The Skein Algebra of the 4-punctured Sphere from Curve Counting
Bousseau, Pierrick
The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL_2 character of a topological surface. I will explain how to realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten invariants of a log Calabi-Yau cubic surface. This leads to the proof of a previously conjectured positivity property of the bracelets bases of the skein algebras of the 4-punctured sphere and of the 2-punctured torus.
 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 their 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 serialization 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 will be talking about knots, links, and related object. In a complex animative algebraic geometry we will be interested in counting configuration of geometric objects satisfying various constraints. And here is a picture of one of the most well-known examples 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 a 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 the fact that sometimes a given supersymmetric gauge series of different string serializations. The point is that string theory often give geometric realizations of gauge theory dynamics and when you get different string theory realizations of the same theory you get a prediction for relation between 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 the Schuyler algebra or the four-point sphere. So 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 interval, open interval 01. The enumerative algebraic geometry part of the talk will be about counting allomorphic curves in complex cubic surfaces. And finally the physics part of the talk is about four-dimensional n-egal-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 part of the talk I will talk about the enumerative 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. 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 cross-section cross interval and some of the abstract theories that you are able to prove such a thing by studying the apriori completely different topic of counting Riemann surface, counting all morphic curves in a complex cubic surface. In this talk I will not focus on the applications to the scan algebra, I 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. 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 innovative geometry, on the left there will be the low-dimensional topology and on the bottom there is the physics. And I will come back to this diagram at the very end of the talk. So the plan is first I will talk about the low-dimensional topology part of the story and so 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 3D-mersion topology via the scan algebra. Then I will talk about the innovative 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 Iogenius curve counting and the main point of the talk is a comparison between the quantization of these cubic surface geometry given by 3D-mersion topology by the scan algebra and the one given by Iogenius mirror symmetry, so the one given by innovative geometry. So Iogenius means you are talking about this current difference between this kind of Iogenius? Yes, Iogenius, let's say Iogenius compact curve. So let me start at the very beginning by saying very generally a few words about character varieties. So if you have gamma, finitely a generated group and g, a reductive algebraic group of a complex number like GLNC or SNNC, then you can construct an affine variety of group morphism from gamma to g. And there is a natural action of g on these affine varieties 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 an 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 we'll be particularly interested 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 G compact orientable surface. And so what is 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 this affine variety. So I will not say what this Poisson bracket is in general, I will just give you some example. So if you just take L equal to 0, 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 by. So ESC space is very 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 where 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 certainly more interesting example which will be the example in which we will be focused in this talk is a case g equal to 0, so you start with a topological some sphere, two dimensional sphere and a equal to 4, so we move 4 points from the sphere, so it is what this picture is supposed to represent, you have some sphere and you remove 4 points out of it and you take for a 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 character variety that I will denote by x, so it is a sl2c character variety for the 4 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 4 parameter family of affine cubic surfaces, so x has dimension 6, complex dimension 6 and it is a family of surfaces depending on 4 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 monogram around loops of the 4 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 you have a 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 monogram 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 7 loops and you will get 7 generators. So there is a1, a2, a3, a4 which are trace around the small loops around punctures, so in this picture there is an obvious loop, for each point to remove you can consider a small loop around this puncture and you get 4 of them because there is 4 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 4 puncture sphere into 2 pair of pints. 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 pints decomposition, pair of pints decomposition. Can I ask a question, 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 singularities and so in this case we can study differential equations on p1 or your 4 singularities and study the monodromy of these differential equations they will produce, I mean the monodromy will be an element of these best acts. So I just said that there are 7 generators corresponding to 7 particular loops and in fact these 7 generators satisfy a single equation which is written explicitly here. So this thing was already known to 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 3 variables gamma v1, gamma v2, gamma v3 and this 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 define you some affine cubic surface in like affine 3 dimensional space. But previously you said that it was a 4 parameter family of cubic surfaces but then it is 8 dimensional right and then here you have it is the 6 dimensional thing. Yeah I say it is at it was a 4 parameter family of surfaces so something 6 dimensional, 4 dimensional, 4 dimensional, 4 parameters in surfaces. So here the 4 parameter a1, a2, a3, a4 and my surface is defined by 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 is an example you can write explicitly 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 were just a product of the corresponding trace function. The point is that in this example it's like true at the leading order of the Poisson bracket of gamma v1, gamma v2 is a product of gamma v1, gamma v2 but then there is a correction term involving gamma v3. So this problem in three variables you cannot split it into problem involving only two variables which will be what happens in the case of the torus. 