 So it's lecture three and the first thing I wanted to do, I wanted to start where I ended last time and explain what are those cluster post song varieties and the requantization. And so last time we started to discuss this, so I remind you that we have notion of a quiver which is a data given by a lattice lambda collection of basis vectors. Let's start from by linear form on this lattice, then collection of basis vectors and a sub-collection in this collection which we call frozen vectors and some collection of integers which are called skew symmetrisers. And so the most important thing about this, that this form takes value in one half z, it's half integers, this valued form. It's not skew symmetric, so phi just by linear form. What there is a condition, the condition is that EI EJ is actually in Z unless both frozen. So this means that this subset of basis vectors we call frozen vectors and so the form takes integral values except the form between the two frozen vectors can be actually half integer. Now then we were talking last time about mutations of this quivers. So there are some people who were not here before, so if you feel that you want me to ask a question, so please do it because that's a basic definition. So then we have quiver mutations, basically it's called quiver because you call this vertices and you measure this bilinear form and you connect two vertices by a number of arrows which given by this bilinear form could be half integer and this vector generates this lattice. So if you start with this vector you can omit the lattice, so just vertices and arrows. Okay now the important formula was, the thing was that we have mutations, so we have a K is one of this unfrozen vertices, so a K is unfrozen, then you start with one quiver and you get another quiver and so this new quiver has everything the same, so this will be the same, this will be the same, this will be the same, this is the only data which changes and it changes by the following formula that EI prime is EI plus, you evaluate your form on EI K, takes a positive value, it takes a value if it's positive and puts zero if it's negative, so you cut it somehow by this plus EK and minus EK, so this is an institution when I is not equal to K and this is an institution when I equals to K, so that's the basic formula. Now we discussed the fact that as soon as you have a quiver you have the square C assigns you quantum torus algebra and so this quantum torus algebra you can say that it has generators x lambda, where lambda is any vector of your lattice and the multiplication rule is that x lambda times x mu is Q to the lambda mu of x lambda plus mu. Now at this moment you can ask, if you were not here before you can ask what this symbol means, it's almost this form but slightly corrected by the skew symmetrisers because we want this expression to be skew symmetric and so you write down definition EI EJ is the form we had multiplied by dj inverse. Okay, now you have quantum torus, it depends, so at the moment Q is a formal symbol but very soon it will be actual number and that's very important. So now we have the notion of quantum mutations, that's a key definition. So quantum mutation you have to say that you start with some C, some quiver let's say C and you go to C prime and you go from here to here so you put H and then you have K which indicates the non-frozen vector which you're going to use for this mutation. And so by definition this is a birectional isomorphism which means it's isomorphism of fraction fields. So it takes the fraction field of the quantum torus algebra which is denoted or Q of T C prime and you map it isomorphically to the fraction field of a similar algebra related to the quiver C. So quantum torus algebra is denoted this way so this is the algebra which we defined and this transformation we denoted it C goes to C prime, K, that's this transformation. Now how this transformation is defined? So it's by definition defined as follows, so most importantly it's a conjugation by the quantum dialog written so let's just write how it works. It takes Y here and it goes to the following expression. So first of all we apply some monomial transformation to Y which I'm going to spell in a second and then we take the conjugation by the quantum dialog written and so this quantum dialog written so far just a Pochamer symbol so just 1 plus Q x times 1 plus Q Q x times 1 plus Q to 5 x and so on inverse. So alright so who is i? Who is this isomorphism i? This isomorphism i from C to C prime x by the rules that it takes any basis vector x e i prime which is basis vector you know that basis vector just defined and map suggest to x defined by that basically by that formula so mu k of E i. So what's happening is that we say that okay look so we have two different quivers this is quiver C prime and this is quiver C so in the quiver C prime we decided to talk about new variables E i prime but this is actually the again by this formula so I'm just writing this down so I'm just writing kind of this notation through this notation so it's kind of identical is it just some kind of monomial transformation which is written there but formally right this way so you take this monomial transformation and then most importantly you conjugate this by the quantum dialog in power series and the remarkable fact that yes you get rational transformation so let's do example which shows how this works so we take quiver which looks this way so it going from two it's two and one are the two vertices and then we do mutation as a vertex one so we get this kind of quiver so it's still one and two and so here's a formulas I just writing to make sure that I use correct conventions and so here the formula x1 prime x2 prime equals q square x2 prime x1 prime so that's the input now we need to make some transformation this means that we have a this means that the this Bollinger form between this and this equals to one and the other way around the other way around it depends on skew symmetrisers but in this particular example equals to minus one because I happened in this case I consider skew symmetric skew symmetric situation so this skew symmetrisers the important only here so this form is skew symmetric by definition and this one not necessarily but in the example it is skew symmetric okay so that's why in this way everything depends on the original form because when you define the mutation you see that this formula depends on the original form that's the key point on the other hand the quantum torus algebra depends only on the skew symmetrius variant of this form so now how this transformation looks like so it's e1 prime equals minus e1 and e2 prime equals e1 plus e2 now we have to do the quantum mutation and it works as follows so we take x1 prime and we map it to psi q of x1 x1 inverse psi q of x1 inverse and of course nothing happens because this is just x1 inverse because all this all this x1 but when you do transformation with x2 prime something is going to happen because you have psi q of x1 and then you have to put the corresponding the the generator corresponding to this vector which is actually this one and then you have to write down psi q of x1 inverse and so the claim is that if you do a calculation and I'm going to do this in a second then you end up with a formula sorry where is the skew inverse coming from who in x1 this is an oh I mean if you say look at this formula as a definition so it tells you that if you take x e1 and multiply it by x e2 you're supposed to have and you're supposed to have this expression now 1 2 is minus 1 so you're supposed to have q inverse x e1 plus e2 this thing in the middle yeah so it's I think I think it's supposed to be oh yeah yeah so it's probably supposed to be q but then that so this tells me let me see again so it's 1 2 so it's minus 1 what I know it should be q so maybe I'll do it myself just a second okay so so so in any case so if you apply this transformation so what's going what you're going to do you're going to take this term let's just not worry how we normalize the definitions if you normalize this way so it's fine so we just wanted to this term going through this way and so in the process of doing this the key point is that when you move it through here you're going to pick up here something which will be psi q of q minus square x1 and then this moves here but then there is a relation that this is 1 plus q inverse x inverse psi q of x1 which is the defining relation for the quantum dialog and because of that you will see that this guy and this guy is going to be cancelled in the end and so you end up with a rational expression so you can calculate which one but you'll get what what I promised so when q equals to 1 you get this class transformations we were talking about that's it so this shows how the transformation works at one mutation now what are cluster transformations so and again so the main kind of point here the main remarkable thing which is happening that you do aftomorphism of the fraction field of the quantum torus algebra and like for the simple algebras like matrix algebras we know that any aftomorphism is inner but this is not quite inner aftomorphism it's inner and only in some strange sense because it's a conjugation so it's kind of inner but it's a conjugation by the element which does not belong to your algebra so it's not inner but nevertheless the result is still aftomorphism so it's a kind of new way to produce aftomorphisms of quantum torus algebra and I claim that they appear here because basically