let's go to a different subject and we'll try to explain those original formulas because remember that we wanted to explain where all those formulas come from and so the subject which we're going to take to talk right now is the notion of a cluster or a son varieties and their quantization this follows our work with Volodypok to zero zero three and this was an attempt to somehow to explain what kind of structure we get when we consider a similar so we considered a similar coordination for PGLM local systems and got some kind of exchange some kind of transition formulas so we wanted to present this in a kind of setup and then we realized that the setup is going to use in a certain way the cluster algebras of Fomin and Zelivinsky but it's actually going to be a different the dual story so it's dual to Zelivinsky cluster algebras you will see the word dual has lots of meaning here and you will see in what sense dual it's a very important later but let's first talk about this notion so first of all let me remind you what is the quiver so remember that when we define coordinates we assign them to the edges and then we use this epsilon exchange matrix this epsilon post on tensor of course between the parts of edges now here is a way to do this in a kind of abstract form it says that the quiver C is a datum which is given by the lattice lambda form a collection of basis vectors and a sub-collection there and collection of so-called skew symmetrisers so here is what we have here so first of all this lambda is a lattice secondly this EI is a basis of lambda and it has a subset which we call EF I is called subset of frozen basis vectors and the second piece of data here is that we have this bilinear form but this bilinear form use values in half integers half integer valued bilinear form on this lattice and the main condition is that the value of this form on the basis vectors is actually integer unless both I and J are frozen it's very very important frozen means they belong to the frozen part of the basis now the third part of this data is that the skew symmetrisers they are positive integers multipliers skew symmetrisers whatever so and the condition is that if you take a new bilinear form which is given as the old one multiplied by D J inverse here then this one is skew symmetric okay and so this is just a quiver or as some people say weighted quiver because what is a quiver oriented quiver I mean so this means that you have some vertices and this vertices is just the basis vectors then from one vertex to the other you have some number of arrows and this number of arrows is given by the value of this form epsilon i J and this form is it is bilinear but it's almost skew symmetric if you multiply this form by a positive number it becomes skew symmetric this means that at least if you change E i E j to E j to E i it changes the sign so there is only one direction between them which is positive direction and that's the direction when you put the arrows but then there are some vectors some basis vectors which are frozen and then the number of arrows from them could be half integrals so we draw this like dotted arrows so this is the frozen variables and then you can put some integers which is skew symmetrisers or multipliers d2, d3 and so on so that's the weighted quiver on a picture and that's the weighted quiver in a formal way exactly the same thing now what we can do with the quiver we can do quiver mutation the second definition and the mutations are done in the direction k who is this k? I can say not k I can just say ek but it's customer just to parameterize the basis vectors by a set and set i and then you can just say direction mutation is direction k so this is mutation mu ek which takes quiver c and uses out of this a new quiver c prime so who is the quiver c prime first of all I insist here that this ek is non-frozen you'll see why in a second we can mutate only non-frozen directions and then this c prime new quiver it has same, almost same date it has same lattice it gains the same bilinear form it has the same set of multipliers, so skew symmetrisers so you may ask what actually is different so the only one thing is different that the basis, the new basis is different and it's given by the following symbol formula by half reflection so the new basis, the mutated basis is old one plus ek plus plus means that it's 0 if it's negative and the number which you see here, if it's positive multiplied by ek this is if i is not equal to k and it's minus ek if i equals to k so that's it, this is the only formula which is relevant to the whole story the only kind of new formula we add now how it relates, first of all the first thing you notice is that if you do mutation twice if you do, let's call it mu k, not mu k, if you do mu k and then mu k, mutation is the same vector then if you apply it to ei so this is your ei double prime this is actually ei plus scalar product ek and so it's a different basis so it's not true that if you do mutation of bases twice, you get the same basis twice in the same direction you get the same basis different basis, but nevertheless but if you take the value of your form on this new basis vectors it is the same as it was before, this means the square of the mutation adds the same direction is actually a transformation which preserves this by linear form okay the next thing we wanted to say parallel to this then we can introduce the exchange matrix let's call it epsilon ij which is by definition is just the value of the form on ei and j and then you also have this multipliers so you have this data and this data you can ask how this data behaves under the mutation and then the formula is precisely that formula which you have here this mutation in the direction k is given by the formula which we had here so remember that this formula was