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Okay, let me start. So I'm talking today, it's the first lecture on quantum geometry of modulus space of geolocal systems in the presentation series. And first of all, thank you for HES who gave this wonderful opportunity to give these lectures. So the lectures are based on joint works with Lin-Hui Shen and Valodya Fock separately. So this is kind of older stuff and this is kind of newer stuff. So in particular here is 19 of 4.1091 and here is 0,311, 149, 0,311, 245 and some other things. So first of all, what do you want to do? We are given the following data. First of all, G has split semi-simple algebraic group over Q and mostly it will be adjoined. So we usually assume that the center of G is trivial and usually plays an important role. Secondly we have S for now which is a surface and oriented surface. This punctures and usually assumes the characteristic of S less than 0, so it's hyperbolic. Then we also have a discrete group, the mapping class group, which is defined as a group of the thermomorphisms of the surface S, the connected component, the different morphisms isotopic to identity. So this is the mapping class group of S and finally we have the character variety. So this can be defined as homomorphisms from the fundamental group of the surface to G, model G conjugation and so it comes with the name character variety, but it also can be defined as model space of principle G bundles with flat connection on S, model isomorphisms or the name look, it also has a name a bit in model space of local systems. G local system is the same as bundles with flat connection, bit indicates that it actually have variety of algebraic structures and so this is one of them, the one we are looking for. Now, why is it called character variety? I don't know, maybe I'm not the one who made this name, you can take characters of, I don't know. So the key point is that the mapping class group acts on this character variety of Gs in an obvious way because it acts by different morphisms of S and whatever we want to do, we want to do in gamma secuvariant fashion and so the goal roughly is to quantize this log Gs character variety, but actually, we immediately have a question, what do we mean by this and first of all I want to explain at least some data which relates to this. So the first comment is that the space log Gs is a gamma S-equivariant force on space. So that's a very classical subject. So it goes back to Atiyah's boat, which is 82, as far as I remember, and then it was X, X, X. I mean this was, sorry, that's a good point, thank you, we'll try to avoid this. So it acts. So it's a post on space, it's a T-boat post on structure, but it was redone in many different ways, very important ways by different people, by Goldman, I guess it's about 84, then Fock-Rosley, it's 90s, also Alexeev, Malkin, Menrenken, it's 98, and we've just followed the Fock and I'll be talking about this later on. So there are many different approaches how we do this and they kind of all give us some different point of view on what's going on, all important. So it's a post on structure, this means that we have a post on bracket on the space of regular functions here. Now as soon as you have post on structure anywhere, the next question, what is the center of this post on structure? Because that's what controls, to some extent, the quantization. And so let me introduce, first of all, the Carton group to describe it. Let's see, group of the group G. So this is an algebraic variety, I would say, yeah, because you can just, I assume that we have a non-zero number of punctures, so this is a free group. And because it's a free group, you just have to give yourself the image of the generators, which is just some matrices if you're in PJLN. And so you have just unrestricted collection of matrices, model of conjugation. Yes? I have a trivial question, maybe. This connection principle G is bundled with the pad connections on the punctures. Do you have any condition? No, no, no. It's a topological data. I emphasize, as a topological surface, there's no complex structure whatsoever mentioned. But if the group adds some kind of variety, it could be quite complex in this space. Yes. Or you take a categorical course. First of all, for the quantization, this doesn't matter. Secondly, you can consider this as a stack, but we will never go to the discussion of, for example, of its special point yield by the trivial local system, which is the most singular point of the stack. And so yes, correct, so you can have different point of views of what it is, but at least the ring of regular functions is a well-defined object there. And so you will just find a variety of sections of the group, and then you take functions and just the variants of sections. Yes, yes, yes. Yes. Okay. So, but not just that. So the most important thing is that the mapping class group acts by automorphism of this. And we want to keep track of this section. So whatever we do, we do it in a gamma-asequivariant way. Otherwise, it makes no sense. Okay, so we have the carton group. And then we consider the following map. So we take local systems and map them to carton divided by the veil group, raised to power given by the punctures of S. Is there a question? No. So what is this map? This map is given by the semi-simple part of the monodrama around punctures. And the claim which comes to this is that if you take the pullback of the functions on h by w, raised to power n, n is the number of punctures, then this sits, of course, in the functions on the space of local systems. And this is the center of the Poisson bracket. So just a little example. If your group G, did you miss function h by w out there? H mod w raised. Oh, yes, thank you, Jan. Functions here. Functions here. Yes. So for example, so what are we doing? So if G is, let's say, SL to C, also I usually consider PGL to C, then if you consider the monodrama around puncture p, it's conjugated to some matrix, typically Jordan block. And so the semi-simple part means that we associate to this this matrix and the action of w means that we don't know which one to take lambda or lambda inverse. So the variable group in this case is Z mod to Z and it takes by taking lambda to lambda inverse. Okay, so this is the first bit of data. The next one is equally, if not more important. So when we quantize, when we say that we quantize some algebraic variety, usually we think about algebraic variety as a complex algebraic variety, but when we come to quantization, we think about some real part of this algebraic variety. And so what is the real part? So this is the high-tech mirror space. And so let me explain what it is. So what are we talking about? Why is it necessary to consider a real part if you want to do quantization? Because we will, first of all, we will do quantization not just of some algebra, but of star algebra. So we need some star algebra structure. It's already a reality condition. So if you consider the space