there are no other ways to to produced in a reasonable way so I don't want to formulate statement now but if you try to construct any aftomorphism quantum torus algebra which preserves of course the algebra structure and then some dual data which is some class in k2 then you bound to consider just composition of this transformations like conjugations maybe by powers of this guy you can take psi q of any element of your quantum torus algebra like any monomial raised to any power and consider any kind of composition you still have rational transformations and the ideological that's all we can get and so that's no surprise that you got it because it's probably there's nothing else but on the other hand yes so this is the main example and kind of the main kind of miracle of the story is that you write down the classical transformations and you see it's a little ugly when you go to the quantum you immediately see that it has much more sense and then you write down classical by going to quasi classical limit and then you understand it better I didn't I didn't do calculation uh to the end I just said that if you just take this guy and move through this one here then this and this not commute they q commute and so if you move any function of x1 through this expression the what's going to happen this is going to move through but here you'll get some coefficient of q which depends how this and this guy q commute okay after that you'll have this moved here and this moves there but then you use this difference relation okay and difference relation will tell you that the transcendental part I mean power series part of this formula will be cancelled and you get just a rational formula so this is the so to speak kind of main point the main kind of essence of what's going on okay now then we can just compose them let's say quantum cluster transformations so we we just want to define them to book to make a bookkeeping of notation so we have some uh quiver c0 then we mutate at the direction k1 and we have a new quiver c1 then we mutate in the direction k2 and have q or c2 mutate direction k3 get something in the end of the day we get to the quiver cm and we arrive to it by doing mutations direction km so this is arbitrary composition of this mutations and so let's call this i and so now we define the corresponding cluster transformation corresponding to this sequence just as a composition so it's my phi from c0 to c1 with this k1 composition so on composed with phi of cm minus 1 to cm or km and I already defined those so that's a definition but if you think about this definition this is conjugation by quantum dialogue monomial transformation conjugation by quantum dialogue and again monomial transformation conjugation by quantum dialogue and so on but we can rewrite this slightly as doing composition first that's a claim monomial transformation first and then conjugation uh second so the claim is that phi of i can be written as follows conjugation psi q of f of x f1 conjugation psi q of fm composed with some monomial transformation which I write as i of related to the sequence so the only question which remains here of course who is this monomial transformation that's easy but also who are those vectors because as I said we can apply a conjugation by quantum dialogue and for any vectors of the letters and so now I just have to specify these vectors which make this true and this is more convenient because now we have compositions of conjugations just collected together at once and okay so the rule for those those vectors is the following so we have the original basis vectors this in the square and then we do this mutation and then we get a new collection of vectors and then we again do mutation and get another collection of vectors and so on in the end of the day we get something like that i m and so uh the rule the the statement is that this vector f sub s the one which you see here is nothing else but the vector e chi s at the moment of time s minus one so you have this you start with the original basis then you transform it transform it transform and transform it and then you look at some particular moment of time s you look at the this this transformed basis and you take the basis vector actually this one is s minus one so it's moment of it's after s minus one mutations and you look basis vectors which parameterized by the index k s which comes to the next step of this mutation process and this transformation this is just a composition of those a monomial transformations related to each of the individual steps the one which written here okay so that's it so that's the kind of so phi of c i to c j k i is a formula that you wrote behind the board so so this this phi of c to c one k one this is a formula you just wrote behind this board let me repeat so so i'm saying that by definition five i is this an eggplant that actually you can write it this way what's the difference the difference is that this is conjugation monomial conjugation monomial transformation so on now i pull back on monomial transformations to the right hand side all conjugation to the left is just technically more convenient well my question was difference just the definition of this phi is just behind the board right this yes okay this yes okay okay so now you wanted to come to the part which is a quantization real quantization of sorry let me finish first full with algebra so okay so now we have many many quantum tori and this quantum tori so you start with the original one it's your we were c then you can do mutations in all n directions in all unfrozen directions you get new rivers and so you get some other quantum tori sitting here original ones here then you do mutations again you get new quantum torus algebras but actually you may just think that this is the same quantum torus algebra because the letters didn't change but you just pick up different kind of generators of this algebra so you get infinite collection of this quantum torus algebra but in order to get the definition of cluster post on variety quantum cluster post on variety you have to identify some of them because it may happen that if you start with some original quiver c and you go to some c tilde it may happen that this quivers are actually isomorphic as quivers then you can identify the corresponding quantum torus algebra so you can take oq of t c tilde and just identify it with oq of c in the way provided by this identification of bases like you take a basis you mutate mutate and take a different basis and now this basis has the same brackets it gives the same quiver means it has the same as the form has the same brackets as here so then there is an isomorphism of this algebras and so you can compose your composition of mutations in this one so in depth is some transformation okay and so this is just some aftermorphism of this original algebra but it may happen that this aftermorphism is trivial and so this way you get this trivial cluster transformations but if it's not trivial there's some aftermorphism of your original of your original data and so if it's not trivial then you get element of the cluster model group that's a definition but if it's trivial it's trivial so we just don't count I'll give you an example in a second so so this way if you want to do this in the quasi classical level when q actually 1 you specialize q equals to 1 then this way you get some split tori many of them and you just kind of birationally identify them but then it may happen that after some process of of this biration transformations you get to torus which is as Poisson torus as a morphic to this one then you also identify so this is the notion of a cluster Poisson variety and let me just give you an example how this works in the classical level you can just take a pentagon and you can put points here we discuss this example and then for whatever two diagonals of this pentagon let's call this diagonals like e1 and e2 we can construct the corresponding coordinates like x e1 it's a general definition so you take this four points and you consider the cross ratio of point x1 in this case x2 x3 and x4 and x e2 is going to be cross ratio of x1 x3 x4 and x5 okay and then you generate let's say quantum Poisson torus or just usual Poisson torus using these two generators you get one chart but you can go to a different chart if you consider different triangulation like this one then you get also a pair of coordinates which related to this ones using precisely the transformation we were talking about before actually just this example but then you can mutate it one more time and this time you get this pair of coordinates then you can mutate it one more time and this time you get something like this and then finally in the end of the day you return to the previous picture let me erase this definition because we had it before and so you get the fifth pentagon and so this is so it now this way and then you mutate it here you get this one so if you can consider a composition of five transformations just formal transformational coordinates then you will see that the result will be an identity map and this is consistent with the fact that this formal transformations they're not just form transformations but they record how this coordinates how this coordinate transform via this coordinate so geometrical it's clear that if you consider this composition five times you get an identity map but from the formal point of view what you do you consider the original quiver then you mutate you can mutate in two directions and so on so far but the