 Okay so thank you very much, thank you very much to the organizers for the invitation to speak at this conference. So in this talk I will report on some work which is available on the archive at this number. So in this talk I will talk about the following topics and the relation between them. So in a very general way the first topic will be low-dimensional topology, the second topic will be complex animative algebraic geometry and the third topic will be string serializations of supersymmetric gauge series. So some physics topic. So on the following slides I have some cartoonish description of each of the topics. So in low-dimensional topology we'll be talking about knots, links and related object. In a complex animative algebraic geometry we'll be interested in counting configuration of geometric objects satisfying various constraints and here is a picture of one of the most well-known example of classical animative algebraic geometry going back to the middle of the 19th century the fact that there is 27 lines on a complex cubic surface in p3. And actually I mentioned this fact not only for historical purpose but in fact this talk will be about the cubic surface and the fact that there are 27 lines on the cubic surface in fact will appear later in the talk. And the last cartoon is the physics part of the story. So what we'll use is a fact that sometimes a given supersymmetric gauge series of different string serializations. And the point is that the string series often give geometric realizations of gauge theory dynamics and when you get different string serialization of the same theory you get a prediction for relation between the geometric at the geometric level. So in this talk we'll focus on a very very particular example so everything will be extremely concrete there will be nothing general and I will not try to address the question of how general it is. So the very specific example we'll focus in this talk so from the low-dimensional point of view low-dimensional topology point of view we'll talk about this kind of algebra of the four-point sphere so is S04 will be the notation for a sphere minus four points and we'll talk about knots and links in the three-manifold which is a product of the surface cross the interval open interval 01. The innumerative algebraic geometry part of the talk will be about counting olomorphic curves in complex cubic surfaces. And finally the physics part of the talk is about four-dimensional n-legal-2 supersymmetric SU2 gauge theory with four hypermultiplets in the fundamental representation. And so essentially the first part of this talk I will talk about essentially the low-dimensional topology story then the second half of the talk I will talk about the innumerative algebraic part of the story and some of the key points will be the relation between the two and maybe only at the very end I will make a few comments on the physics part of the story. And so the main point of the talk will be a connection between these two topics and this connection has some non-trial mathematical consequence once you know it you can prove some positivity conjecture about this kind of algebra which were done by people studying this kind of thing. And so essentially you can prove this result about the kind of algebra so some result about topology something about knots and links in the four-point transfer cross the interval and some of the abstract ways that you are able to prove such a thing by studying the up theory completely different topic of counting remand surface counting all morphic curves in a complex cubic surface. In this talk I will not focus on the applications to the scan algebra we will very focus on this connection between these two topics and once you have this connection then you can get non-trial applications out of it. Okay so here is a diagram so you are not supposed to understand what is in this diagram so the only thing is that on the right there will be the innumerative geometry on the left there will be the low-dimensional topology and on the bottom there is a physics and I will come back to this diagram at the very end of the talk. Okay so the plan is first I will talk about the low-dimensional topology part of the story and though the main character will be the cubic surface and some affine version some affine cubic surface which will appear as sl2c character variety and then I will talk about this quantization given by three-dimensional topology via the scan algebra. Then I will talk about the innumerative geometry part of the story which involves mirror symmetry and curve counting on cubic surfaces and then here we have a different quantization story which involves iogenous curve counting and the main point of the talk is a comparison between the quantization of these cubic surface geometry given by three-dimensional topology by the scan algebra and the one given by iogenous mirror symmetry so the one given by innumerative geometry. So iogenous means you are talking about this current the difference between the this kind of iogenous? Yes iogenous, let's say iogenous compact curve. Okay so let me start at the very beginning but I think very generally a few words about a character varieties. So if you have if you have gamma a finitely a generated group and g a reductive algebraic group of a complex numbers like glnc or slnc then you can construct a affine variety of group morphism from gamma to g and there is a natural action of g on these affine variety given by a conjugation action of g on representations of gamma in g and then you can take the quotient in the sense of geometric invariant theory so here is simply some affine quotient and you get something called the character variety so it's the affine variety whose ring of function is a space of function on this affine variety of representation of gamma in g and you take the g invariant part of this algebra function and you take spec of that you get some affine algebraic variety and so it will be particularly interesting in the case where gamma is a fundamental group of some interesting topological space and even more specifically we'll focus on the case where sigma is what I denote by sgl which is a topological surface which is a complement of l points in a generous g compact orientable surface and so what is a specific in this case of character variety for surfaces is that this space this character variety admits a natural Poisson structure there is a natural Poisson bracket on the algebra of regular function on these affine variety so I will not say what this Poisson bracket is in general I would just give you some example so if you just take l equal to zero so just a compact topological surface and if you take your group g to be simply gl1 just c star then the character variety is simply c star to the 2g essentially because c star is a billion your map from pi1 to c star factors through first homology which has wrong 2g and so you get your character variety c star to the 2g and