of all local systems and taking a set of complex points, then of course you have inside the real locus. But this is not the one we are really looking for. It turns out that inside there is a smaller part, which is a connected component called plus. And I also denoted the high-tech mirror space. So this guy is the high-tech mirror space I'm talking about here. So the first comment is that the mapping class group acts everywhere, in particular on this part of the real locus. Secondly, if you want to understand what's going on, we better do some example. First, the question. Teichmuller tau. No, plus is the real part. So I didn't really understand the question. So log gs plus. It's a notation. It's unseparated from the rest of the story. No, no, no, no. So my goal is to explain how what this is in the simplest example of PGL2. The story is you can ask a question what it is going to happen when g is a multiplicative group. When the answer is very simple. So let's start from here. So if you take g, for example, to be a multiplicative group. So I said the group is adjoint, but this will be very often the simplest running example. Then in this case, this log gm, how do you know gms is just a set of homomorphisms of the first dimensional homology group of s with coefficients in z to the multiplicative group. And so just you can say this is first dimensional homology with coefficients in gm or c star if you're over complex numbers. And then here lives this. Okay, so let's take, let's take immediate. Okay, this is true, but let's take complex points. We have log gms of c. This is just first dimensional homology of gamma with coefficients in c star. log gmr, you have first dimensional homology of gamma with coefficients in r star. But here you can take a positive locus. You consider homomorphisms not to c star or r star, but r star plus. So you can consider here log plus given by first dimensional homology with coefficients in the group r star plus. And so it's indeed smaller. Okay, so, but that's a very trivial example. So let's do the non-trivial example. So let's take g to be pgl2. In this case, this log pgl2 s plus is by definition. So it has a variety of definitions. So I'll give you four of them. So you can say this is a set of all. So first of all, when I put plus, this means that we're already sitting in the real locus. I said, I mean, my story starts with an algebraic group over q, so it's pgl2. And then I'm going to restrict to real locus and in the real locus take even smaller components. So yes, it's pgl2r, but I start with pgl2. So we can take faceful and discreet homomorphisms from the fundamental group to the group pgl2r. Sometimes we say psl2r, same thing, by default, a modular conjugation. And I have to add of type s because you cannot distinguish from the fundamental group of one's puncture torus or three puncture sphere, but this is two different things because the difference s involved here. And so type s means that because it's discreet, we can take up a half plane and take it model the image of rho of pi1 of s. And this is supposed to be homomorphic to s. This gives you the type. Okay, the next description is it's the collection of pairs. So we have sigma and amount phi from s to sigma, which is a homomorphism, where the sigma is a surface with complete hyperbolic metric. Hyperbolic means coracea minus one metric. And I emphasize the word complete here. And phi is a homomorphism. So phi, a homomorphism, defined model isotope, so you can deform it continuously. And you get the same object by definition. So some kind of rigidification of this surface with metric. But hyperbolic metric, this means curvature is minus one by definition. Or you can say, this is the same thing as a surface with a hyperbolic metric and geodesic boundary. So here's what's going on. So if you consider a hyperbolic surface, so for example, once punch retours, then I pretend that I'm trying to embed this to the three-dimensional space, then somehow goes to infinity because it's complete. But then there is a unique geodesic around the neck here. And what you can do, you can cut out this neck. Then you get to the picture like that. And so this provides the equivalence between the description and this description. So we either cut out the neck or we include the neck, the same thing. But important invariant of this is L, which is the length of the geodesics, Lp with respect to every puncher p around p. And relating to the previous discussion, this Lp is just logarithm of eigenvalue of the monodrama around p, the one which gives you the positive number, or at least non-negative number. Okay, so we have collection of non-negative numbers associated with the punchers. And why this definition is our equivalent? Just because we consider the upper half plane, we consider universal cover, which is over the surface with complete metric. And then we can take the fundamental group of the surface s acting by deck transformations on upper half plane. But then this means that we have a map from this group to pgl to r. And so this gives you equivalence of all these definitions. Okay, so that's what it is in this case. And again, the comment is the Mapping class group X on all these things. So I wanted to note that this tau s, which is the Tachmure space, which is in our current location, current language, this guy, it's a manifold with corners of depths less or equal to n. So it looks like that when you have, for example, two punches. So why this is so? The point is that when you go to the limit when this next shrinks to zero, you go into the boundary. If you have two cusps, then each of the next can shrink to zero, and then you go into the corner and so on. So if you look at the dimension of this guy, then this is three times order characteristic G's number of handles. And so in particular, you see that this is dimension over r. This implies that tau s is not a complex space, complex manifold. It cannot have complex structure because it can be odd dimensional. For example, for the puncture tau s, you get three dimensional space. However, if you consider a subspace of, let's say, representations with unipotent monodrama, then it is algebraic, I mean, at least complex manifold, and the mapping class group acts, and the quotient is mgn. And so that's the modulate space of genus G curves with unpunchers. So it's a definitely algebraic variety, and this is its universal cover. So in this case, it's algebraic. Okay, now the next claim is that all this has analog for any group G, not just for PGL2. All I mean is the spaces and some of the descriptions, not all. So this un is unipotent monodrama around the punctures? You said what? This tau u means unipotent monodrama around the punctures. The monomial is 1, 1, 1, Jordan block with ungin value 1. And that is the pre-imaged identity end of this. Exactly, exactly, exactly, pre-image of the identity, yes. So this is the algebraic guy inside, and the guy we're usually talking about when we talk about Teichner space, but