point is that if you consider this kind of sequence of mutations and you also apply some you you relabel this variables each time then if you do this then you get precisely the original pair if you don't relabel you'll have to do this twice so it's a composition of this class transformation and relabeling of the variables and so that's a typical situation how we get the make the the cluster model group so in this case this map is element in this case cluster modeler group is isomorphic to z mode 5z and it's generated by by this change of variables combined with renaming first diagonal as a second second as a first but when you consider five transformations like that you get identity so the mapping class group is z mode 5z and so that you can kind of adjust to the general definition just keeping this example in mind okay so and if you want to get cluster Poisson variety you just glue five c squares using these transformations and you get some two-dimensional space all right so but what do you want it to do why we like this structure because it's because we want to quantize it and quantize I would say in real sense I mean quantizing in the sense of producing operators in Hilbert space and so on so far and so let's just move into this direction to explain what we are going to do just remind you that on the first lecture we were talking about quantization of cluster Poisson Tori and we ended up is understanding that you can assign to each quantum Poisson Tori functions and l2 of r with the action of the corresponding quantum torus actually it's modular double and so we had five unity operators which actually the same kind of operator and it's the main statement that its fifth power equals to some constant times the identity map so but what we did we started with this cluster Poisson Torus quantizes using Baylor representation go to Hilbert space and then what we are going to do now I said we did but we didn't we want to construct some unitary operator which relates these Hilbert spaces this is this so constructing this Hilbert space this is as I explained just Baylor representation construction but constructing this unitary operator this is a key point this is a kind of non-linear value presentation which we wanted to get so let me get to this point okay so after this kind of preparation let me do it and so the first question is so how to proceed there so let me recall what we did basically on the first lecture that you have this Heisenberg group related to lattice and and the form and actually I now wanted to consider the Heisenberg Lie algebra so this is for now is Heisenberg Lie algebra and so what is Heisenberg Lie algebra its extension lambda Heisenberg lambda and then here you have just some central element and so you can tensor this with R to get vector spaces now to any but I really need just the lattice to any vector P in this Heisenberg algebra so in lambda I can associate its lift to Heisenberg Lie algebra and then the standard Heisenberg formula that this lifts commute now non-trivally it's 2 by i by 1 by 2 times this identity so that's a recollection about the Heisenberg Lie algebras but then what we did we said that we wanted to consider the quantum torus algebra related to t lambda we wanted to tether it with the quantum torus algebra for q check related to t lambda and I remind you that this q now such as question now it's a number i pi h and q check is exponent i pi divided by h so we have this tensor product q and q check are numbers now and so we call this a sub h of t lambda or just a sub h and then this a sub h is generated you can say is generated by the two pairs of variables it's variable x sub p p is the element of your lattice which is just exponent of beta p hat and y sub p is just exponent of beta divide sorry p p hat divided by v so as we did in the first lecture you can check that these two operators commute with between them I mean x's commutes with y's but they do not commute between themselves so this gives you this generates or q of t lambda and this beta is a number sorry I said this in the first lecture but right now just some complex number okay and so and this generate or q check of t lambda and beta related to h by the formula that h is beta squared okay okay so this modular quantum modular double algebra you can think about this way and so now if you have any representation of the Heisenberg algebra you go to representation of your of your modular double just by those formulas okay sorry it seems no q t lambda you just had generators corresponding to lambda now you rescale them by complex numbers yes I rescale them by complex numbers and they still correspond they still corresponds to some any vector of lambda but now they they commute by q to something so you fix better yes yes yes I fix better better is a given number yes yes yes yes yes in the first lecture lambda was the c2 and then the two generators of this quantum source one of them was a translation and one of them was okay okay so so what gel said correctly that in the first correct in the first lecture lambda was just c2 so this was an example now lambda is anything secondly it's a very good comment so on the first lecture we had a very concrete way to write down the Heisenberg representation it was translation by 2 pi i and multiplication by exponent of t and then when we do the strict multiply by beta and divide by beta we get translation by 2 pi i beta and 2 pi divided by beta and multiplication by e to t beta and e to t divided by beta that's exactly the formulas we had so they depend on the original input which is which of the realization of the Heisenberg representation you take okay and so let me just repeat what i just said because just need this for keeping so let me remind you how the Heisenberg representation works so let's assume first this form is not degenerate then we pick some decomposition of lambda into some l plus l prime but this guy suggests Lagrangian subspaces for the form and then this allows us to construct star representation of the algebra we occur about a h of t lambda and it acts in l2 of this l tensor r in this vector space it's just half of the lattice and it acts in a very standard way so vector l in this l goes for t to 2 pi i l and then we are scaled by t to 2 pi i beta l or t to 2 pi i divided by beta l if you wanted to go through the presentation of h but the the original thing is how you construct the Heisenberg representation this way and then if you take any vector called l star from the second Lagrangian subspace then it goes to a multiplication by so you can consider the corresponding linear functional and if you construct the group representation they take it exponents so so anyway so this way you get the presentation of the Heisenberg Lie algebra okay so what's important to take home is that it depends on the choice of polarization choice of the decomposition in l and l prime and when you change you have to do some transformation and Andrei well explained how to do that so we got this representation of h in general and now in the case usually because this is more or less what is usually considered in theory of non-commutative theory right this is this bi-module of one torus and other torus k is by meter per meter h i don't know what means usually but this is the representation of this a h of the model or double and that's what it is it's it's a well known construction so in general i wanted to say that we have now kernel of our balanar form and therefore we get a projection from the Poisson torus to the one related to the kernel and so this describes the Poisson structure the fibers are symplectic leaves and so now we can write down that h lambda now in this case will be semi-forms on this t lambda zero r plus these values in Hilbert space which we already defined which is as the one assigned to the fiber if you call this map mu let's say mu inverse of zero so we get just semi-forms the base as well as in the Hilbert space which you see fiber wise and then the main claim is that this h lambda is of course integral of some more elementary Hilbert spaces over lambda and the important thing here is that lambda belongs only to points of this torus as well as in positive numbers that's the spectral decomposition that's the classical stuff okay now we've done again so we assigned to to the quiver some Hilbert space and we understand that this means that this Hilbert says actually depends how we realize this how we take this decomposition l and on prime and if you take a different one you have to put intertwiner into the game so when i said that any transformation is obtained by conjugation by quantum dialogues i mean quantum dialogues they're powers or monomial transformations so this is what you do if you have monomial transformation now let's do the main step so far it was rather trivial preparations a definition we wanted to say how we relate these Hilbert spaces if they're assigned to different quivers and so given a mutation mutation from c to c prime in the direction k we define an intertwiner and so this is an operator k which you know from c to c prime somehow relates to c and c prime i would say and this k and so we define this as a composition of two just as we did with the notion of classical transformation so it's a composition of two operators and they go from the Hilbert space which we just assigned to c prime following Heisenberg and Veil to Hilbert space assigned to c this is the main point so we have now two different quivers and as i related by mutation and we want to construct