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 this geometry appear in many different context. So first of all it's a character variety and as any character variety by Raymond Hilbert there will be some analytic asomorphism with a modu space of flag connection with regular singularities so which is related to my answer to Kiosk question. It has to do with second order linear differential equations. And then by the non-nabian odd correspondence C space in fact homomorphic to a modu space of parabolic X mandals and then you have some etching elliptic vibration story and in a physics language this thing will be the sub-adjoint return geometry for this particular SU2 engagement theory. Something very much related is that the smooth fibers of this family of surface, so this affine cubic surfaces admits complete hypercalametrics and something which is maybe a bit more specific to the four puncture sphere, this space is the 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 the 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 as goes through three dimensional topology and the second is not well known and it involves some Iogenous version of mere symmetry. And the non-trial result of the talk will be that these two ways to quantize this geometry will agree. So I guess this 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 natural questions like does such a thing exist is it unique and the kind of thing we will 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 or the character variety and in particular for the very specific example which is of interest for us in this talk X the character variety SL2 character variety on the four punctures sphere. So here is an example of deformation quantization going back to the simple case where I have no puncture and where my group is simply GL1 simply C star. In this case I explained that the character variety is simply a complex algebraic torus simply C star to 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 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 this 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 become rational in q. So H is real or complex? So in this talk everything 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 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 the coupled 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 algebra and which is coming from three dimensional topology and so because I will only consider this SL2C story I will only describe this scan algebra story for SL2C. So here I'm describing some the way I've presented things I introduce is character variety SL4SL2C for the four puncture sphere it's an example of Poisson variety you can ask and then you can ask the algebraic problem or 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 Kaufman bracket scan module of any oriented stream manifold it's a module of a ring Z bracket A plus minus so long 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 of the 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 follow up shape so so each time locally your frame links look like that you can replace it by C's validation 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 previous size of framing just a knot with a nowhere vanishing section of its normal bundle. More concretely it's one way to view your knot as a boundary component of some annulus. So this Kaufman bracket scan module is one way starting from a three manifold you take all possible frame links you impose this relation you get some module. And so for example if your three manifold is R3 you can show that this module is simply the base ring of Laurent Poinomial 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 would be an element in the base ring and it is a so-called Kaufman bracket polynomial of this frame link which is essentially equivalent to the data of the joints polynomial of your link in R3. Okay so the point is that if you take R3 this Kaufman bracket scan module is just the base ring and it is just where essentially the joints or Kaufman bracket polynomial is leaving. So some more says Kaufman bracket scan module it's come out historically from trying to generalize the joints polynomial story to more general three manifold. And so in this talk we'll focus on three 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 and 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 things 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. So here's the question. 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 some more agree with the story described. 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. I don't know the pictures. Can you please show the pictures? Stack one after another you say. I cannot imagine just by looking at the thing. It can even draw something so that I can learn. I don't think I can draw anything but I can try. So here is my surface. Here is my interval and at some level I put some link. 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 L1. And by definition the product of L1 L2 is simply the union of these two nodes. So just a link in your three manifold and this thing is a product. And why it's not commutative? Sorry, I just still don't understand. Yes, so here 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. You have to pass. You have some underlying assumption. So I don't know if it's related but all the things are considered up to isotopy, up to continuous deformations, embedded continuous deformations. And the claim, so in some cases in fact you can move L1 L2 without crossing but it's like the non-generic situation. Generically you will have to cross and then it will be because of this crossing you will get non-commutative. 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 dish-joint 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 so 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 3. And you can put some obvious framing coming from the vertical third direction. And the claim, Soci Serem of Pristiki tell you that other topic classes of multi-curves in fact form a basis of a 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 this scan 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 bar over 4, then this thing is a deformation quantization of the algebra of regular function on the SL2C character variety of your surface. And essentially the isomorphism when you specialize A to minus 1 is obtained by sending a multi-curve, so which as I say are basis of this kind of algebra. So this collection of curves drawn on the surface, the corresponding function of the character variety is just a trace along the various components of your curves. So most of these multi-curve in the scan algebra are some kind of lift of the 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 this multi-curve to trace function, the scan relation