we understand it in a little bit wider context. Okay, now the second part, the second point is that this space can be defined for higher-end groups. So what's the story about this? A little later, you actually will see an honest definition of this space. Now I'll just give you some description of what we have. So first of all, it has several definitions. So it has an analytic definition due to Hitchin, it's a paper in topology 92, and technically he was considering the case when n equal to 0, but then it was extended to puncture cases. And so he was using a solution, the existence of solutions of some nonlinear PDE, and the existence of these solutions was guaranteed by theorems of Donaldson and Corlett. So it's an existence fact, and so what Hitchin proved, he proved that there exists a component in serial locus, which is defiomorphic to R2 dimension of G times the Euler characteristic. So as I said, he considered the case when n equals to 0, so in general we have corners, in this case we do not have corners, but the dimension is the same, 3 is dimension of PGL2, and even if you have punctures, the fact that it's topologically trivial survives. Okay, so another take on this subject. Dimension of G times the negative Euler characteristic, 2G minus 2. So in the case of PGL2 we get the usual number 6G minus 6, this is dimension of the tachometer space. So it's 3G minus 3 complex dimensions. This is why because we have no punctures, it's a consideration right now. Now another take of this was an algebraic geometric definition. This is our joint work with Volodypok, and what it gives, so it's an analytical algebraic geometric, it gives an explicit combinatorial definition and parameterization of the space. What it gives, it gives something which looks entirely, actually I shouldn't put here plus, sorry, plus it's only when I use, when I talk about local systems, it gives actually entirely different definition and then we prove that it's actually Hitchin's component, that they're compatible, but it has one more feature which is very important for the quantization and actually the quantization was the original code. Here is this feature. So in the ring or algebra of all functions on this model space log Gs, and now this is algebraic geometric notion, so there was no field mentioned, so there is a cone or plus of log Gs which has a structure with semi-ring. So the semi-ring means that we have the operations of addition, multiplication, division, but we do not have the operation of subtraction. So it has a semi-ring and there is a statement, theorem, that a point x belongs to this high-tech Mueller space if and only if for any element of the semi-ring the value of the corresponding function at the point x is bigger than zero. So this condition holds. So you can phrase this in other terms. You can say that this locus tau Gs can be obtained by taking this log Gs and taking the locus of positive points. So usually in algebraic geometry we are not allowed to take positive points, it does make sense, but this space has a particular structure, so it's a cone, and this structure allows it's semi-ring, so you take homorphisms to any semi-field, and this is a semi-field. I said semi, what did I say? Semi-ring, but I mean, okay this is semi-ring, but it's actually semi-field, particular semi-ring. Semi-ring doesn't have operation of division, sorry, but actually I want to say semi-field, so it has this structure, which is sorry, his state, so let me stick to semi-rings. But if I say semi-field it will have also division, so it's a semi-field, and so you can just consider the homomorphisms from this O plus of log Gs to this semi-field. What kind of semi-field it is? So we have three operations, so we have plus, which is by definition let's say minimum, we have multiply, we should buy the, sorry, in this case it, so in positive, so when you have positive numbers, so we can add them up, we cannot subtract them, we can multiply them and we can divide them, and so, but we also can take a maximum, and so we say that okay, most importantly we have the operation of multiplication, which corresponds to usual addition, we have the operation of division, sorry, I beg your pardon, I'm trying to confuse myself. So this is obviously a semi-field, so what I was trying to say comes a little later. So, but what you can do, you can take any semi-field, okay, and the one I was talking about second ago was z sub t, so this is, as I said, it's just z, and then you have operations of the addition, multiplication, and division, and this is minimum addition and subtraction, so this is tropical numbers, and so it turns out that this high-tech mirror space can be realized instead of positive points of this semi-ring, but it also makes sense to consider various of this semi-ring in other interesting tropical fields, for example, in integers, and so what comes out of this also very interesting and plays an important role in the story, so in the case, for example, of pgl2c, minus, if you take log pgl2 as, as well as in this real tropical semi-field or integral tropical semi-field, then we get again a space which is recognizable by x-percentacular sphere, this is the space of Thurston's measured laminations, and this is the space of integral laminations, so let's keep this in mind because this will come out later, so that not only real real positive points of this semi-ring are important, but also this one will play very important role in the story. All right, so we have this high-tech mirror spaces, but there is also the third definition of the same thing, which was about the same time developed by Francois Laboury, and this is dynamical system definition, which I just mentioned, I'm not going to talk about this, it's the same time maybe later, it's dynamical systems. Okay, so yes. What is the definition of integral laminations? I did not give you the definition, I will give it to you later, I don't need it right now. Okay, so for now you can forget these features of the technical space, but what is important is that what we quantize is a triple. Is the real dimension of this particular space equal to the complex dimension of this? Definitely, definitely, it's a component, yes definitely. Okay, so back to quantization, so we want to quantize not just this algebraic variety, but the following data. So we have log gs, it has inside this high-tech mirror locus, and on the top of this we have the mapping class group, which acts on this pair, and so what we really quantize is this data. Now there is a surprising feature of what's going to happen, and so quantization is the following one, that it makes manifest a Lenguance model of duality. So what do we mean by this? So I consider group g, but there's also the Lenguance dual group, and I consider g I joined, so I actually have to consider here the corresponding enjoying group, but still it's Lenguance dual groups, it could be entirely different group, and I also have the plan constant, the parameter of the quantization, and the