unit transformation which identifies these Hilbert spaces canonically in a way that it's in some sense has to be explained commutes with the algebras acting here and here equivalent on the section of those algebras so okay so here is the definition which will be a little bit cryptic because i'll have to define the main player and so you say the following that okay you have the pk hat this is Heisenberg operator which corresponds to a base vector ek we do mutations the base vector ek and so before when we're doing algebra you're saying okay let's take psi q of x ek and let's conjugate by this now we cannot do something like that because the problem is that this series is divergent if absolute value of q i would say divergent unless absolute value of q is less than one and if you remember we actually wanted to have story which makes sense when absolute value of q equals to one so we cannot use at all this transformation it doesn't make sense to us when q is actual number unitary number and so what we say we say that we apply some function which depends on beta to this operator which is actually self adjoint operator under some assumptions if beta is real or absolute value of beta is one so we will talk about this later on but we wanted we wanted to find some nice function which replaces the quantum and which is going to give us maybe I'll erase this now give us a map from h c to h c and I still have to tell you what this function is the main point and the other part of the story is completely trivial so the second operator is the quantization of the monomial part of the transformation and this is just an isomorphism h c prime to h c isomorphism induced by basis mutation h Hilbert space related to c is a Hilbert space which we assign following Heisenberg and under a veil to the quiver q using some polarization some decomposition so you have to understand that this is not just some l2 it's many many l2 related by canonical isomorphism which not completely canonical if you you know it's a metaplectic representation not symplectic but we keep this as a block and say okay this is a Hilbert space now this is entirely different story it's entirely different Hilbert space which corresponds to nonlinear change of variables and so what we want to do we want to quantize this nonlinear change of variables and the main point is that yes we can do it how we can do it so the first attempt would be to use this function quantum dialog which we use for conjugation but this cannot work so we have to end up with some more clever device how to do this and here's how it goes so that's where actually the quantum dialog actually appears the the actual one i would say noted power series version so we consider the following function so phi beta of z is defined as follows so we take the power series psi q of e to beta z and divided by psi 1 over q check it's a modular transformation over exponent of z divided by beta and now you can complain that this makes sense only when absolute value of q is less than 1 because this makes sense only when absolute value of q less than 1 and this makes sense only if absolute value of q less than 1 but remarkably if you take the ratio then you can write this as exponent of some integral and it makes sense for any q this is integral over zero of e to phi z divided by sinus hyperbolic of pi p beta inverse and sinus hyperbolic of pi b z dp over p and so this function is well defined for example it's well defined if beta is real or beta has absolute value one and that's where the exponent will be has best absolute value one or even something else so it's actually defined for all betas and so this is the function we wanted to play with so this is the quantum dialog and it has the property that if basically that phi beta of z bar is phi beta of z bar but basically what you wanted to say you don't even need this so we need just to say that if beta is a real number or absolute beta value of beta is one let's say that for now beta is not plus minus i then the absolute value of phi beta of z is just one this means that in the regime we really care about for example beta is real means that h is positive so our functions not only well defined but it's also unitary it takes values of absolute the numbers of absolute value one z so far z can be anywhere as you can see in this formula but in a moment we are going to substitute the self-adjoint operator into this formula and then if beta is real it will be real if okay so we just going to use this formula so but pk is a self-adjoint operator under the assumptions that's exactly where the assumptions we're coming from is it beta z or beta p in the denominator it's p p p of course i'm sorry thank you Giovanni there is no there is no z involved in any kind of integration just stands as a parameter integration over p thank you and this function has beautiful properties that it does satisfy two different equations you can shift by 2 pi beta or as usual you can shift by 2 pi i divided by beta and in this case you get 1 plus q a check exponent of z divided by beta multiplied by phi beta of z so that's the key point about these functions unitary when when we need it and it satisfies two uh different equations related to beta and beta inverse all right so after this we are basically done because after this uh my uh intertwiner is defined so i can define for any mutation and more generally i can compose them so wait so why is it an intertwiner i didn't say in what sense it's an intertwiner i said this in the first lecture but didn't say this right now it's actually uh very uh i thought with respect to this algebra a right yeah yeah just a second just a second just a second so i'm just saying that i got an i got an operator okay now this operator in some sense quantizes nonlinear transformation and so in what sense it's quantizes nonlinear transformation so uh but before it goes there i just want to formulate a statement so uh so let me just do it so so more generally uh any cluster was on transformation um i'll be called uh i don't know we use already the c that's called kappa from c to c prime so it's a composition of many mutations uh gives rise to the corresponding uh intertwiner which goes from a Hilbert space assigned to c prime to Hilbert space assigned to c so we just compose those elementary transformations so when you define k prime do you kind of have some compatible polarizations no no no i i don't have to so this is this is taking care of by previous discussion once again if i have a quantum torus which mean lettuce with bilinear form let's say symplectic then there are many many many infinitely many Hilbert space i can produce they correspond to polarizations okay if you choose one of them here and one of them there i produce some unity operator if you choose a different one here i will precompose it with this isomorphism and then uh use the isomorphism which i have so i have infinitely many operators so to speak which each corresponds to a pair of Lagrangians here for each choice of yes and i don't say this absolutely and i don't say this because the presentation will be too heavy i'm just saying that you have to understand that the Hilbert space means not one concrete Hilbert space but infinitely many of them and they are laid by canonical unit transformations and for any choice of this one and this one there is a unit transformation and they're compatible i just said but uh so what this uh transformation does uh it tells you that uh basically uh you have some monomial transformation and so you have this some monomial transformation and if you happen to write down the first one in using certain bases then you kind of tilt it after this so you have to do something about it you have to use in situating the operators of for the representation yes that's what it does yeah yes yes yes yes yes yes yes exactly so that's very uh trivial and this is the only point okay now what is the statement uh so we wanted to say that this uh uh unitary operator commutes with the representations of those algebras but this uh statement on the nose doesn't make any sense whatsoever because uh this uh this operates this our algebra x by non-bounded operators and you can't really say that i mean this is kind of warning if you have some transformation like you have x and y and x goes to let's say x inverse and y goes something like y times one plus qx inverse inverse something like that so and you call this x prime and y prime so what do you want it to say you want to say that when we that uh we have this unitary transformation and so if you apply x first and k second then you get x you know call it x y let's do it with y then you get y prime k you want to say that we have this formula but this formula makes no sense because you have to deal with this inverse of one plus x of something you can define this operator no problem but you in order to make this formula to be true you have to specify some vectors in your Hilbert space on which this operator can be applied and after this unitary transformation this operator can be applied and that's of course a very complicated issue and so is that not visible how to do this and so if you just say this is true for any vectors in the Hilbert space this is plain wrong because this makes no sense this makes no sense so we have to worry about some domain where all these operators are actually defined and so here's what is going on that's there's a theorem it's my theorem that first of all before I go there let's make notation so x is this original cluster cosine