again because c star is a billion conjugation is trivial so there is nothing to divide but okay so yes this space is really simple it's just a complex algebraic torus and so geometrically function on this torus monomial you can think about them as being monodromy around element of a basis of homology so the element of the form z to the gamma while gamma is some homology class so if you want the class of a curve on your surface and then the Poisson bracket is given by the following formula the Poisson bracket of z to the gamma i with z to the gamma j is equal to pairing between gamma i gamma j times z to the gamma i z to the gamma j where the pairing is simply the intersection number between the corresponding homology class so some slightly more interesting example which will be the example in which we will be focused in this talk is a case g equal to zero so you start with a topological some sphere two dimensional sphere and equal to illegal four so remove four points from the sphere so it is what this picture is supposed to represent you have some sphere and you remove four points out of it and you take for group g the group sl2c so something a bit so unlike the previous example where I was only taking c star something abelian here I take something which is non-commutative sl2c and in this case a corresponding a character variety that I will denote by x so it is a sl2c character variety for the four puncture sphere it is an object which is known since the end of the 19th century and it is known since the end of the 19th century that is x can be described as a four parameter family of affine cubic surfaces so x has dimension six complex dimension six and it's a family of surfaces depending on four parameters and I will tell you explicitly what it is so one way to get functions on this character variety is to take trace of the monogamy around loops of the four puncture sphere if you have a loop in this picture and if you have a representation of your fundamental group you can just look to the image of your loop if the fundamental group you apply your representation you get an element of sl2c and then you take the trace of this element and you get a number so trace functions of monogamy around loops are ways to produce functions on these character varieties and the claim is that you can generate this is algebra so as an algebra by essentially taking seven loops and you will get seven generators so there is a1 a2 a3 a4 which are trace around the small loops around punctures so in this picture there is obvious loop for each point to remove you can consider small loop around this punctured and you get four of them because there is four punctures and then there are more interesting loops that I will call gamma v1 gamma v2 gamma v3 which are the loops which separate the four puncture sphere into two pair of pumps so gamma v1 I guess for example will be the horizontal cycle here gamma v2 will be the vertical cycle here and gamma v3 will be some kind of diagonal thing producing a third pump decomposition pair of pumps decomposition can I ask a question should why did Fulton frequently care about the character variety yeah so essentially they were studying linear differential second order linear differential equations and so with I guess with regular singular singularities and so in this case we can study differential equation on p1 where you have four singularities and study the monodromy of these differential equations they will produce I mean the monodromy will be an element of this space x okay so I just said that there are seven generators corresponding to seven particular loops and in fact these seven generators satisfy a single equation which is written explicitly here so maybe so this thing was already known to say in the 19th century and so now what is important here is that you should think to a1, a2, a3, a4 as parameters and then once you fix them there remains three variables gamma v1, gamma v2, gamma v3 and see the equation is cubic in gamma v1, gamma v2, gamma v3 because it starts by gamma v1, gamma v2, gamma v3 equal something which actually is at most quadratic in the gammas so if you fix a i's you see that this thing defines you some affine cubic surface in like affine three dimensional space but previously you said that it was a four uh four parameter family of cubic uh uh surfaces but then it's eight dimensional right and then here you have it's the six dimensional thing yeah I say it is that it was a four parameter family of surfaces so something six dimensional four plus two four dimensional four parameters and surfaces so here the four parameter a1, a2, a3, a4 and my surface is defined by c's cubic equation and here and you can write down explicitly what the so I claim that on character variety there is always a natural Poisson structure and here in this example you can write explicitly uh what it looks like and so maybe some remark for why a c-story will be a non-trivial is that the bracket of gamma v1, gamma v2 will be equal to gamma v1, gamma v2 so this thing looks exactly what was happening in the case where my group was c star if you remember can I go back enough here you see roughly the Poisson bracket of two trace function or just a product of trace of the corresponding trace function yeah and the point is that in this example it's it's like true at the leading order the Poisson bracket of gamma v1, gamma v2 is a product of gamma v1, gamma v2 but then the correction term involving gamma v3 so these problem in three variables you cannot split it problem involving only two variables which will be what happens in the case of the task okay so this thing is a summary of the geometry that we have with c-space x this is sl2c character of the four puncture sphere and as I already repeated it is a family of affine cubic surfaces over a four-dimensional affine space parameterized by a1, a2, a3, a4 and I guess this slide is just some advertisement slide for c's geometry which is that these geometry appear in many different contexts so first of all it's a character variety and as any character variety by Riemann Hilbert there will be some analytic asomorphism with a multi-space of flag connection with regular singularities so which is related to my answer to Kirchhoff's question it says to do with second order linear differential equations and then by the non-nabian odd correspondence c-space in fact homomorphic to a multi-space of parabolic x-manuals and then you have some etching elliptic fibrations story and in a physics language this thing will be the sub-adjoint written geometry for this particular su2 hunkage theory something very much related is that the smooth fibers of these family of surface so these affine cubic surfaces admits complete upper calor matrix and something which is maybe a bit more specific to the four puncture sphere c-space is a so-called phase space of the part of a six non-linear differential equation which is as a monodynamic condition for these