quantization survives this kind of duality when h goes to 1 over h, and so these features of the quantization will appear as a result of the construction, so they are not, you can say that, you know that you anticipate this, but it so happens that they appear on their own. But now let me explain what are the structures we get when we quantize this model spaces, and then do the simplest possible example when the group is actually the multiplicative group, when you see how all this, oh this was unfortunate. Wait a second, we have more interesting questions than h-bar. Yes, h-bar is not necessarily real, you'll see. So let me list on the blackboard the data which quantization provides, so we will do the following. First of all, we define a Q deformation of the algebra of regular functions on this model space. Now as usual, when I say we define something, this means that the mapping class group x natural on this something, and so this deforms to the action we were given by construction, the original action of the mapping class group here. Secondly, the center, eventually Q will be just a number, but there is another incarnation of this when the parameter Q, when Q belongs to the ring of Laurent polynomials with positive coefficients. So the center, if Q is not a root of unity when the center is much bigger, is given by what it should be given, namely by the preimage of the regular functions on h mod w to m. So it was the Poisson center and it's going to be the quantum center, unless we had roots of unity. So that's number one. Number two is that we consider a big algebra which we call the Langland's modular double. So I denote this by a algebra, it depends now on plant constant h and we assign to the pair g and s. And so we define this as follows. So first of all, we introduce now numbers. So Q is exponent of i pi h, now Q check is exponent of i pi divided by h and h is any complex number away from zero. So now what we do, we take what we get in the first part or Q of log gs and tensor this over complex numbers, because now we just over numbers with all Q check of the local systems for g check, but we consider a joint group. So I put this into the game. So that's the Langland's modular double. The Langland's is apparent because we have here g check, modular reflects this h to one over h behavior. And so then assuming that this number h plus its inverse is real and this condition is just equivalent to saying that h is a real number or absolute value of h is one. So h is not necessarily a real number, that's the answer to Grob's question. So assuming this, we introduce a star algebra structure on this algebra a sub h gs. And this is already something which remembers about the real structure. And finally we define what we call principal series of star representations, the star algebra a h. And as usual everything has to be gamma equivalent. And so in this setup it looks as follows. It's going to be gamma s hat equivalent in some extension representation of this guy. So I emphasize that whatever we do, we do it in an equivalent way. Now what is gamma s hat in this case? It's some central extension so it looks as follows. Oh there is a central extension and it's extended by z to n plus one. I'll define it later. Okay so this blackboard is a kind of road plan what we're going to do. So we're going to construct this data. Now there are some compatibilities here. And also the word we construct representation of star algebra which is gamma equivalent. I have to explain what does it mean. And so let me do it first and then I will proceed. So I guess it's a good idea to have something like 10 minutes break. So I'll do 10 minutes break a little later. Oh maybe I just do it now. So let's do 10 minutes break and then 10 minutes later return to discussion. So I list on this big blackboard the data which is a little complicated but actually I still have to say something to explain what do we mean by constructing representation. So here the key words we say that we construct a representation and so what does it mean that we construct representation and representation of whom. So first of all we define the representation space but it's not a single space. We define a triple of spaces and a little later I'll give you an example which shows exactly how all this works for GM which is already not quite trivial. So we give a triple of spaces AGS and then some subspace here and it's dual. And so this is a Hilbert space. This is a topological space. It's topological Rache space and so it basically has the same structure as some kind of Schwarz space. So it is some kind of Schwarz space. And this is the dual. So that's number one. So that's where the representation leaves. Now how it leaves there. So we have two guys. So first of all we have this extended mapping class group and it takes on each of the spaces. So it takes on HGS. It takes on the Schwarz type space and takes on this dual. But on the other hand so it acts. But on the other hand we have this algebra which quantizes the algebra functions on local systems. And actually naively you want it to say that you have just a Hilbert space. So you don't consider this SS. You want to say that you have a unity representation of the mapping class group, extended mapping class group on the Hilbert space and you want it to say that you have the action of this algebra here. But in reality you don't get it. So in reality what you get you get some action on the Schwarz space and therefore you have action on the dual space. But this one you do not have. And so this diagram is what actually happening. So once again so you have the group, the algebra and they act on the spaces and they are compatible with compatibility. First of all notice that the mapping class group does act on this algebra. And so I can write down that if you take any element of the Schwarz space and act on it by some algebra element and then act by representation of gamma this is the same as acting by element gamma of A on rho gamma of S. So this is compatibility of the discrete group action and the algebra action on the spaces. And also don't forget the fact that this is the star representation which means just that. Okay so now this is a complete package of what do you want to do and so now let's do this for GM. Yes. What is the meaning of that A H bar does not act on HGS? Is it means that it is not gamma gamma have equivalent way or something what does it mean? It means that it does not act. Can you realize it as a module over A H bar? It means that it does not act. It does not act. You will see in a second. It does not act. So the whole design was to deform algebra of functions and construct its representation in the Hilbert space and there is no such representation. Just give me a second you will see. So the problem is that it kind of does act but it acts by unbounded operators and unbounded operators are not really operators. They have some domain of definitions but you cannot apply them to any vector. And so you have to find the maximal