variety and we created this quantum model or double I remind you is tensor product of oq x tensor oq of x actually have to take language here and what is this guy I didn't quite define this so I need to do this now so definition we define this oq of x as follows we say that f belongs to oq of x if for any cluster cosine coordinate system c which comes with coordinates x1 and so on xnc this f is a low run polynomial in this variables x1 and so on xn with coefficient in z of qq inverse you can put actually positive integers if you want you can consider variance but the most important property is that you do the following so you have your cluster Poisson guy and so it glues from infinitely mandatory or finely mandatory you take one of the torus it's a quantum torus and you consider the algebra of laurant polynomials in this quantum torus very good then I go to a different coordinate system and some of those laurant polynomials may become not laurant polynomials then you throw them away and you keep only those which remain laurant polynomials after the first mutation and after the second and after all mutations and so our priority is not clear that you have a single element except constant in this algebra it's very easy to see that at least the center of this algebra is there as if you have the the center of the formal lambda then you can get some center of this algebra but if lambda is not degenerate nobody our priority guarantees you that you have elements because you have infinitely many conditions on that that's number one but we have to deal with this algebra and we take it quantum model or double and then this is kitchen oh I didn't write this is kitchen yes thank you so extra is a language dual quiver I didn't tell you what it is and so let me not do this for now okay but it's important here but let me skip it so hmm so first of all there is a subspace which is called s sub c in the field of space hc for each quiver c such that the algebra a h or acts on the subspace so for each quiver I can find some domain of definition for all those operators which I care about which forms the algebra the model or double the algebra functions on my cluster post on variety so again I define this guy this algebra I didn't as I said it could be very small but still I define it and now I consider I claim that there are some natural subspaces in my Hilbert spaces we're all unbounded operators from this algebra can be actually applied and the result will be still sitting in the l2 which do not take you outside of l2 and so and now the main statement is that the intertwiners kc0 induces an isomorphism of these schwarz type spaces there are some conditions so first of all is definitely this subspace is definitely the definitely dense with respect to the Hilbert metric here in in this space so it's big enough that's number one but more than that I'll I'm about to go to there wait a second so I'm going to list the properties there are these subspaces with some with some properties and so so first of all the algebra x you can add it as c is dense in hc this space has a definition it's not just some subspace this is the domain of the definition for this algebra it consists of all vectors in the Hilbert space this has the properties that if if you apply that that you can apply a prater sumi algebra to this vector so you're still sitting in the Hilbert space okay it's like if you consider instead of the algebra a h if you consider the algebra of polynomial differential operators that's the definition of the schwarz space it's all infinitely smooth functions which decay faster than any polynomial smooth vectors yes yes yes this means that you can apply any element of this algebra and you're still in l2 so that's the definition of the space so this is the space the claim is that it big enough that's the dense and that intertwiner we talked about this there oh it was kappa sorry thanks kappa is any class of transformation that one so it defines a map from the schwarz type space assigned to c prime to schwarz time space to c and that's an isomorphism and the main point is in what sense it commutes that if you take this cup k of kappa zero and you take some vector let's call vector s in the schwarz space and you apply some element of your algebra this belongs to a sub h of x then what you get is k kappa zero of s applied to gamma c zero of a acting here and so now I have to put quantors so first of all this is true for any for any a in h and gamma is the action of minor linear transformation so this is a basically the cluster transformation and so the statement is it knows that they're literally commutes you know that you can pull a here but when you pull it you're picking up it's it's intertwined by the action of the of the of the quantum cluster transformation so that's the main point so it commutes in this sense so I emphasize once again that this is definitely what we wanted to have but we absolutely cannot have it and absolutely cannot state it in the situation when first of all s is any vector from h to it makes no sense secondly we have to be very carefully to put commutation commuting conditions intertwining conditions for which element of the algebra so if we consider not algebra a h which is basically the algebra of this universal Laurent polynomials but if we put into the game any kind of element which has denominator we are immediately dead so we cannot do anything we cannot prove anything there's no statement there's nothing so the only algebra which allows us to put this condition is the algebra of this universally Laurent polynomials which is the algebra of regular functions this quantum cluster variety and that's the only kind of conditions we can write down so once and once again we absolutely cannot write down this condition if you have some kind of transformation which has denominators and so the literature is full of statements that we impose something like that condition and prove wonderful theorems and they this is completely wrong and has to be unfortunately ignored we assume what better we had before is real right real or absolutely better is one yes this was who are the you said that some of these operators they're all unitary this is this case they're all unitary if better is has absolute value one or better is real and no they're self-adjoint yes cluster variables are self-adjoint and the intertwiner say unity operators which transform you know this non-linear action but emphasize again you absolutely cannot say that you take a cluster there is this cluster variable you absolutely cannot say that your transformation commutes with the action of the cluster variable because most of the cluster variables has a property that if you apply any class transformation to them then you get denominators like 1 plus y in the denominator as soon as you get denominators you cannot say anything you cannot prove anything there is no statements so this means that you don't really make claims about cluster variables you make claims about some Laurent polynomials which turns out to be regular functions on your manifolds but your variables very rarely belong to this class and so that's it this is this is the correct way to state this and so this is a theorem this is a non-trivial theorem I mean it's non-trivial so I didn't actually even formulate it completely so this is the center twiners and so all this is equivariant with respect to the cluster modular group now I want to explain what do I mean here so if you happen to have some sequence of cluster transformations like here you have some sequence of a cluster transformations which is the same cluster transformation basically with this little symmetry such that if you take it five times you get the identity transformation then a priori nobody guarantees that if you apply the sequence of your quantized unit transformations you get identity or actually multiple of the identities scalar multiple of the identity it shouldn't be identity so nobody guarantees this and it's a quite quite very non-trivial theorem that actually this does happen and so it doesn't follow immediately from any kind of properties that it commutes with the action of the algebra you have to be careful again here but it's true it's a non-trivial fact and proving it it indeed uses full force of this you know working with these schwarz spaces and this algebra and so on so far and so in the end of the day we do get the statement that it is equivariant on the section of the mapping class group and what it really means that if you happen to have sequence of cluster transformations which is trivial sequence of quantized unit class transformations which sequence of this unitary intertwiners will be almost thrilled will be a multiple of identity operator so this is kind of generalized pentagon relation and this is a very essential part of this theory so now that's it so I stated the main result so I explained what is cluster Poisson variety so how it can be quantized how it leads to some Hilbert spaces related by this unitary intertwiners that we get the schwarz spaces and one more comment here that this s sub c has a natural structure of so-called free space which means in simple words that it has different topology we don't use topology on s which is induced from the from the topology on the Hilbert space it's like in the usual schwarz space when we consider generalized functions we don't take functions which are continuous with respect to l2 norm we take functions which are you know convergent with all derivatives and so on so far so we