second order linear differential equations and so in this talk we'll focus on the question which is slightly different from all these relevance facts which is a question of quantizing x and as I will review in one moment is quantization in the same sense as in the previous lecture it's in the sense of deformation quantization so I explain x is an affine variety its algebra function is a Poisson algebra and so you might try to produce one parameter non-combinative deformation of this Poisson algebra which will be the definition of quantization and I will explain that there is two possible approaches to address this question of quantizing x the first one is in some sense well known is goes through three dimensional topology and this kind of algebra and the second is I guess not well known and it involves some iogenous version of mirror symmetry and the natural result of the talk will be that these two ways to quantize his geometry will agree okay so I guess his definition essentially already appeared in the previous lecture that if you have Poisson algebra then a deformation quantization is a flat formal one parameter family of associative algebra a h bar such that when you set h bar equal to 0 you recover your starting algebra and such that if you measure the non-competitivity at the first order in h bar you recover the Poisson bracket and so once you have this general deformation quantization question you can ask various various natural questions like that such a thing exists is it unique and so kind of thing we'll focus on in this talk is can we find nice deformation quantizations and I will say a bit more one nice means in one moment so you can ask this question for any Poisson algebra and so in particular you can ask here for the algebra function on the character variety and in particular for the very specific example which is of interest for us in these talks x the character variety sl2 character variety on the four puncture sphere okay so here is an example of deformation quantization going back to the simple case where I have no puncture and when my group is simply gl1 simply c star in this case I explain that the character variety is simply a complex algebraic torus simply c start with 2g and I wrote explicitly this quadratic Poisson bracket and in this case a nice deformation quantization is provided by the so-called quantum torus so which is associative algebra which are the linear basis given by monomial z hat to the gammas and while the product is given by the yeah and we and the this monomial z hat to the gamma satisfies this commutation relation z hat to the gamma i is z hat to the gamma j is equal to the same thing with the two factor reversed to the price of a power of q to the power of the pairing gamma i gamma j and so in this formula this formal parameter q is related to the h bar parameter in the definition of formal quantization through the change of variable q equal exponential h bar so it's an exercise to take these quantum torus algebra you write q got exponential h bar and to check that this thing is indeed a deformation quantization of the complex algebraic torus and it is nice in the sense that the dependence in h bar is not any formal power series but it is through these simply exponential function and once you express everything in terms of these q equal exponential h bar everything becomes rational in q so in this talk everything is is formal but in fact once here in this picture once you I mean in the general definition everything is formal but here once you know that the dependence is only through q equal exponential h bar then you can take h bar anything such that exponential makes sense so you can take real or complex and as I said so very concretely the problem we have if we consider this sl2c character variety I wrote previously explicitly the Poisson bracket it's not so obvious how to write the analog of what we did for the quantum torus precisely of this issue I was talking about that the variables are not decoupled the three variables some are all mixed and so it's not clear what to write down so in fact there is a general way to construct deformation quantizations of character variety which is given by the series of scan algebras and which is coming from three dimensional topology and so because I will only consider this sl2c story I will only describe this kind algebra story for sl2c okay so here I'm describing some the way I've presented things I introduce these character variety sl4 sl2c for the four puncture sphere it's an example of Poisson variety you can ask and then you can ask the algebraic problem how do you produce a quantization and now I claim that one solution to this problem that people found back to the maybe end of 80s beginning 90s is that three dimensional topology provide one way to do that and I will review that now so I'm talking about three dimensional topology so we'll talk about knots and links and a frame links is a link with a nowhere vanishing section of this normal bundle so very concretely it's a link each component of the link is realized as a boundary of an annulus in your stream manifold and then there is something called the Kauffman bracket scan module of any oriented stream manifold it's a module over a ring z bracket a plus minus so lower polynomial in a formal variable a and it's a module that you obtain by taking all possible frame links in your stream manifold so you just take the free module over all possible isotopic classes of frame links it's a very very big modules and then you quotient by scan relations which have a C's following shape so so each time locally your frame links look like that you can replace it by C's relation or each time your link is union of a knot which is not linked to the link you can remove the another component to the price of adding this factor and in this picture because everything is framed everything has vertical framing so there is everything has vertical going out from the screen what is the definition of framing yeah so it was written on the precise of framing just a knot with a nowhere vanishing section of its normal bundle or more concretely it's one way to view your knot as a boundary component of some annulus okay so so this Kauffman bracket scan module is one way starting from a stream manifold you take all possible frame links you impose this relation you get some module and so for example if your stream manifold is r3 you can show that this module is simply the base ring of Laurent polynomial in the variable a and in fact if you take a frame link in r3 and if you look to its class in the scan module so it will be an element in the base ring and it is a so-called Kauffman bracket polynomial of this frame link which is essentially equivalent to the data of the Jones polynomial of your link in r3 okay so the point is that if you take r3 this Kauffman bracket scan module is just the base ring and it is just where essentially the Jones or Kauffman bracket polynomial