domain of the definitions of this prospective action and that's exactly the space, the Schwarz space. Okay. So let's do the simplest example. So remind you that we have the modular space related to multiplicative group which is home from first dimensional homology to GM and this is actually a split post on torus and let's do it more generally. So more generally let's suppose we have just a lattice. Let's lambda be a lattice with a skew symmetric form like that and so the main example is of course for us lambda is first dimensional homology of Z and the form is intersection pairing and then we can assign to this the split torus t lambda which is just home from lambda to GM and to any element of the lattice we have the corresponding regular function X sub lambda on the torus and we can define the Poisson structure. The Poisson bracket between X lambda and X mu is given by the pairing of lambda mu X lambda times X mu and so it's a quadratic Poisson structure. Now we want to quantize as usual we want to quantize not just this complex points of the torus but real positive points of the torus. So let's see how this works. First of all so we need to proceed following the plan. So first of all we need to define the Q deformation or Q of t lambda. So how we define this? So I remind you that we have the Heisenberg group it's just a central extension called H lambda. So it's a central extension by Z and with the standard group law so if we are encoding elements here is like a1 lambda 1 and want to multiply them by a2 lambda 2 we get a1 plus a2 plus pairing lambda mu and lambda 1 plus lambda 2. Now we just say that our deformed algebra is just a group algebra of this so OQ of t lambda by definition is a group algebra of the Heisenberg group or A is an integer? Sorry what? A is an integers. A integers yes it's a pair integer and element of the lattice multiplied by another integer pair of the lattice so you get this formula. So or equivalently you can say that OQ of t lambda is a free Z Q Q inverse model linear basis X lambda and multiplication law is that X lambda multiplied by X mu equals Q to lambda mu X lambda plus mu. Okay so we have the quantum torus now we take the double of the quantum torus exactly as we see on the blackboard and now we want to represent this in some Hilbert space so we wanted to construct the representation so I can maybe for completeness write down the part 2 that A H of t lambda is by definition this OQ of t lambda tensor OQ check of t lambda and now the key part is the theorem which is nothing else but adaptation of the real representation real representation of the integral metaplectic group and so it says exactly as we were talking about here that there exists a triple of spaces this S lambda in some Hilbert space lambda and the dual to the Hilbert space such that first of all there is a unitary action of the metaplectic group which is automorphism of this lattice preserving the bilinear form on the Hilbert space also preserving the other two and secondly if we have this reality condition h plus h inverse is real then there exists a star algebra structure on this A H and sp of lambda till the equivalent representation now how this goes in order to see how this works we just do the simplest possible example so we just consider the case yes yes yes yes yes yes so in this case so as always we have algebra and we have a discrete group so the discrete group in this case is a metaplectic group it's basically a symplectic group by extended by z mod 2z the algebra is a double model or double of the quantum torus algebra now representation comes as follows so the simplest case is when we have lambda to be z square so it comes with the basis e1 and e2 it's a basis a symplectic basis this means that the pairing symplectic pairing between e1 and e2 is 1 and it's convenient to introduce square root of plant constant and then we consider the Hilbert space h to be just functions on real line l2 of r and we think about them as just collection of functions of variable t and now we need to define the action of the tensor product of two quantum torus algebras this quantum torus algebras they're generated so they're generated by two generators and so the generators is x1 and x2 which corresponds to the basis and they act as follows this is translation by 2 pi i beta and this is multiplication by beta multiplied by t and the second operators of y is translations by 2 pi i divided by beta and multiplication by t divided by beta beta is a number h by this complex yes it's a number you choose a number so for example you can take assumption real part of beta is non-negative okay it doesn't make big difference because you just rename the generators of the algebra otherwise so this is a generators and what you immediately notice you notice that axis commutes with y's and that x1 and x2 generate this oq of t lambda and that y1 and y2 generate oq check of t lambda because exponent of 2 pi i beta square is exactly if you commute them you'll see that you'll get exactly q square not yet not yet but the second thing which you notice I said that they act in l2 of r and they obviously do not act there at all because you multiply function in l2 by exponential function and so if beta for example real it doesn't make sense or you shift to complex domain you also cannot do this so they pretend to act but they do not act so now actually it acts somewhere it acts on a space of functions like exponent of minus t square plus at plus b multiplied by some polynomial and so at least there is some infinite dimensional space on which they actually do act and then we actually define the space on which they do act so this is a schwarz space so the schwarz space inside of h which I remind you is just l2 of r is the collections of functions f of t such that f decays faster than exponent e to ct for any c and if you consider the Fourier transform it has exactly this property and so you can also say that s is the maximum subspace of l2 of r such that for any element in this a sub h the negative action on a function does belong to l2 of r so if you replace this a h algebra with the algebra of usual differential operators with polynomial coefficients this would be the definition the schwarz space so this is the functions which are smooth which means you can apply any differential operators and decay faster than any polynomial so that's the conditions that if you multiply by any polynomial different by any operator with polynomial coefficients you still live in l2 of r so in this case is the algebra is different the algebra is this tensor product of two quantum tori and that's a sufficient that's actually the condition which guarantees that everything works now the key question so what are the conditions on h so so far what we did works for any h so but the point is that if you want to introduce a star algebra structure then you do need some conditions you want it your representation be not just representation of algebra but be representation of a star algebra and