have to do it here and the correct way is to say that the all the semi-norms this is Hilbert norm and this a belongs to a sub h of x so we have to consider all the semi-norms and the topology induced by the semi-norms that's what free space means so this is topology like in the space of classical schwarz space but now it uses our algebra algebra of regular functions I insist again on this cluster Poisson variety which is analog of the algebra of polynomial differential operators on the line and we can now define the space of distributions yeah it's it's it's part of that but yes but actually it even if oq of x is small there is some kind of tricks how to make all this true it's true for any cluster Poisson variety in spite of the fact that in some exceptional cases it could be smaller than you might think so it's a good point but so now we we can introduce a space as dual which is just the continuous dual to s for its fresh out apology okay so now we basically constructed our main working horse this is the quantized class of Poisson varieties and the rest of the lecture select today in the next lecture is kind of downhill we're going to have all the applications because now I got the data which was announced in the first lecture I said that we are going to have triple of spaces so now we have them and we're going to have mapping class group acting on them and we're going to get this algebra also acting on them now we have all this data which was in the first lecture so I'm going to do a break now like for I don't know 10 minutes and after this after the break I'm going to talk about the main example the I'm going to explain that the model space of this this model space we introduce which relates to a group G split adjoint semi-simple group G and any decorator surface does have this structure and therefore is quantizable and therefore produce all this kind of representation so I'm going to remind you what the space is and then explain how to construct this structure there and therefore quantize and get all the applications which we'll discuss today in the next lecture so once again so if you some kind of vanget lack or oversleep for another legitimate reasons all this discussion that's completely okay so what you have to take out from here that there exists a way to quantize it and we are going to use it and we are going to produce on kind of model spaces we are looking for we are going to produce the structures and therefore use this kind of device quantize okay two three fifty on my break this valodi fork and it's long ago it's uh two zero zero three and complete some one more paper two zero zero seven this is minus part absolute beta equal one story but okay I mean everything else is is from there now now I emphasize this is joint work with link we shan from unismich and glancing and uh this is reason so this is like 19 something in the archive 19 or something okay so let me proceed to this so what are the key features the first question is what are the key features of the modeling space pgs and again there are some new listeners here so I will briefly remind you what we're talking about so we have a decorated surface which means that we have a surface which has boundary and on each boundary component by default we have at least one special point maybe more but finally many and in addition to this we have punctures and they they together we can call them mark points and so here is the surface now we put some data so first of all so we have this model space pgs so it parameterizes remind you this pgs parameterizes as triples l beta p and so l is just g local system as usual on the surface s so it's 2d data so I mean it's better to think about this as a kind of data given to you in 2d 1d and 0d so in 2d you have topological data g local system now this 1d is what you see between these red points and you see it's on the boundary and this is this notion of pinning I'll remind you what this is in a second but you put on boundary intervals some kind of data and 0d data you put at punctures and at red points so 0d data is just what was called previously framing which means invariant flag near mark points I kind of very quickly write this down but remind you that you have to have you have to restrict your local system in the neighborhood of the puncture of the puncture and you have to consider the corresponding local system of flags and take invariant section you can just say that this is just invariant flag this means if you go monodrama you get the same flag and this is for generic local system just w n factorial for pglm choices so when you take local system you exclude the punctures yes and another date of this framing means that you put some flags here like b1 b2 so you put some flags there but besides these flags this spinning which is the main kind of new hero you put here pinning means the following that here you put some decorated flag let's call a1 and a2 such that the so-called h distance between them is one so I explained in the previous lectures what h distance is and the point is that there is a map h from the space a cross a divided by g to the carton group and so if you take this h map for the pair of decorated flags sitting there it's one and the group g is adjoint because no center center of g is trivial no center means trivial center so that's the data so is that clear or should I repeat what is h little h carton z carton it's just name for the map what so I mean I define this map I'm not going to do it but you can okay so you can define this very quickly you can just say that this of course the same as g divided by u and u and this is and use bryad decomposition and you write down this w h u and u and this is h to take and this is some specific leaf to the very group to the group and so on some natural invariant which is canonical and for the adjoint group symmetric respect to the permutation all right yes yes yes so the point is okay let's look what do we have for the triangle we will have this anyway so let's just repeat for the triangle we have no local system whatsoever it's trivial but we still have flags and so we have to put three flags which means three baryad subgroups here b1 b2 and b3 and on the top of that you have to put this six integrated flags so we can call them a1 minus a1 plus a2 minus a2 plus oh sorry it's a3 and a2 minus a2 plus so we have six decorated flags but there is a condition on them then of course these flags are just lifts of this leg b2 they project to it okay there is a canonical projection from g mod u which is called the principal affine space to g mod b which calls the flag variety and so these two elements of principal affine space go here these guys go here and so on but what is most important for us is that there is this h distance here and it's one here's also one and here's also one this is z the definition that you use and this is the correct definition okay all right so now what is the property of this modulus spaces so the main thing is that there are many functions and many groups which act in this modulus space and so let me prepare the soil for the introduction so first of all we have z carton group and we also consider it's some kind of quotient which is z carton group divided we take coin variance of the involution which takes this to w0 of h inverse this is my h star secondly we have the braid group for the dinkin diagram of g and i hope you remember the definition so it's generated by a simple reflections as one so on sr which corresponds to simple roots so r is as many as elements as simple positive roots and it has braid relations namely you have s i s j equals s j s i is the carton matrix between this i and j is zero then you have s i s j s i equals s j s i s j if c i j equals c j y equals minus one and so on meaning that you have identity c i c j square equals c j c i square when you have two there more precisely if c i j equals two c j i equals minus two and similarly you have for g2 s i s j cube equals s j s i q okay there was this this group and now there is a map mu a satiristic map from the veil group to the braid group and it has a property that mu of s s prime is mu of s times mu s prime if the length of s s prime is the sum of the lengths of s and s prime and now the important first definition is that we have some kind of variant of the braid group b g star which consists of all elements let's call this x of the braid group which has a property that they commute with their mu of longest element w zero all right right and so now we can use this in order to introduce the following notation so for each boundary component pi of s we have the corresponding flavor of the braid group so i call b j pi so this is you can write this as a single formula this is okay let me just write it without deciphering what it is it's sometimes b g is a whole braid group and sometimes the centralizer of mu of w zero and so this is the case if the number of marked points is even and this is the case if the number of marked points is odd all right this is the flavor of the braid group assigned to the boundary component and also there is the flavor of the carton group which is h divided by this kind of involution but taken to the power given by the number of special points so this is either h or h star the h star is the one which i define there and so if h is dp is even and as h stars dp is odd all right that's the data which you assign to the created surface braid group to each boundary component and also you have versions of the carton group now comes the main theorem which