will reproduce this kind of identities. Okay? That's the classical limit of h bar goes to 0. Yes, yes. Okay, so the previous theorem is like general for any SL2C character variety of your surface, the theorem tells you that the scan algebra produce 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 escape some detail and essentially you just compute the scan algebra and people doing not have completed the scan algebra, the four puncture sphere 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 cubic relation. And now this cubic relation in the scan algebra in the non-combative level, this cubic relation is deformed in this cubic relation. So there are also this non-commutative deformation of these cubic surfaces or final surfaces more generally. Can you, does that match in any way? What do you mean? I don't. I think if you have like P2 for example, you can have non-commutative deformation and more generally also final surfaces. Can you view this 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 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. So the point is I see things are just deformation of the previous equation except now everything depends on the variable big A. When we said big A equal to 1, you recover the ordinary equation of the cubic surface. And what was not obvious was where to put the correct powers of A. For example in the original equation I wrote, there was a minus 4 and the claim is that in the quantization this 4 should be replaced by minus 2 times A to the 4 plus A to the minus 4. And so some of the Scandinabian story tell you that it's correct to replace 4 by C's polynomial in A and not something else. And so this non-commutative surface appear in many, many different places from many, many different point of view. Is there any question about this story? So there is this character right here of the four-point sphere. I want to summarize it. There is this general Scandinabian story which works for any SL2C character right here. You just apply it for this geometry and you get some explicit answer, some explicit presentation but you have relations of this kind of algebra which produce some non-commutative version of the cubic surface. If there is no question, I will somehow change topic and start talking about the different way to produce a deformation quantization of the cubic surface and just do with an automotive geometry. Maybe first it has to do with murosimetry. So very generally speaking murosimetry is some connection between let's say two Calabi varieties. It has 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 Calabi varieties. Or precisely the case of log Calabi varieties which are 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 Calabi variety. And so for this talk essentially I will use nothing about murosimetry except that it's a way, essentially it's a, it's predicts that you should be able to construct algebraic variety. If you have a variety, its mirror should exist and people studying murosimetry think about ways to construct this mirror. And in particular we will use the fact that in dimension two, this kind of, this kind of thing we came about in dimension two, there is a mirror symmetry construction for log Calabi surfaces due to growth like in Cal and here this 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 murosimetry 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 Calabi surface YD by some appropriate counts of rational curves in this pair, then it's possible to construct its mirror of family. And its mirror of family here I wrote it curly V goes to spec C bracket N of Y. So N of Y is a cone of effective curves on Y and C bracket is a corresponding monoid algebra. So essentially the mirror of construction gives you a mirror of 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 symplactic or color parameter or the original thing which has to do with essentially H2 homology. And so the claim is that it is possible to recover the particular geometry we care about since X, this family of affine cubic surfaces over A4 can be obtained as an output of such mirror of symmetry construction. And this mirror of family is obtained by applying the mirror of 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 of configurations. And you take that for D. So here there is something which is possibly 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 of family but the place where the curve counting would be happening would be the mirror cubic surface. So shall we complain that some of the cubic surfaces H2 equal to 7? Yeah, you should complain. And so essentially there is a way if you do things properly you get a 7 dimensional family but there is a natural action of a 3 dimensional torus coming from the fact that this D has 3 components and when you quotient by this 3 dimensional torus you get this 4 dimensional family. Okay, so here is 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 an anti-canonical cycle and we assume a fractional curves. And then if you fix a curve class on your surface there is 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 are the property to intersect the anti-canonical device or D in a single point. So maybe there should be a picture somewhere. So here is a picture, let's say on my cubic surface with 3 line D1, D2, D3 on it and then I am looking in Y to rational curves which are the property to intersect this triangle at a single point. And this 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 there is this 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 device order. If you take some integral point which is on one of these ray here and these 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 the 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 are 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. So roughly there is some kind of space of integral points which parameterize all possible ways your curve can intersect the triangles at a single point. You are counting all morphic or algebraic curves so when I say effective class, it is some class which is represented by some all morphic or algebraic curve. I mean it is in play in the fact that maybe at this point it is not fully clear what we are doing. So the point is that we are counting these curves so for every beta curve class you will solve this innovative problem. Counting this curve which is given class, which is given contact order, meeting the boundary in one point and then you will get numbers. So this thing is just innovative geometry on the surface and you get a lot of numbers out of solving these innovative problems. And maybe before answering your question I just give an example in our concrete geometry where I was a smooth projective cubic surface in P3, d a triangle of lines on the cubic surface and I said at the beginning there is 27 lines on such cubic so already 3 of them form the boundary d so there is 24 of them remaining and the claim is that each of the 24 lines not contained in d intersect d in a single point. So you have this kind of picture where for each side of the triangle