so in l2 of r you have the natural structure and so it induces the following one so there is a star for real and star for unity so to speak two kind of stars and so this one works when h is a real number and in this case you have the operate you have the x lambdas and star of x lambda is again the same x lambda and you have a y lambda and again serial with respect to this star denoted by r so generators of this quantum torus algebra both of them are real and also star of q is just q inverse and also star of q check is q inverse check okay so this is a one reality condition which works but you can also consider absolute value of h equals to one that's another one and here it works differently so you have this x lambda and y lambda and the star interchange them and you also see that it acts on q okay you can write it the same way that you have q and you have q check and the star u interchange them as well and so in the case of this real structure you clearly see that you cannot say that you are you have a representation of the algebra or q of tlandy you cannot say that you have a representation of quantum torus because you have a representation of such a star algebra where involution interchanges the two factors of the torus and so it's not representation of the torus algebra and so in some sense it's a relation to this real components kind of a little bit more elusive but still okay so these are the setup where we already defined the action of the algebra now the question is how we define the action of the group and that's a typical story of the value representation that the group x by automorphisms of the lattice with the form so it preserves all the structures but when we define representation we broke the structure we broke the symmetry and so value representation tells you how you so to speak reconstruct the symmetry so the claim is that in this case the metaplectic cover s l to z hat acts on this h which I remind you is just l2 of r by unitary operators of the following shape so how it goes so if take any element gamma in s l to z so you assign to this some operator which takes test function f of x and terms it in the function kappa gamma x y f of x dx and so that's a new function of y now this k gamma needs to be defined but you need to define it only for the two generators so you have t generator this type when important and s this one then the corresponding kernel k t of x y is just exponent of minus x square over 4 pi i delta function of x minus y and ks of x y is 1 over 2 pi i exponent of x y divided by 2 pi i so that's it so you have it on the generators and basically tells you how to define in the protocol generators with the caveats that you have representation of the central extension but still okay right now I said that this comes from the series of a representation but where actually this operator is coming from so the separators are forced on us by the condition that represent representation of our algebra which I already defined a supposed to be gamma equivalent if you just write down the condition that the gamma equivalent then we see that this kernel say uniquely defined if exist so let me just give you one example so I remind you that we have conditions that rho gamma of a acting on s is the same as gamma of a acting on rho of s or gamma of s and so 4 ks for the simplest generator this condition means the following equation so if you look at the kernel then if it translates this by 2 pi i beta and keep y this is supposed to be exponent of beta y ks of x y that's what this commutation relations means for the first of the generators then for the second well still on x or y plus 2 pi i beta gives you the same but also if you write down the second set of the generators and equivalence with respect to this you get the dual set of conditions now is 2 pi divided by beta shift ks of x y plus 2 pi i divided by beta is exponent of x divided by beta ks of x y now this reflects the two sets of the generators and if you look at this you'll see that this kernel of two variables is defined uniquely because we know what happens when we shift by basically by beta and we know what happens when we shift by beta inverse and if beta is irrational number that's it so we cannot have more than one function which satisfies this property still it's not given to us that such a solution exists and so here you see the solution but at least if it exists unique and so the shows that for example when we try to see what happens when you multiply the generators so we basically up to scalar have no choice so that's why we are bound to get some projective representation and so here you clearly see why this model or double matters because it works in such a way that it produces kind of double system of difference equations which determine your function as good as a single differential equations in in usual story another comment about this representation is that if you happen to miss the second part of the generators for example you just have this one so how you recover this one so you can just say that you want to consider all operators which commute with a given given generators of quantum torus algebra and then you recover this the second algebra so the fact is that if you take this oq of t lambda in this case okay this particular lambda two-dimensional uh then it centralizes is the centralizer of oq hat of t lambda in our representation and vice versa so one is centralizer of the other in the other centralizer of one so that's where the statement so originally I said look at this blackboard I said that we are going to get representation of oq of log gs and oq check of log g hat of s so a priori we start this constructing just the left hand side representation of the left hand side but then when we look at uh on all operators which commute with this representation so we find out that actually we know this algebra and this is just oq of log g hat and so in this sense it's not kind of we're not designed to be language invariant it's so happened that it's language invariant and if you go to the other side if you start with not from the group g you start from the group g hat you do a different construction related to the group g hat and you represent oq of log g hat and then you look what kind of representations you got and you realize that actually this is the same representation you got the previous time but this time the centralizer of the right hand side is the left hand side is this structure related to the fact that the topical points for when the primary case is the function is not in my head actually it's very so joel is very far ahead of this discussion so there is several incarnations of some kind of dualities which intervened in this business which all interchange the g and the language of the group g and joel asked how they are related to each other we will see actually more relations on the course but it's not completely clear how one is dictated by the other okay so now we constructed this representation for we can we defined