written in a very specific setup of the triangulation but the formula itself works for arbitrary mutation of quivers if you understand mutation this way and so this is kind of simply writing down this formula for mutation of the exchange matrix now the history of this formula is a following so this formula is the Fomin-Zelevinsky mutation formula for the exchange matrix the only difference is that they denote it by bij not epsilon ij but there is essential difference which looks like a little difference but actually very essential that in the Fomin-Zelevinsky have no values epsilon f1 f2 if f1 f2 are frozen this is the same setup but there is no values between the frozen variables okay but in yours there is yes and you notice that in this formula if you mutate in the unfrozen direction this is still an integer and therefore it takes the basis of the latest to the basis of the latest if you apply formula for if you take mutation with unfrozen with a frozen k and this will be frozen this is half integer so strictly speaking you are already out of the latest you can say alright big deal you are out of the latest but you will see that it actually leads to problems later on so you see it a little later on so there is no the epsilon matrix like rectangular matrix and ours can be against square matrix and also it's very interesting to note so this Fomin-Zelevinsky this is about 2000 I think they discovered like in 2000 and also the same formula was discovered by Naden-Zeiberg in 95 Zeiberg is a thesis of course in a completely different setup but it was exactly the same setup that you have quivers and you mutate quivers so he mutated quivers for for different reasons and that is the same formula so it's very interesting eventually to see why this two subjects going in the same direction but we are not going to talk about this alright so now we kind of we get this formula on the blackboard now and it's kind of explained by the mutation formula for the basis vectors which looks kind of a little simple now the last challenge which we have we want to explain this formula and in order to do this as I said we are supposed to go to the quantum world so we have to quantize the story so if we start with quiver C then first of all we can assign to this the cluster Poisson torus T sub C which is just home from the lattice lambda to GM basically the game we are playing now we want to get to recover as much as we can from this PGL2 story or Tecumuler story in the setup of arbitrary quiver so basically we start from arbitrary quiver rather than from triangulation but we are going to get the same notions as we like and we have for the classical Tecumuler theory so the question is how far we can go and actually we want to get more we want to get the quantum portion so the quantum, so first of all this is Poisson torus but how to explain this Poisson torus because first of all it gives rise to quantum torus which we denoted by as in the first lecture OQ of Tc and it has linear generators over ZQQ inverse X lambda where this lambda and mu corresponds to L vectors of the lattice and the multiplication rule as we discuss this is Q2 by linear form lambda mu with coefficient 2 or next lambda plus mu but now it's completely defined in our setup because we have this Q symmetric form in the game you see it there and so we can define the quantum torus algebra now we can define the quantum mutation hmm so we can quantize we can define immediately the quantum mutation in the direction of any unfrozen base vector and it's done as follows Poisson, if you said quantum you already got Poisson by quasi classical limit, yes so the main definition is that quantum mutation at ek and when I say ek this means that it always comes ahh so you have some here c and here you have c prime and so mutation goes this way and has label here so this is the data which gives you the mutation but this mutation is a vector ek is by definition an isomorphism of non commutative fraction fields so it's phi corresponding to mutation which takes us from c to c prime in the direction k and this is a map from the fraction field of OQ of T c prime to isomorphically to fraction field of OQ of T c and is defined as follows so this is is given by the following operation so this is just a conjugation by some power series which is called the quantum dialogic power series evaluated as a variable x ek remember that every basis vector produce an element in your quantum torus algebra whose psi will tell in the second but this has to be precomposed with certain isomorphism from c to c prime so now I need to define both gadgets here but this is a formula that's how the quantum mutation works now who is who here first of all this i from c to c prime it's just an isomorphism from t c prime to c t c this is just a story you can say you can say this is isomorphism from the corresponding quantum torus algebras which works in a very simple way so the generator which corresponds to x ek prime goes to the generator x ek this ek prime is a new basis which is obtained by mutation and so basically this is the isomorphism which takes mutated lattice to the original lattice it's a very simple monomial transformation now psi is more interesting so psi q of x is an expression which is known from 19th century and it's called the pohgamer symbol the only difference slightly is that I want to take the inverse of that and so now we call it the quantum dialog written power series and so what we do we take the conjugation in our quantum torus algebras by using this infinite power series so it's like in-automorphism but not quite in-automorphism because it's not given by element of your algebras it's given by element of some extended algebras and so the main key observation there was a word ad ad means conjugation adjoined this means that I'll write you in a