you have like 8 lines which will intersect this side in exactly one point and nothing else. So these lines on this cubic surface are examples of curves we are counting in this problem. Can you say again what does that mean by quantization? Quantization is whether you have a Poisson algebra, so it's a commutative algebra with a Poisson bracket and here it's quantization in the sense of deformation quantization so it's meant to construct a one parameter family depending on veg bar which deforms your commutative algebra into some algebra which is typically non-commutative and with non-commutativity is controlled by the Poisson bracket. So if you're saying it sounds like deformation so what does that mean? In some sense here is just an example of deformation as associative algebra. It's just a particular type of deformation as associative algebra. So the point is the Poisson bracket and the deformation is constrained by this Poisson bracket. So this becomes on the last line. Yes. As a crossing point of the three lines corresponds to what object on the left hand side. Sorry. The crossing point on the three lines on the right hand side. Here. 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 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 the C's ray and so this thing corresponds to curve intersecting this side of the triangle at a single point with contact order one. And 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. So the crossing point is has no corresponding to the left hand side. You mean the point here? Yes. 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 on pole 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 innovative 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 involves 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 innovative problems. So the most interesting is the point which would sound slightly mysterious but it is what more or less mirror symmetry tells you to do to use these numbers this kind of rational curves you construct generating series out of them and more or less you use the generating series to define gluing transformations between simple pieces. So I will not give more details here is some rough picture 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 involves in that definition all these counter-fragile curves I was describing previously. Should we view this mirror variety as an actual family or is it a variety with a regular function? Yes, you get some 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 invariance is naturally to have a variable t to the beta keeping track of the curve class beta. So all series of 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 family over these t variables and in general there are convergence issues but for example I am describing everything converges and you get some honest algebraic family. My question was more like why do you get a family of a single variety? Yes, you get a family because you have these variable t's so the specific value of t is like fixing one particular fiber. So the hodge number of the mu is related to y or the hodge number of y is related to mu? The hodge number of mu? Yes, because it is non-compact it is certainly subtle to phrase. 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 here some of the mirror is itself a family of affine cubic surfaces. So some of these kind of two dimensional situations are kind of two sides of the mirror at least topologically in fact they are the same. It is like a bit similar to the fact that in dimension 2 a mirror of a k3 is again a k3. That is some of the construction of this new family do you have to come across which touch the corner? Yes, I pray 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 is cleaner to say that you have a fixed divisor and a fixed variety and you can't curve which possibly go into the corner. Because here you somehow say you primitive and you can't do it to this thing. But this is when three copies glue together then you have to come to it. Yes, I pray you need to count all of them. In fact in the example there are really things going in the corner in fact there are a lot a lot of them going into the corner. So the upshot of this slightly fake story is that starting from a pair YD surface with an anticanonical divisor you count these genital zero 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 this mirror family naturally comes with a Poisson structure. So 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 that's our Poisson structure. In fact the output is really a mirror, it's really a family of non-compact Calabria. So yeah like Fabriola of Dimension 2, it's a family of olomorphic 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 this construction using animative geometry you can ask is there a way to produce a deformation quantization by deforming the construction coming from your symmetry. And we deform this game of counting genital zero curve which produces a mirror family, can you in some way deform it to produce deformation quantization of the mirror family? Maybe I'll skip that. And the claim is that you can do it if you replace your count of genus zero curve by count of iogenus 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, genus 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 edge bar to the power 2g minus 1 where g is a genus. And here the fact to call it edge bar is not a coincidence, the claim will be this deformation at the level of animative geometry replacing genus zero curve by iogenus curves is replacing our previous generating series by new generating series depending on edge bar. And the claim is that you can deform the mirror symmetry construction by now gluing simple non-combative pieces, you can glue them together using this generating series of iogenus invariants and you produce a non-combative algebra which is a deformation quantization of the mirror family. Is it associative? Yeah that's right, associative, yeah so it's part of the quantization condition, you produce some associative algebra, non-combative which is a deformation of the commutative algebra formula function. How do you translate associativity in that case into properties of the numerical deny? Yeah that's right, so 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 fit together in a way to produce some associative commutative algebra and the way it happens is not obvious at all, it's something you need to prove and it's part 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 it's even more true, the question why going to iogenus, 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 some other black books. Can you go back to the formula, formula you wrote for high-g scale? Yes. If you put g equal to zero for the disk, so it scales as h bar, scales as inverse power, but previously is it the same? Yeah, so this thing is like some normalization convention, so indeed here I'm starting the learning term as being one over h bar and so this thing will 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? Yeah, so roughly yes. And it's roughly because