quantization for for the group gm and so now we want to define the quantization for other groups than gm and so let's at least describe some strategy so how we what do you want to do in general so before I proceed any further I just wanted to say that what we do here is is it just a standard Heismar construction but in a different star algebra structure and so people played with this construction before so not sure who played first but for example cones con Alan con played and especially important for Dave played with this construction and so as as we will see this kind of model of double construction will go everywhere through the story I emphasize once again that's what's surprising here is the language duality comes naturally so we kind of didn't expect it but we are getting it okay so what's the next step so you want to have a strategy to quantize log gs for any G as before and it seems that discussion suggests what to do because whenever we quantize something so usually what happens you find some kind of derby coordinates and then you quantize this derby coordinates and you declare that that's your quantization and so let's try to do the same here so idea is first of all to find a vibrational isomorphism between the space we want to quantize log gs and certain split torus for a certain lambda this is some split torus so let's call this i and the second part of the problem would be to quantize till lambda as before and then declares the victory that's okay we quantized the model space of local systems there are some objections to this plan you know there's an anecdote so when napoleon took vienna so he has a burgemeister why there was no salutation to my you know victory and the burgemeister said there are 14 reasons for that and napoleon said okay name is them said number one we didn't have gunpowder said okay enough go so in this case also there are some kind of issues and there are many of them at least two of them the first one is that such such a vibrational isomorphism generally does not exist that's a theorem you can say that if you consider some general s and general g then you get some algebraic varieties this algebraic variety is not rational period so you cannot find such vibrational isomorphism should even be Poisson you would want it to preserve Poisson structure yeah definitely should be Poisson but even before that so we should have you write we should find some Poisson isomorphism Poisson it's even much stronger but forget about Poisson just finding vibrational isomorphism possible does not exist so the story ends here so but even if it would exist there is actually a more serious issue that the group gamma s acts on this model space log gs in nonlinear way and the claims that it acts nonlinearly means that it acts through a big very large quotient so when it acted on the split torus in the case of the gsgm it acted through the quotient which is just simplect a group it's a very small quotient of the huge mapping class group but when it acts already on the model space of pgl2 local systems it acts basically facefully so as the action is huge and so it's certainly nonlinear and this means in particular that it cannot preserve the chosen coordinate system so you have some kind of nonlinear situation nonlinearly if and only if g is nonabilia if you can rewrite the nonlinear understand what you say properly what do you mean by non-linear this means that's a good question this means that even if you would find some some isomorphism with split torus the action of the mapping class group is going to destroy it because if it keeps it means that it acts through a quotient which is simplect a group acts by monomial transformation of the split torus even if you find one but it cannot because the action basically is you know it doesn't factorize through anything that it cannot factorize through the small quotient okay so this means for example that if you happen to find it it does not exist but if you happen to find such birational person isomorphism with one torus your action of the mapping class group will generate infinitely many other such coordinate systems and so you don't you certainly do not have any preferred one that's for sure but what's worse I said it does not exist and so there is a big issue comes up that you need a nonlinear analog of the value representation so that's the second problem which we need to deal with and so how are we going to deal with these problems so let me first of all so that's our strategy first of all we want to define we cannot get away with log gs that's for sure because it doesn't have a parameterization we want to define a new model space so we call p gs and what is extremely important that this label of the surface change the flavor now it's different as and so this means that it's defined for any surface with corners so for example it could be like that or you could have some holes here or you can have some handles so we want to be able to consider surfaces with corners it's number one you will see why a little later secondly we want to prove that this model space p gs carries again emphasizes gamma s equivariant cluster Poisson structure and so I put in red what we don't have so we don't know what this means at the moment and we don't have this definition yet and then I'll explain that this implies that it can be quantized and in particular it means that it does have this birational isomorphism with actually infinite infinitely many Tory we were talking about and so it looks like we quantized what we wanted but it seems that we quantize a different space so here comes the third part of the story so we use b plus additional symmetries of this model space p gs to deduce from this quantization of log gs in the case of the surface with punctures because we didn't quite again so our goal is quantizing local system on surface with punctures but we are going to extend the notion of the model space of model space of local systems in such a way that it enables to do this okay to implement this that's a question so gamma s will see the what uh i think classical surface does it respect the punctures uh it can interchange punctures that's okay okay and we have this uh corners does it rotate you can rotate the corners so for example you can consider polygon and rotate it there's a gamma both phase s maybe the both yes this is both phases that's right sorry but again your question for example about include punctures or not include punctures so it kind of made of taste because uh works both way okay and so now if you want to implement this plan first of all we need to define this new model space and it's actually indeed a new model space so it wasn't considered before new because of what's happening at the boundary of the surface otherwise we had it we had it so then the next step is the model space pgs so the first question is who is s so for s we have the following picture so we have some surfaces boundary then the surface can have punctures it can also have marked points on the boundary so