is living so some of these Kauffman bracket scan module it's come out historically from trying to generalize the Jones polynomial story to more general stream manifold and so in this talk we'll focus on stream manifold which has a very particular form the other one which are product of the oriented two manifold by an interval in this case your scan module has an algebra structure given two frame links l1 l2 in your surface s cross interval you essentially get a product by placing your link l1 on top of l2 you stack the link on top of each other as they think define you a product and it's a product which essentially by construction is associative because if you stack thing on top of each other it's associative operation but it's not commutative in general because essentially when you try to push one link through the other you will have crossing and to know what happens you need to apply scan relations and so something non-trivial can happen here's the question yes do these counter varieties emit quantization by viewing them as cluster algebras and passing to the quantum cluster algebras yes roughly yes and so there is some up to some technical details which has to do with sl2 versus psl2 the answer is roughly yes and it will somewhat agree with the story describing so is that the same as this scheme thing sorry is that the same as the scheme thing yes so the claim will be whatever you construct in the cluster thing will be the same thing that I'm describing here in fact as consequence for some reason I do not want to focus on the cluster point of view here but as consequence of the full story you can prove stuff which were conjecture in the cluster language so I have a question yes you made the remark that this is not commutative I did not know about the pictures means can you please show the pictures means stack one after another you say yes how if you can show the pictures you say I cannot imagine just by looking at the thing it can even prove something so that I can learn I don't think I can draw anything but I can try I mean so so so here is my surface here is my interval and at some level I put some link I don't know some I don't know how to draw some link something here and then at some up so this thing is whatever l2 and then here is another I don't know l1 and by definition the product of l1 l2 is simply the union of these two knots so just a link in your three manifold and this thing is a product and it's not commutative it means sorry it's still that you don't understand yeah so here is in this picture l1 is placed on top of l2 and it's not commutative because you cannot pass continuously from the picture with l1 on top to the picture with l2 on top in general so you have an underlying assumption of homotopy kind of means you have to pass you have some underlying assumption so I don't know if it's related but all the all the things are considered up to isotope up to continuous deformations embedded continuous deformations and the claim so in fact in some cases in fact you can move l1 l2 without crossing but it's like it's like a non-generic situation generically you will have to cross and then it will be because of this crossing you will get non-commutative okay so the claim is that in the case so I was describing a surface cross interval so just from this picture you get an algebra and the claim is that for this kind of geometry in fact this algebra has a natural basis at the linear level a linear basis given by the so-called multi-curves so the idea is that the multi-curve so it's a curve living on the surface no longer in the three-fold just on the surface and it is a union of finitely many disjoint compact connected embedded one dimensional some manifold of your surface so that none of them bound the disk so it's just a collection of curves drawn on your surface which are not intersecting and such that none of them bound the disk and it's clear that if you have such a curve on the surface if you view the surface embedding in this three-fold product in particular it is an example of link in dimension three and you can put some obvious framing coming from the vertical third direction and the claim so the theorem of pristiki tell you that other topic classes of multi-curves in fact from the basis of the scan algebra as a module so from some kind of linear point of view module point of view you kind of understand reasonably well what the scan algebra is and now here is a claim that the scan algebra is a solution to the deformation quantization question I was asking previously the claim is that it's kind algebra so in general it's an algebra over this ring of Laurent Poignol in a variable big a and if you set big a equal minus exponential h power four then this thing is a deformation quantization of the algebra of regular function on the sl2c character variety of your surface and essentially uh the isomorphism when you specialize a to minus one is obtained by sending a multi-curve so which as I say are basis of this kind of algebra the collection of curve drawn on the surface the corresponding function of the character variety is just the trace along the various components of your curves so it's almost this multi-curve in the scan algebra has some kind of lift of this trace function of the character variety and so some more so okay so the long story of people approving this statement and one way to understand why it's true is to look to the classical limit of the scan relation and the claim is that the classical limit of the scan relation recovers this kind of relation between traces at the level of sl2 so this thing is just identity for matrices in sl2c and the claim is that when you pass from these multi-curve to trace function the scan relation will reproduce these kind of identities okay so the previous theorem is like general for any sl2c character variety of your surface the theorem tells you that the scan algebra produces a deformation quantization of the character variety in particular we can apply it to the very specific case of the four puncture sphere and maybe I skip some detail and essentially you just compute this kind of algebra and people doing not say we have computed this kind of algebra the four puncture three and they just wrote down the answer so here is the answer so before gamma v1 gamma v2 were commuting and now they no longer commute so here you have some kind of commutation relations which tell you how they no longer commute and previously the commuting variable gamma v1 gamma v2 gamma v3 were satisfying some some cubic relation and now this cubic relation in this kind of algebra in the non-cumulative level this is cubic relation is deformed in this cubic relation so they wanted this non-commutative deformation of this cubic surfaces or final surfaces more generally can you does that match in any way when they mean I don't I think if you're like p2 for example you can have non-commutative deformation and more generally also final final surfaces