taking classical limit even in the most simple quantum mechanical example it's not as simple as just setting h by equal to zero, you need to take an appropriate limit. But if you do this correctly, yes. So for example I could have for simplicity just for get minus one, then I set h by equal to zero and I recover the previous formula. If it scales as one by eight, yes that's right, but this thing is just a choice 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 still, that's the question. So 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 set h by equal to zero naively, sometimes it's a wrong thing to do. But if you do this properly, everything works. Okay, 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 second kilzibbert 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 geochirm of lines. So you count your rational curves, you run the symmetry machinery and the output is exactly this family. And now running this quantum Iogenes version of the symmetry, running the machinery counting Iogenes 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 of algebra of business, this low 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 the left hand side here we realize your geometry has a character variety and then you use the fact that it is a character variety to run the scan algebra story to produce a deformation quantization. On the other side you just use the fact that it is a family of non-compact calabino surfaces which appear as a family of something. And then deforming this remark, the mirror symmetry construction you cook up a deformation quantization. And the theorem is that these two deformation quantizations 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. So at the very end there was some physics slide that probably I will not really go into. So the claim is that roughly seems a slightly surprising connection between low dimensional topology on one side. So on one side you are really studying links and knots. So you are drawing curve on a four puncture sphere. Whereas on the other side you are counting all the morphic 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 one morphic curve story is more natural. So here is the side already put at the beginning. It is a summary side or conclusive side. So on the c side adds low dimensional topology. On c side adds animative geometry. And 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 deformation quantization. And the two ways agree. And something I do not stress in c-stalk is that once you know that then you can derive non-trial application. Because you know about all the morphic curves in cubic surfaces, somehow you can prove stuff about curve on the four punctures field. Which is a bit surprising but it is 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 the same equals to 2 is zero written theory. So as far as I recall, zero written theory has a finite model space. It means finite number of points. How this is related to this plus, sorry, vector variety. This one is finite. It means finite number of points only count them to get an ingredient for the general choice of matrix. What do you mean by finite many points? So you said zero written geometry. Yeah, so. So between geometry means it satisfies this zero written equations and it keeps your model space which is finite number of points at the end. Yeah, so here I am using cybeg written geometry in some kind of physics sense. So this is not a model space of solutions with cybeg written equations. It is a different terminology. 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 cybeg written that given some n equal to four dimensional gauge theory. There is a corresponding geometry attached to it which in physics language controls the low energy effective theory. So in fact, so cybeg written arrived to the mathematical cybeg written equations through this kind of story but it's a bit long to explain the connection. 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 long story. Can you do that and the formation continuation with the geometry for other surfaces? Yeah, so in fact you could imagine realising the story. So in fact there are other examples where the character variety is still a family of surfaces and they are still this kind of delpato geometry complement of some divisor in the delpato geometry. And the ones which are very interesting are difficult. So definitely you can run this construction, you can run this deformation quantisation through innovative geometry but I don't approve that it's agree with a version of the sky and algebra. But forgetting the character variety side, so do them... Yeah, so certainly the... Do the pyrogene environments satisfy the right properties to give you... Yeah, yeah, that's right. They produce one deformation quantisation. What is difficult is to compute the numbers explicitly to know what is explicitly the locomotive algebra but yeah, so construction makes sense. But in this case you computed explicitly on your... Yeah, that's right. In some sense the story I just compute explicitly. You said that character varieties always emit a person structure and you gave us like the formula for some specific examples. Can you write down like a formula in general or I mean this is like a non-constructed proof? Yeah, so I think maybe in the world of character variety maybe some key word, it may be God band bracket, Poisson bracket. So I think you can write a formula in term of... For the torus I explained just how to do with intersection of homology classes. In more general you can still I guess more or less write down a formula in term of intersection of curves drawn in your surface but you need to be slightly careful about some kind of geometric intersection numbers. But 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... Here you are considered non-oriented links and nodes. What happens if you consider the oriented links and nodes? Yeah so the story I'll be describing is related to this Kaufman bracket, Skyin module, which somehow is formulated in terms of unoriented framed links. Like for R3 it produces the Kaufman polynomial but up to some easy transformation it's more than the same thing as the Jones polynomial and usually the Jones 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 something's Kaufman bracket scan algebra in which I think you consider oriented links and they are just different objects. So somehow the one which appears in this story is Kaufman bracket scan algebra but maybe in some not so unrated 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 SGN surfaces? So definitely in the example where I compute everything explicitly for these four puncture spheres the answer is yes and somehow it's one consequence of the story I'm describing. So for general 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 is it like very specific case or the story proved it. Is there no further questions? Let's just finish it again. If there are no further questions let's just finish it again.