second so let me since you're also I'm about to finish today so the transformation is the following one so if you wanted to know what happens with a element of your quantum torus algebra what you do first of all you apply your monomial transformation from c to c prime to this vector y new element of quantum torus algebra related to the same lattice so it's basically the same quantum torus algebra just with a different basis as a label but then you do something highly non-trivial you take this infinite power series psi q of x e k and you conjugate this guy with this infinite power series so this is z this thing this transformation and a priori this transformation doesn't make any sense because it looks like it takes you to the world of infinite power series but the main observation is that actually this transformation is rational is a birational isomorphism this is the key fact and I'll give you an example already going over time what I suppose to do today I'll give you an example next time but the key point is that this is the rational transformation and then if you apply this transformation the case of the lattice you're talking about you'll get exactly the formula star so the last claim that not only this rationality you can compute it and see what it is, we'll do it next time and so the claim that was a classical limit when you do first the transformation quantum case and then you said q equal to 1 the classical limit is when q limits to 1 but what you do, you first do it when q is generic you get some formula and then specializes formula in q equals to 1 and then you get precisely the formula star for if you started this ideal triangulation quiver if you start with some other quiver you'll get some similar formula which looks a little bit more complicated which I'll be talk about next I'll give you this formula next time but it's very similar to this formula actually maybe as a last step I'll just give you this formula and that's the last thing I wanted to do today hmm yeah so the formula you get is that x i prime equals it's x i inverse if i equals to k it's x i times 1 plus x k to epsilon k a epsilon k a is bigger than 0 and x i times 1 plus x k inverse epsilon k a is less than equal to 0 and actually I know that what I'm telling you is it's little little light because you have to change from epsilon to minus epsilon to get this simple formula there is a little in the setup which I was giving you the correct formula will be obtained by changing from epsilon to minus epsilon but that doesn't change too much so you get this kind of formulas which are very similar to the one you should have on the blackboard but again the main point is that this formulas look kind of ugly it's very strange but if you go to the quantum world then you see that actually they are very simple and beautiful because this is just a conjugation by one function and this function is universal you always conjugate by one function how you do this conjugation in quantum torus but it's not really a part of the definition because the definition is much simpler and so you kind of see the beauty of this transformations only when you go quantum and you don't see it on the classical level it looks a little strange so that's basically it so what you want to say now we say that okay we have collection of quantum torus and we have them for every quiver and then what we do quiver then we mutate in all possible directions we take all other this quantum torus algebras their fraction fields then mutate again in all possible non-frosable directions and then we get many, many fraction fields and we identify them in this bioreational transformations and then what we get is the kind of field of functions on this cluster variety which we're going to talk about actually more, we want to talk not about the field of functions we want to talk about the ring of functions and I'll tell you next time how to do this but basically the output is that we will recover what we had before as the algebra of regular functions in this model space PGS now is recovered by this general construction in a very general setup when you start with arbitrary quiver and it actually comes not as a usual algebra of functions it comes as a q-deformed algebra of functions okay so next time is Monday how does this form that are related to the Fomin-Zelivinsky? there is no Fomin-Zelivinsky here because the usual Fomin-Zelivinsky story is a dual story and it's actually not quite, you can't quite obtain it immediately this way so you can you can eventually get to cluster algebras formulas in this setup but not immediately not exactly, so this the point is that you have this cluster algebras you can take the spectrum, this is some variety you have this cluster Poisson variety and it's also some other variety so this one has Poisson structure this one has a different dual structure it has two forms so this one is canonically quantized canonically means that all construction all quantization construction invariant and all symmetries of the problem it's not quite, it's not possible to quantize symmetrically because it's possible to quantize many different ways and Bernstein-Zelivinsky did this but then you lose the symmetry of the problem so because there are many quantizations the kind of symmetry group of the problem it moves them and so there is none which is actually preserved and it shouldn't be one because this is there is no Poisson structure in the space canonically again so it's there are many of them so because this story is quantum canonically quantum so you can define this transformation formula using the quantum torus and if you want to do this to the other ones then you have to adjust it basically you have to invoke you have to consider a bigger space which is also Poisson and get the cluster algebra story by some specialization