can you view this some sort of as a non-commutative surface I mean definitely you should think of this thing as a non-commutative surface and then maybe your question is how to compare it with other notions of non-commutative surface and and let me say I'm not sure it depends how precisely maybe you can ask me later which notion of non-commutative surface you are taking okay so the point is I see things are just deformation of the previous equation except now everything depends on the variable big a okay when we said big a equal to 1 you recover the ordinary equation of the cubic surface and what was not obvious where was where to put the correct powers of a for example in the original equation I wrote yeah there was a minus 4 and the claim is that in the quantization this 4 should be replaced by minus 2 2 times a to the 4 plus a to the minus 4 and so some of the scan algebra story tell you that it's correct to replace 4 by these polynomial in a and not something else okay and so it's non-commutative surface appear in many many different places from many many different point of view is there any question about these stories so there is this character right here of the four-point sphere I want to quantize it there is this general scan algebra story which work for any sl2c character right here you just apply it for this geometry and you get some explicit answer some explicit presentation but it has relations of this kind algebra which produce some non-commutative version of the cubic surface okay so if there is no question I will somehow change topic and start talking about the different a different way to produce a deformation quantization of the cubic surface and you just do with the animative geometry and maybe first it has to do with a mirror symmetry so very generally speaking mirror symmetry is some connection between let's say two calabiol varieties there's a property to exchange symplectic geometry and complex geometry and let's say from the point of view of this talk it's better to think about the case of non-compact calabiol varieties more precisely the case of log calabiol varieties which appear yd where y is a smooth projective variety and d is an anti-canonical divisor and in this case the complement y minus d is a non-compact let's say if d is non-empty is a non-compact calabiol variety and so for this talk essentially I will use nothing about mirror symmetry except that it's a way it's essentially it's predicts that you should be able to construct algebraic variety if you have a variety its mirror should exist and people studying mirror symmetry think about ways to construct this mirror and in particular we will use the fact that in dimension two is this kind of this kind of thing we care about in dimension two there is a mirror symmetry construction for log calabiol surfaces due to growth I can tell and here's the thing is wrong it's probably 2011 and this construction involves animative geometry so this thing is a part which is slightly surprising and which is coming from mirror symmetry which is that to construct some algebraic variety you do that by doing animative geometry in another algebraic variety the one from which you are trying to construct the mirror of so the idea is that if you have a log calabiol surface yd by some appropriate counts of rational curves in this pair then it's possible to construct its mirror family and its mirror family here I wrote it curly v goes to spec c bracket any of y so any of y is a cone of effective curves on y and c bracket is a corresponding monoid algebra so essentially the mirror construction gives you a mirror family whose base is roughly let's say h2 of y and it's expected because from your symmetry your complex parameter here should be related to the symplectic or color parameter or the original thing which has to do with essentially h2 a chronology and so the claim is that it is possible to recover the particular geometry we care about this x this family of affine cubic surfaces over a4 can be obtained as an output of some of such pure symmetry construction and this mirror family is obtained by applying the mirror symmetry construction to a pair yd where y is a smooth projective cubic surface and d is a triangle of lines on y so y smooth projective cubic surface from the very beginning I said there is 27 lines on such cubic surface and in fact you can find a triple of lines which form a triangle configurations and you take that 40 so here there is something which is possibly a very confusing y is a cubic surface and then I construct its mirror and then its mirror is a family of cubic surfaces so you should not mix the two sides so somehow the object we want to quantize is the mirror family but the place where the curve counting will be happening will be the mirror cubic surface so shall we complain that some of the cubic surface has h2 equal to 7 yeah you should complain and so essentially there is a way if you do think properly you get a seven dimensional family but there is a natural action of a three dimensional torus coming from the fact that this d has three components and when you quotient by these three dimensional torus you get this four dimensional family okay so here's a short description of the kind of animative questions you need to study to solve to make this construction so y is a smooth project surface and d is the anti canonical cycle let me assume of rational curves and then if you fix a curve class on your surface there's a kind of animative question which enters in this mirror construction rational curves so really honest general zero compact remand surfaces in y of class beta which has a property to intersect the anti canonical device at d in a single point so maybe there should be a picture somewhere so here's a picture let's say on my cubic surface with three line d1 d2 d3 on it and then I'm looking in y to rational curves which are the property to intersect this triangle at a single point and see single point could be somewhere in the smooth part of the triangle or it could be somewhere in the corner of the triangle and it's possible to parameterize the choice of possible essentially contact orders or tendency conditions with the boundary components of the triangle so here so there is a set called b of z so essentially for each corner of this triangle you construct a two dimensional cone so here from this triangle you get these three cones each corresponding to one corner of the triangle and while you are doing that because each integral points in this cone you can think about it as defining a contact order for your curve along the divisor if you take some integral point which is on one of these ray here and see the ray correspond to this divisor and if this point is some integral multiple of the primitive generator here let's say d times the primitive generator of this ray then you interpret this point as specifying your curve to have contact order d with a divisor at the unique point of