 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 their 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  serialization 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 will be talking about knots, links, and related object.  In a complex animative algebraic geometry we will be interested in counting configuration  of geometric objects satisfying various constraints.  And here is a picture of one of the most well-known examples 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 a 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 the fact that sometimes a given supersymmetric gauge series of different  string serializations.  The point is that string theory often give geometric realizations of gauge theory dynamics  and when you get different string theory realizations of the same theory you get a prediction for  relation between 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 the Schuyler algebra  or the four-point sphere.  So 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 interval, open  interval 01.  The enumerative algebraic geometry part of the talk will be about counting allomorphic  curves in complex cubic surfaces.  And finally the physics part of the talk is about four-dimensional n-egal-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 part of the talk I will talk about the enumerative 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.  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 cross-section cross  interval and some of the abstract theories that you are able to prove such a thing by  studying the apriori completely different topic of counting Riemann surface, counting  all morphic curves in a complex cubic surface.  In this talk I will not focus on the applications to the scan algebra, I 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.  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 innovative geometry, on the left  there will be the low-dimensional topology and on the bottom there is the physics.  And I will come back to this diagram at the very end of the talk.  So the plan is first I will talk about the low-dimensional topology part of the story  and so 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 3D-mersion topology via the scan algebra.  Then I will talk about the innovative 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 Iogenius curve counting and the main point of the talk is a comparison  between the quantization of these cubic surface geometry given by 3D-mersion topology by the  scan algebra and the one given by Iogenius mirror symmetry, so the one given by innovative  geometry.  So Iogenius means you are talking about this current difference between this kind of Iogenius?  Yes, Iogenius, let's say Iogenius compact curve.  So let me start at the very beginning by saying very generally a few words about character  varieties.  So if you have gamma, finitely a generated group and g, a reductive algebraic group of  a complex number like GLNC or SNNC, then you can construct an affine variety of group morphism  from gamma to g.  And there is a natural action of g on these affine varieties 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 an 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 we'll be particularly interested 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 G compact orientable surface.  And so what is 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 this affine  variety.  So I will not say what this Poisson bracket is in general, I will just give you some example.  So if you just take L equal to 0, 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 by.  So ESC space is very 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  where 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 certainly more interesting example which will be the example in which we will  be focused in this talk is a case g equal to 0, so you start with a topological some  sphere, two dimensional sphere and a equal to 4, so we move 4 points from the sphere,  so it is what this picture is supposed to represent, you have some sphere and you remove  4 points out of it and you take for a 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 character variety that I will denote by x, so it is  a sl2c character variety for the 4 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 4 parameter family of affine cubic surfaces, so x has dimension  6, complex dimension 6 and it is a family of surfaces depending on 4 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 monogram around loops of the 4 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 you have a 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 monogram 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 7 loops and you will get 7 generators.  So there is a1, a2, a3, a4 which are trace around the small loops around punctures, so  in this picture there is an obvious loop, for each point to remove you can consider a  small loop around this puncture and you get 4 of them because there is 4 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 4 puncture sphere into 2 pair of pints.  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 pints  decomposition, pair of pints decomposition.  Can I ask a question, 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 singularities and so in this case we can study  differential equations on p1 or your 4 singularities and study the monodromy of these differential  equations they will produce, I mean the monodromy will be an element of these best acts.  So I just said that there are 7 generators corresponding to 7 particular loops and in  fact these 7 generators satisfy a single equation which is written explicitly here.  So this thing was already known to 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 3 variables gamma v1, gamma v2, gamma v3 and this 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 define you some affine cubic surface in like  affine 3 dimensional space.  But previously you said that it was a 4 parameter family of cubic surfaces but then it is 8 dimensional  right and then here you have it is the 6 dimensional thing.  Yeah I say it is at it was a 4 parameter family of surfaces so something 6 dimensional, 4  dimensional, 4 dimensional, 4 parameters in surfaces.  So here the 4 parameter a1, a2, a3, a4 and my surface is defined by 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 is an example you can write explicitly  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 were just a product of the corresponding trace function.  The point is that in this example it's like true at the leading order of the Poisson bracket  of gamma v1, gamma v2 is a product of gamma v1, gamma v2 but then there is a correction  term involving gamma v3.  So this problem in three variables you cannot split it into problem involving only two variables  which will be what happens in the case of the torus.  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 this  geometry appear in many different context.  So first of all it's a character variety and as any character variety by Raymond Hilbert  there will be some analytic asomorphism with a modu space of flag connection with regular  singularities so which is related to my answer to Kiosk question.  