tells you what's the features of this model space pgs so for those people who were not here on the first and second lecture this model pgs is a kind of true analog of the model space of local systems on a surface which has punches and boundary points so if surface has no punches no boundary points just consider plain local system but if you do have some extra data in 1d on 2d we have to it's a good idea to move to this model space which i claim is the right choice so serum a says that the model space pgs has the following features well these features are kind of stratified by the geometry of the surface and so for each puncture p it has a full and extra data first of all there is w group acts on this model pgs birationally and secondly there is a canonical w equivariant map that's called mu sub p relate to the puncture from the model space pgs to the carton group now let me put here somehow something like constructions sorry near the punctures you have usual flag no no near the puncture you have the usual flags and near the red points the special points you have not only the usual flags but you have the left and right lift to the decorated flags and so if you do have a flag near the puncture and monodrama of your local system is generic then monodrama is a generic conjugacy class so you take regular conjugacy class and there are exactly w of borrel subgroups which contains this conjugacy class exactly w and so if you have and this space of borrel subgroups is the principal homogeneous w space that's where the action is coming so the action just changing from one borrel to the other okay now what is the canonical projection so this is a construction so a one a i claims that w acts on the set of borrel subgroup containing generic regular element that's a that's a just effect and now take the monodrama around the puncture p so then you get the data which you need to get the section of w by changing it's like you have eigenvalues and they're all different then you can just permute them that's very simple that's how w do you require the space that the monodrama is generic no no no no no i don't require but the generic point you have yeah yeah so so i'm just saying that i you you more you may concentrate to generic parts so that's how it acts on generic part now if you talk about local systems monodramae goes to h divided by w but in the presence of invariant flag it goes monodramae kind of corrected monodramae goes precisely to h so that's where this part comes from okay now the next thing is b sorry part two that for each special point s we have the following so first of all we have canonical potential functions and so this is a functions omega s i which corresponds to positive simple roots which are regular functions on this model space and so i is just the set of words so the dinking diagram and secondly we have a canonical projection i called rosa bass from the model space pgs to carton and so i wanted to stress from the very beginning that there is some kind of similarity between this data so it's one two one two projection to carton projection to carton here you don't see a priori any relation but it exists okay this is for the special points now for every boundary component what let's note that trivial to say this in one word so you will see it so for each boundary component of s we have the following data so first of all again there is a discrete symmetry so the corresponding variant of the braid group acts on pgs and b there is a projection we called mu related to boundary component from the model space pgs to the corresponding flavor of the carton group related to the sponsor so here we use this data which you see upstairs and again there is a parallel there is on the bottom you always have projection to carton and on the top of all this we have the following which kind of data for something d pi is just a number of special points of the boundary right d pi is just a number of special oh yes yes yes yes yes yes yes yes so it's not the end of the theorem just part four that the discrete group which i call the group gamma gs which by definition is the following group you take the group of auto aftomorphism of the group g it always has at least two elements which you can produce then you take a product with the mapping class group and then you take semi direct product of this guy with the following group you take the product of the weight groups over the punctures and the product of the appropriate flavor of the braid group over the boundary components and so the claim is that this group acts on pgs and now this is clear from what i said because i already introduced the action of the braid i already say that i have the action of the braid groups in part three veil groups in the part one i certainly have the action of the mapping class group because everything is natural with respect to aftomorphism of the topological surface and i definitely have aftomorphism of the group g acting on the whole story it's also clear so the only kind of extra data which the statement for adds that the action of the veil group are different punctures commute that's clear by definition that the action of the braid groups of the different components commute this is i didn't give you the definition so we cannot argue about this that the action of the veil group and braid group commute well they're far away punctures from special points and that basically the other groups commute with them to be precise the mapping class group can commute for example punctures and that's why i put the semi-direct product but other than that they're kind of independent so we have semi-direct product acting okay so we have this data and of course there is a main theorem about this the main theorem is that this model space has a cluster Poisson structure which has all these properties and all the symmetries but let me start with just a part of the main statement this is theorem b the moduli space pgs has this gamma gs equivalent cluster Poisson structure and this is of course true for any adjoined g and for any decorated s now this theorem immediately implies that this moduli space can be quantized because we quantized cluster Poisson varieties before and that this can be quantized in the equivalent way with respect to this big group of discrete symmetries of the space which is very important so why is this important yes so do you now have a better understanding of the cluster mapping class group of this object what do you mean better understanding well before you if we so you had this conjecture that you set some modularity properties for gluing that's a difference it's a different story it's not you can tie it to the mapping class group but it's not related to the theorem so we will talk about this but not at this moment so first of all i wanted to have an application sir do you need to have punches and come back like if you have a closed surface if i if i have closed surface then i get nothing there is no statement no nothing yes it has to be it has to be non-empty it's not actually true i say this emphatically nothing just to emphasize how the different situation is but actually it's not true that we get nothing we get a lot but this has a lot has to be obtained with a kind of with a big pain and with lots of work we get we get we get basically all we want and i'm trying to explain how we at least start to proceed so remember that in the first lecture the main point was that we want to quantize but we want to quantize the modulo space log gs of local system not anything else because this is really what is geometrically interesting i mean geometrically mean an applications or at least you might think this and so now the question is how we go from there to to the supplications so let me just make a kind of corrected definition so let's say that log gs is the moduli space of g local systems on s plus pinnings at boundary so this means that we parameterize the data which consists of g local system and the pinning and we have no framing so there is no framing in this definition so when i say local system this means that it's a puncture so i don't choose any other data and for example if i have a surface without boundaries and that's the classical space of local systems so now we can just say that we can define remember that in the first lecture i said that if you consider moduli space of g local system on a puncture surface we cannot approach it quantization right away because it could be non-rational so no coordinate could possibly exist and so but now we can because we just say by definition that oq of this log gs is by definition oq of p gs and then we take just w and invariance acting there so the point is that when we forget the framing it's like we go into the quotient by the action of the w to n and so that's indeed what happens and it happens the most kind of classical way just taking variance of the action of w on this on this algebra now the this extremely important point is that this theorem says that it's it's gamma gs if you vary in class percent variety which means in particular that so in particular means that this group gamma gs acts by cluster Poisson transformations i have to i don't want to do this right now but there are very few exceptions to this fact for example the case of puncture torus ssl2 has to be excluded if you talk about the action of the w group then the action of the w group is not exactly clusters there are so very few special cases which i don't want to bother you right now but so i just say it like that that that this group acts