it looks like that and more formally we say that s is a decorated surface uh which means that it is an oriented 2d surface with uh marked points who are the marked points they by definition either punctures or special boundary points so we have two kind of marked points and we assume that the total number of marked points as positive so we do not allow surface which doesn't have boundary and doesn't have punctures and we also assume that on each boundary component we have at least one uh special point this means that if you consider boundary component which does not have special points then actually we think about this just as a puncture so you can say alternatively that the puncture is a boundary component without uh special points but in this case topologically we can just shrink it to the puncture and that's it but if the boundary does have this red special points we cannot shrink this to a puncture because there is some additional data on the boundary and so we just call them special points to distinguish from punctures okay now uh so the simplest example of this decorated surface as I said is a polygon so in this case there is no topology but there are special points okay now in order to introduce the corresponding modeling space I need to remind you a little bit about uh general facts about semi-simple league groups just a tiny bit so first of all uh we have a so-called flag variety which is uh defined as a collection of all Borelso groups in G and uh if you choose one of them then you can identify this with G mode B uh secondly it has a principle of fine space and uh it's defined actually as G mode U uh where U is a maximal unipotent and uh in the case we're really interested in in the case when G is adjoined there is a much more satisfactory definition so this one is not this one is this is the definition but the problem with this definition is that we say okay we pick some uh maximal unipotent subgroup like here but here we have a definition as a modeling space it's a modeling space of some sort of Borelso groups and so if G is adjoined we also can define A as a modeling space and so in this case we say that A is just a modeling space of pairs U and psi where U is a maximal unipotent and psi is an additive character which is a non-degenerate character character now what non-degenerate means first of all let me give an example so if you take uh SL3 like A12 A13 A23 then a character psi uh could be association to this the following number lambda 1 times A12 plus lambda 2 A23 where this is a non-zero number and this is a non-zero number that's what non-degenerate character means and in general we say that H which is B divided by U where B is a normalizer of U it acts on such size and non-degenerate by definition means the action is simply transitive okay so this is the definition we're going to keep in mind because we work with adjoined group now the next piece of data is that we have of course canonical projection from principal affine space to flag variety which is principal H bundle and it works by assigning to unipotent subgroup and a character the normalizer and because uh we forget about psi and the psi's form principal homogeneous space over H that's why we have the principal H bundle and finally let me start with a notation let's suppose that X is a G set then uh we want to denote it by conf N in X just collections of N points meaning X to N divided by the diagonal action of G because configuration space of points so sorry the notation G by B what is B small b what is the difference between D did that P and G by B so we have a maximal unipotent subgroup like the subgroup of upper triangular matrices okay and B is a normalizer so if you take all this is strictly upper triangular matrices with ones on the diagonal this is U's unipotent matrices and then we can have solvable matrices we can have matrices which are upper diagonal but can have anything on the diagonal here so this is a normalizer of U it's called B it's a maximal a maximal solvable subgroup it's by definition a Borel subgroup and so this B contains U and so U is a normal divisor there so we can take a quotient the quotient is a group of diagonal matrices isomorphic to the group of diagonal matrices it's isomorphic to Z carton group okay and so if you look what happens here so the collection of pairs U and psi is a G set because the group G acts by conjugation on the whole thing now how it acts by conjugation what is the stabilizer of this point the stabilizer first of all if you want to stabilize U as I just said you have to be in B so the stabilizer of U is B but then if you want to stabilize the non-degenerate character then the quotient B mod U have to act texturally so this means that actually your stabilizer reduces to the subgroup U therefore the stabilizer of the G action on this pair is just U therefore the homogeneous space itself is G mod U that's why it's identified with a principle of fine space defined as G mod U okay I actually think that I better stop here because I'm supposed to stop at 230 and I probably don't want to give you the main definition at the very end of the lecture so the very beginning of the next lecture so what are we going to do next in the very beginning of the next lecture I'll finish this talking about this definition then I will explain the key properties so the main thing which happened that we defined this new model space related to decorated surfaces and it enjoys some properties which the usual model space of local systems certainly do not have so and we will talk about these properties and then we will talk about the main property so there will be discussion of examples of pgl2 and then there will be the main property of the spaces that this spaces have lots and lots of coordinate systems in which the Poisson bracket the main point is that this space turns out to be a Poisson space and has lots of coordinate systems rational coordinate systems which has the properties as the Poisson brackets between the generators are quadratic and given by some skew symmetrizable skew symmetric matrix and so the main property of this space is that you have infinitely many of these coordinate systems but if you know this Poisson coordinate systems but if you know one you know all the other so this is called cluster Poisson structure so its main feature they said that you have infinitely many coordinate systems but it's enough to know a single one to reconstruct all the others to to at least to pretend that you know all the others to kind of get them you can get you can get in theory all the infinitely many of them so that's the main point and so after that I will be talking about the general structure cluster Poisson variety I will quantize this and so this way we will be basically down to our list what we have to do in order to quantize so next lecture maybe begin of the the suit lecture we will have already the space quantize and then we'll start we can start talking about the application of this construction okay thank you