It has to do with second order linear differential equations.  And then by the non-nabian odd correspondence C space in fact homomorphic to a modu space  of parabolic X mandals and then you have some etching elliptic vibration story and in a  physics language this thing will be the sub-adjoint return geometry for this particular SU2  engagement theory.  Something very much related is that the smooth fibers of this family of surface, so this  affine cubic surfaces admits complete hypercalametrics and something which is maybe a bit more specific  to the four puncture sphere, this space is the 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 the 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 as goes through three dimensional topology  and the second is not well known and it involves some Iogenous version of mere symmetry.  And the non-trial result of the talk will be that these two ways to quantize this geometry  will agree.  So I guess this 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  natural questions like does such a thing exist is it unique and the kind of thing we will  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 or the character variety and in particular for the very specific  example which is of interest for us in this talk X the character variety SL2 character  variety on the four punctures sphere.  So here is an example of deformation quantization going back to the simple case where I have  no puncture and where my group is simply GL1 simply C star.  In this case I explained that the character variety is simply a complex algebraic torus  simply C star to 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 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 this 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 become rational in q.  So H is real or complex?  So in this talk everything 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 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 the coupled 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 algebra and which is coming from three dimensional  topology and so because I will only consider this SL2C story I will only describe this  scan algebra story for SL2C.  So here I'm describing some the way I've presented things I introduce is character  variety SL4SL2C for the four puncture sphere it's an example of Poisson variety you can  ask and then you can ask the algebraic problem or 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 Kaufman bracket scan module of any oriented stream  manifold it's a module of a ring Z bracket A plus minus so long 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 of the 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 follow up shape so so each time locally your frame links look like that you  can replace it by C's validation 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 previous size of framing just a knot with a nowhere vanishing  section of its normal bundle.  More concretely it's one way to view your knot as a boundary component of some annulus.  So this Kaufman bracket scan module is one way starting from a three manifold you take  all possible frame links you impose this relation you get some module.  And so for example if your three manifold is R3 you can show that this module is simply  the base ring of Laurent Poinomial 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 would be an element  in the base ring and it is a so-called Kaufman bracket polynomial of this frame link which  is essentially equivalent to the data of the joints polynomial of your link in R3.  Okay so the point is that if you take R3 this Kaufman bracket scan module is just the base  ring and it is just where essentially the joints or Kaufman bracket polynomial is leaving.  So some more says Kaufman bracket scan module it's come out historically from trying to  generalize the joints polynomial story to more general three manifold.  And so in this talk we'll focus on three 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 and 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 things  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.  So here's the question.  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 some more agree with the story described.  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.  I don't know the pictures.  Can you please show the pictures?  Stack one after another you say.  I cannot imagine just by looking at the thing.  It can even draw something so that I can learn.  I don't think I can draw anything but I can try.  So here is my surface.  Here is my interval and at some level I put some link.  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 L1.  And by definition the product of L1 L2 is simply the union of these two nodes.  So just a link in your three manifold and this thing is a product.  And why it's not commutative?  Sorry, I just still don't understand.  Yes, so here 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.  You have to pass.  You have some underlying assumption.  So I don't know if it's related but all the things are considered up to isotopy, up to  continuous deformations, embedded continuous deformations.  And the claim, so in some cases in fact you can move L1 L2 without crossing but it's like  the non-generic situation.  Generically you will have to cross and then it will be because of this crossing you will  get non-commutative.  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 dish-joint 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 so 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 3.  And you can put some obvious framing coming from the vertical third direction.  And the claim, Soci Serem of Pristiki tell you that other topic classes of multi-curves  in fact form a basis of a 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 this scan 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 bar over 4, then this thing is a deformation quantization of the algebra of regular function  on the SL2C character variety of your surface.  And essentially the isomorphism when you specialize A to minus 1 is obtained by sending a multi-curve,  so which as I say are basis of this kind of algebra.  So this collection of curves drawn on the surface, the corresponding function of the  character variety is just a trace along the various components of your curves.  So most of these multi-curve in the scan algebra are some kind of lift of the 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 this multi-curve to trace function, the scan relation will reproduce this kind of  identities.  Okay?  That's the classical limit of h bar goes to 0.  Yes, yes.  Okay, so the previous theorem is like general for any SL2C character variety of your surface,  the theorem tells you that the scan algebra produce 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 escape some detail and essentially you just compute the scan algebra and people  doing not have completed the scan algebra, the four puncture sphere 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  cubic relation.  