intersection and now if you have a point in the interior of the cone so specified by two integers inside this cone then it means that you're considering some curve going into the corners and the contact orders along the two components of the triangle meeting at these corners are given by these two integers okay so roughly there is some kind of space of integral points which parameterize all possible ways your curve can intersect as triangles at the single point yeah i mean you're accounting like on a morphic or algebraic curves so when i say effective class it just some class which is represented by some more morphic or algebraic curve i mean it is in play in the fact that maybe at this point it's not fully clear what what what we are doing so so the point is that we are counting these curves so for every beta for every beta curve class you will solve this innovative problem counting this curve which is given class with given contact order meeting the boundary in one point and then you will get numbers so this thing is just innovative geometry of surface and you get a lot of numbers out of solving these innovative problems and maybe before answering your question i just give example in our concrete geometry why was a smooth projective cubic surface in p3 d a triangle of lines on c the cubic surface and i said at the beginning there is 27 lines on such cubic so already three of them form the boundary d so there is 24 of them remaining and the claim is that each of the 24 lines not containing d intersect d in a single point so you have this kind of picture where you have for each side of the triangle you have like eight lines which will intersect uh c side in exactly one point and not and nothing else so so this curve i example so these lines on this cubic surface are examples of curves we are counting in this problem can you say again but this what do you mean by quantization quantization whether you have a poisson algebra commutative so it's a commutative algebra with a poisson bracket and yeah it's quantization in the sense of deformation quantization so it's meant construct a one parameter family depending on veg bar which deform your commutative algebra into some algebra which is typically non commutative and with non commutativity is controlled by the poisson bracket okay so what you're saying sounds like deformation so what does i never know the word quantization yeah i mean in a sense here just an example of a deformation as associative algebra it's just it's just particular type of deformation as associative algebra so you know the point is the poisson bracket the deformation is constrained by uh this poisson bracket on the last slide yes the crossing point of the three lines corresponds to what the object on the left hand side sorry the crossing point on the three lines on the right hand side here yeah so i was trying to explain before that this picture is like a cone for each corner and that the integral point on this picture correspond to to various tendency conditions for your curve along the boundary of the triangle so here's a point v1 here is like a primitive generator of this ray and so this thing corresponds to curve intersecting this side of the triangle at a single point with contact order one and in all these lines are examples of such a thing they all intersect this side with contact order one so the corresponding contact order or tendency condition is this point v1 you mean the point here yes so this point will be like zero it will be a curve like touching no boundary of the triangle and such a curve in fact do not exist because the boundary is ample and so we can forget about it okay i will skip that and so here i will maybe go quick so as i was saying from solving these primitive problems you get collection of numbers sorry you get collection of numbers this count of rational curves so what does it have to do with constructing a mirror geometry the point is so this construction of which is in the paper of growth i can kill involve constructing a mirror geometry essentially by gluing a simple local models and you glue these simple local models by some non-trivial gluing maps and these gluing maps are essentially generating series of the numbers we computed by solving the primitive problems so the most interesting is the point we should sound slightly mysterious but it is what more or less mirror symmetry tells you to do to use these numbers these kind of rational curves you construct generating series out of them and more or less you use these generating series to define gluing transformations between simple pieces so i will not give more details here is some rough picture or how it works so roughly for each vertex of this triangle there will be a simple algebraic variety which has some very explicit description and these three varieties will give you essentially three patches on the family you are trying to construct and you glue these patches together according to some gluing transformation which involved in that definition all these kind of rational curves i was describing previously yes should we view this new variety as a as actual family or it's a variety with regular function yes yes so i mean you get some variety with in fact you get some affine variety with a ring of regular function and the claim is that this thing naturally comes as a family over what i was describing the spec of the cone of effective curve and the reason for that is that when you form generating series for your environs it's naturally to have a variable t to the beta keeping track of the curve class beta so all these generating series have parameters for curve classes which is why what you construct is something which depends on these t parameters and which is why the resulting thing is a the family over these t variables and in general there are convergence issues but for the example i'm describing everything converges and you get some honest algebraic family yeah my question was more like why do you get the family of the single variety yeah so you get a family because you have these variable t's so the specifying one value of t is like a fixing one particular fiber sorry yeah so yeah because it's like non-compact it's subtle to phrase so okay so here i was starting with a projective surface a triangle of line and i construct this family so i start with a projective cubic surface and a triangle of line so the complement is a affine cubic surface and yes somehow the mirror is itself a family of affine cubic surfaces so somehow it is kind of two dimensional situations the kind of two side of the mirror at least topologically in fact they're the same it's like a bit similar to the fact that in dimension two a mirror of a k3 is again a k3 and you that's some of the construction of this mirror family do you have to concur which touch the corner your yes i pray yes yes so i mean there are technical ways if you don't want to work with corners there are technical ways to go around that but