by class transformations because it acts by cluster transformations we can and we do this a little later we would be able to quantize log gs quantize in the you know actual sense because we would just consider the Hilbert space assigned to this space and just consider the actions there of the sinter twine in operators which just construct which correspond to w and and so this pair is the quantization of the model space of local system so it's it's like Hilbert space with additional data is the action of this group w so okay so now the next thing so so far i explained that at least one part of this cluster nature of this date is useful because the w action allows you to to to quantize in a strong sense the model space of usual local systems but before we proceed to any kind of quantization we need to ask that to address the main question what is the center of this algebra oq because that's what governs the series of representations if you're going to get and so that's the next part of the general data which we have so it's i would say this was a yeah so b is describing the center of oq of pgs and here i introduce so when i describe the discrete symmetries this was group gamma gs now i want to introduce the group hgs it's a torus which is given by the product of z carton groups over the punctures multiplied by the product of the carton group but actually flavors of the carton group related to the boundary components so now pi right so all boundary components so let's take this group so let me give you an example of how all this looks like this is going to be major example because this is the simplest example where you have all this kind of data so you can take a punctured disk you have punctured disk it has two special points and that's it now in this case if you're talking about this group then first of all so in this case the group hgs is what so first of all this is the carton which comes from the puncture and you can actually make it in two different colors if you want it so we have the carton group which comes from the puncture and we have the carton group which comes from the boundary they are very different so to speak in their nature and so that's the second factor red and the first factor kind of greenish okay now what's the statement the statement is that there exists canonical map to this hgs mu which is map from pgs to hgs now i have to break the lecture because now i remember something important which i forgot to tell you so the next lecture in july 12 i believe this is friday yes so the next lecture is eight nine yes the next lecture is not Wednesday it's friday okay there is a canonical map to this group which i write as a map which has two components the component related to the punctures and the component related to the boundary components and now we actually halfway to know this map because what is the map from pgs to the product of carton groups this comes from the data one b so this map it comes from one b part now how's the map mu p so the boundary components you see it in the part three and indeed there is something which corresponds to map to some kind of carton group so this is just the product and so this comes basically for the part one three b and so this is the where the projection to carton comes from and now the serum let's call serum c but first of all if you consider the pullback of the algebra of functions on this torus this is the center of the Poisson algebra of functions on pgs which i did not by the way introduce so we did not yet introduce the Poisson structure in this space because the very existence of this Poisson structure follows from serum b which says that the model space pgs has clustered Poisson structures it's Poisson but why it's Poisson I mean what kind of Poisson structure we still have to discuss so we don't yet know but still is going to be center and b it's of course quantum version that if you consider the same sub algebra this is the center of oq of pgs if q to n is not equal to one if q to n if q the roots of unity the center is huge and not just coming from this kind of torus related to gns but if it's not then this is the center and then you can ask the question what is the center of the local systems and for the local systems you can take u star of this o of hgs but you can put w invariance and this is the center of oq of log gs okay now we know the centers of all the spaces which means that when we go into quantize we know that the corresponding family of Hilbert and the related schwarz and space of distribution spaces the corresponding triples of spaces will be parametrized by the real real points of those torus that's the main corollary therefore the spaces of quantization are parametrized by either hgs of r plus or this hgs of r plus divided by the action of w to n but i insist that i'm stressed that it's parametrized not just by real points of the carton group they're parametrized by real positive points of the carton group all right so now let's look what do we have so we have we handle one we are not going to handle three today but i can probably explain where does the two coming from and so in the statement two there are two kind of statements of the first question is about this potential and the second statement is about the this projection to carton so let me explain the simpler one so where do we get the projection to carton so this will be 2b so so when we talk about 2b we take a part of the boundary where you see this red special point and then we have this extra data so we have two decorated flags like call them a minus and a plus sitting near this point and then this clearly means that there exists unique element of the carton group such that it is the ratio of these two decorated flags meaning the second is h times the first and so we just define this map rho sub s related to this special point s applied to the triple given by the local system framing and pinning to be just this h and so this is the first time you really see the application of this framing so if we didn't put the framing we would not have this projection to carton group here but since we do have framing it's completely obvious that we have projection to carton and it does play a major role in the description of the center and then the other thing and so maybe like I supposed to finish maybe in one minute what if I finish like in five minutes just thinking oh maybe maybe maybe I just maybe I just stop here oh no one thing yeah one more thing which I can say easily so what is about 3b so we had 2b so we define projection to carton but in 3b we have for every boundary component so for example you can consider boundary component like that as we did on the or you can consider a boundary component with two special points or with three special special points we still have to get to projection for the carton group and so for boundary component pi we define the projection mu sub pi to be the following one so we consider this element of the carton group which corresponds to the first point but then we have the second point the third point the fourth point and so on and so we take projection to carton for the first one and then multiply by projection to carton to the second one but with a star star is this evolution of the carton group then we multiply by the projection to the third one and then to projection to the fourth one star and if you happen to have odd number of points so this is if you have even number of points if you have odd number of points then we have rho s1 rho s2 star rho s3 but now it actually lives where it's supposed to live so it lives in h sub p in the in the space of covariance i mean in h we call h star this just lives in h so why is this is so you can argue in many ways but there is you have to start for some from some point and you can easily see that this definition will be okay this will not be quite okay if you project to carton so we are forced to do this so that's it so now we have one a b uh two b and three b uh done so we still have to do two a i will do this next time and and three a and i have to explain the the main representation theoretic and uh geometric applications of the quantization and uh in the end if i have time i will explain how i introduce the coordinates sexual takes not that long time the main theorem why is this i just want to stress one thing that uh it is not i mean it's not it takes not that much time to introduce coordinates using some data but uh it is quite difficult to prove that this coordinate systems they all sit together in this very strict way that they form plaster poston variety and the main difficulty of the statement lies on the boundary so if you would settle for the situation when we don't have a boundary that for example consider just triple of flags the proof will be much much much easier i mean basically quite easy but because for example this is the case of strangles basic case because we have to to handle the boundary uh as well the proof is much much more complicated because of that and so in a sense there really is a the whole difficult sits in the boundary and we actually that's what we use when we glue so when we glue you we really use the boundary components to glue and so i emphasize again and again that putting extra data to the boundary was key uh for for the whole thing was key for localization of the notion of local system key for introducing right notion making its work on triangle otherwise we cannot do this and it's a major problem in order to prove something okay so we'll do this next time that's it for today