it's cleaner to say that you have my fixed divisor my fixed variety and you can't curve which possibly go into the corner because here you somehow say you primitive in a cone dual to this thing oh but this is when three of the three copies glued together then you have to count it yeah that's a pretty neat to count all of them in fact instead of examples there are really things going in the corner in fact there are a lot a lot of them going into the corner okay so the upshot of this slightly fake story is that starting from a pair yd a surface with an anti-canonical divisor you count these generate your curves touching the boundary in a single point and out of these numbers you cook up generating series you use these generating series as growing functions and then you glue simple patches to get your mirror family and the claim is that in fact these mirror family naturally come with a Poisson structure so i am not i did not explain why but some of the simple pieces have a Poisson structure and the gluing transformation preserves this Poisson structure so in fact the output as I saw Poisson structure so in fact the output is really a mirror it's really a family of non-compact calabiow so yeah when like fiber of dimension two so it's a family of holomorphic symplactic varieties and the total space is Poisson and so in particular you can ask is there a way to produce a deformation quantization of this mirror family and here because this mirror family comes out of these construction using animative geometry you can ask is it is there a way to produce a deformation quantization by deforming the construction coming from your symmetry and we deform this game of counting general zero curve which produces a mirror family can you in some way deform it to produce deformation quantization of the mirror family maybe as keep that as a claim is that you can do it if you replace your count of general zero curve by count of higher genus curves so you will consider exactly the same kind of problem curve of a given class touching the triangle in a single point but rather than to consider rational curves general zero curves you consider curves of arbitrary genus and before the game was to put this number into generating series and now you still do the same except now you have an extra variable which is a genus and you put an extra variable h bar to the power two g minus one where g is a genus and here the fact to call it h bar is not a coincidence the claim will be this deformation at the level of animative geometry replacing general zero curve by higher genus curves is replacing our previous generating series by new generating series depending on h bar and the claim is that you can deform the mirror symmetry construction by now gluing simple non cumulative pieces you can glue them together using these generating series of higher genus invariants and you produce a non cumulative algebra which is a deformation quantization of the mirror family is it associative sorry is it associative yeah that's right associative yeah so it's part of the quantization condition you produce some associative algebra uh non cumulative which is a deformation of the commutative algebra regular function how do you translate the associative in that case into properties of the yeah that's right so somehow in this mirror symmetry business there is something a bit miraculous which is happening which is first even in the classical case you count rational curves and some of these numbers uh they fit together in a way to produce some associative commutative algebra as the way it happens is not obvious at all somehow it's part of the I mean something you need to prove and it's part of the of the non trivial part of the story so maybe if you want you can ask me later but I don't want to say more here and so yeah yeah it's even more true the question why going to iogenis why does it replace commutativity by non computativity of the algebra and why is it still associative somehow it's even more non trivial so I will not try to explain why does it come out yeah it just take on somehow the black box can you go back to the formula since it's formula you wrote for iogen's curve yes if you put g equal to zero for the disc yeah so it scales as h bar scales as inverse power but previously is it the same with yeah yeah so this thing is like some normalization uh convention yeah so indeed here starting the leaning term has been one over h bar and so so this thing would be like one over h bar times the previous thing so when h bar in the classical case when h bar goes to zero is it the same h do you recover the same h as previously you said or yeah so roughly yes and it's roughly because taking taking classical limit even in the most simple quantum mechanical example it's not a simple I just setting h by equal to zero you need to take appropriate limit and so but if you do think correctly yes so for example I could have for simplicity just forget minus one then I set h by equal to zero and I recover the previous formula but it scales as one by eight first yeah that's not but this thing is just a choice that because I like to write it like that but I could have written it without minus one and putting the one over h bar somewhere else which is less disturbing so you don't need a regularization or you need a regularization to have this h bar goes to zero then this kind of contributes eight zero coefficient finite yeah I mean indeed there is a tricky point somewhere that in general fact when you go from quantum to classical if you just take h by equal to zero naively uh sometimes it's a wrong thing to do but if you do think properly everything works okay so so so now how is it related to our starting question so we have our sl2c character right here the claim which is the same of growth I can kill zebert is that this thing is what I claimed before this family is the output of the mirror symmetry construction applied to wire smooth projective cubic surface and geothermal outlines so you count your rational curves you will run the mirror symmetry machinery and the output is exactly this family and now running sees a quantum iogenous version of mirror symmetry running the machinery counting iogenous curve will produce a deformation quantization of this family so now I arrive to the final point of this talk which is that in the first part of the talk I explained this kind algebra business is slow dimensional link story which was producing one deformation quantization of the character variety and here I've just explained you a different way to construct a deformation quantization of the same geometry so what we're doing in the two sides are really completely different on on the left hand side here we realize your geometry as a character variety and then you use the fact that it is a character variety to run this kind algebra story to produce a deformation quantization rather than the other side you just use the fact that it is a family of non-compact calabino surfaces which appear as more family of something and then deforming sees a remark the mirror symmetry construction you cook up a deformation quantization and the theorem is that these two deformation quantization they agree and somehow here the claim is that they agree in the strongest possible sense in the sense that here you have explicit generators here you also have explicit generators map one to the other and this map is an isomorphism okay so at the end there were some physics slides that probably I will not really go into so the claim is that roughly seems slightly surprising connection between so low dimensional topology on one side so on one side you are really studying links and knots so you are drawing curve on the four puncture sphere whereas on the other side you are counting on amorphic curves of any possible genus in a complex cubic surface so these two sides seems a bit unrelated and somehow you can at least argue that such connection should exist from the fact that the same physical theory some particular n-egal 2 for dimensional gauge theory has essentially two different realization for string and m theory and in one realization the scan algebra story is natural and on the other realization the amorphic curve story is more natural so here is a side already put at the beginning it's a summary side or conclusion side so on the c side adds the low dimensional topology on c side adds the animative geometry and as the main result of this talk was the low dimensional topology give you one way to produce deformation quantization the animative geometry give you another way to produce the deformation quantization and the two ways agree and something I do not stress in this talk is that once you know that then you can derive natural application somehow because you know about all amorphic curves in cubic surfaces somehow you can prove stuff about curve on the four puncture sphere which is a bit surprising but it's one application of this kind of connection and sorry for being a bit of a time and I will stop there thank you are there any further questions so previously you remarked about this n equals to 2 is zero written theory yes so as far as I recall zero written theory has a finite modeling space means finite number of points how this is related to this plus sorry vector variety I didn't this one is finite means finite number of points only count them you get an ingredient for the general choice of metric what do you mean by finite many points you have so zebra you said zebra written geometry uh-huh uh yeah so so in geometry means we satisfy this zebra written equations and it gives you a modeling space which is finite number of points at the earliest yeah so so yeah I'm using cybeck written geometry in some kind of physics sense so this is moduli space is not a moduli space of solutions to cybeck written equations it is some a different terminology what in what sense in physics sense I understand but what do you mean by zero written geometry yeah so it's part of the story of cybeck written that given some n equal to four dimensional a gauge theory there is a corresponding geometry attached to it which in physics language controls a low energy effective theory so so in fact so so cybeck written arrived to the mathematical cybeck and written equations through this kind of story but it's a bit long to explain the collection and for the purpose of this talk I don't know an obvious direct connection if you want I can tell you more afterwards but it's a bit a long story so can you can you do that uh the formation continuation with what happened in the geometry form other aspects of yeah that's also in fact so you could imagine generalizing this story so so in fact there are other examples where the character variety is still a family of surfaces and they are still this kind of del pezzo geometry complement of some divisor in the del pezzo geometry and but the ones which are very interesting are difficult so definitely you can run this construction you can run since deformation quantization through innovative geometry but I don't know proof that it's agree with a version of scann algebra but forgetting the character right this side so do them yeah so so certainly the hygiene environment satisfies the right properties to yeah yeah that's why they produce one deformation quantization what is difficult is to compute the numbers explicitly to know what is explicitly the locomotive algebra but yes the construction makes sense but in this case you computed explicitly on you yeah that's why in some sense his story I just compute explicitly you said that character variety is always made of proson structure and you gave us like the formula for some specific examples yes can you like write down like a formula in general or I mean this is like a long constructed proof yeah so I think maybe in the word of character writing maybe some keyword it may be gold band bracket poisson bracket so I think you can write a formula in term of for the tourists I explained just how to do with intersection of homology classes in more general you can still I guess more let's write down a formula in term of intersection of curves drawn in your in your surface but you need to be slightly careful about some kind of geometric intersection numbers but yeah it's possible to write some kind of explicit like geometric description and there are various points of view on slightly related like but okay let me just have here you are considered the non-oriented links and notes what happens if you consider the oriented links and notes yeah so yeah the story a bit described is related to his Kauffman brackets kind module which somehow is formulated in terms of unoriented framed links and like for R3 it produces the Kauffman polynomial but up to some easy transformation it's more of the same thing as a joint polynomial and usually the joint polynomial is for oriented unframed links so I would guess that at least in this R3 model you can go from one description to the other so I don't know yeah so definitely there are things called like umfly pta scan algebra which are like different some things Kauffman brackets can algebra in which I think you consider uh oriented links and they are just different objects so somehow the one which appear in this story it is a Kauffman brackets kind algebra but maybe in some not so unrelated story maybe the other one will appear we have one more question in the chat so are the trace functions of sl2 character varieties the same as theta functions of ghkk as your last theory implies for general sg services yeah yeah so so so definitely in the example where I compute everything explicitly for these four puncture sphere the answer is yes and somehow it's one consequence of the story I'm describing so for general uh SGN as far as I know it is still a conjecture so yeah so in fact I think even in this case it was not known so it's like very specific case all the story proved it somehow