And now this cubic relation in the scan algebra in the non-combative level, this cubic relation  is deformed in this cubic relation.  So there are also this non-commutative deformation of these cubic surfaces or final surfaces  more generally.  Can you, does that match in any way?  What do you mean?  I don't.  I think if you have like P2 for example, you can have non-commutative deformation and  more generally also final surfaces.  Can you view this 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 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.  So the point is I see things are just deformation of the previous equation except now everything  depends on the variable big A. When we said big A equal to 1, you recover the ordinary  equation of the cubic surface.  And what was not obvious was where to put the correct powers of A. For example in the  original equation I wrote, there was a minus 4 and the claim is that in the quantization  this 4 should be replaced by minus 2 times A to the 4 plus A to the minus 4.  And so some of the Scandinabian story tell you that it's correct to replace 4 by C's  polynomial in A and not something else.  And so this non-commutative surface appear in many, many different places from many,  many different point of view.  Is there any question about this story?  So there is this character right here of the four-point sphere.  I want to summarize it.  There is this general Scandinabian story which works for any SL2C character right here.  You just apply it for this geometry and you get some explicit answer, some explicit presentation  but you have relations of this kind of algebra which produce some non-commutative version  of the cubic surface.  If there is no question, I will somehow change topic and start talking about the different  way to produce a deformation quantization of the cubic surface and just do with an  automotive geometry.  Maybe first it has to do with murosimetry.  So very generally speaking murosimetry is some connection between let's say two Calabi  varieties.  It has 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 Calabi varieties.  Or precisely the case of log Calabi varieties which are 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 Calabi variety.  And so for this talk essentially I will use nothing about murosimetry except that it's  a way, essentially it's a, it's predicts that you should be able to construct algebraic  variety.  If you have a variety, its mirror should exist and people studying murosimetry think about  ways to construct this mirror.  And in particular we will use the fact that in dimension two, this kind of, this kind  of thing we came about in dimension two, there is a mirror symmetry construction for log  Calabi surfaces due to growth like in Cal and here this 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 murosimetry  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 Calabi surface YD by some appropriate counts of rational  curves in this pair, then it's possible to construct its mirror of family.  And its mirror of family here I wrote it curly V goes to spec C bracket N of Y.  So N of Y is a cone of effective curves on Y and C bracket is a corresponding monoid  algebra.  So essentially the mirror of construction gives you a mirror of 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 symplactic or color parameter or the original thing which has to do with essentially  H2 homology.  And so the claim is that it is possible to recover the particular geometry we care about  since X, this family of affine cubic surfaces over A4 can be obtained as an output of such  mirror of symmetry construction. And this mirror of family is obtained by applying the  mirror of 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  of configurations.  And you take that for D. So here there is something which is possibly 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 of family but the place where  the curve counting would be happening would be the mirror cubic surface.  So shall we complain that some of the cubic surfaces H2 equal to 7?  Yeah, you should complain.  And so essentially there is a way if you do things properly you get a 7 dimensional family  but there is a natural action of a 3 dimensional torus coming from the fact that this D has  3 components and when you quotient by this 3 dimensional torus you get this 4 dimensional  family.  Okay, so here is 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 an anti-canonical cycle and we assume a fractional  curves.  And then if you fix a curve class on your surface there is 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 are the property to intersect  the anti-canonical device or D in a single point.  So maybe there should be a picture somewhere.  So here is a picture, let's say on my cubic surface with 3 line D1, D2, D3 on it and then  I am looking in Y to rational curves which are the property to intersect this triangle  at a single point.  And this 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 there is this 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 device  order.  If you take some integral point which is on one of these ray here and these 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 the 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 are 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.  So roughly there is some kind of space of integral points which parameterize all possible  ways your curve can intersect the triangles at a single point.  You are counting all morphic or algebraic curves so when I say effective class, it is  some class which is represented by some all morphic or algebraic curve.  I mean it is in play in the fact that maybe at this point it is not fully clear what we  are doing.  So the point is that we are counting these curves so for every beta curve class you will  solve this innovative problem.  Counting this curve which is given class, which is given contact order, meeting the  boundary in one point and then you will get numbers.  So this thing is just innovative geometry on the surface and you get a lot of numbers  out of solving these innovative problems.  And maybe before answering your question I just give an example in our concrete geometry  where I was a smooth projective cubic surface in P3, d a triangle of lines on the cubic  surface and I said at the beginning there is 27 lines on such cubic so already 3 of  them form the boundary d so there is 24 of them remaining and the claim is that each  of the 24 lines not contained in d intersect d in a single point.  So you have this kind of picture where for each side of the triangle you have like 8  lines which will intersect this side in exactly one point and nothing else.  So these lines on this cubic surface are examples of curves we are counting in this  problem.  Can you say again what does that mean by quantization?  Quantization is whether you have a Poisson algebra, so it's a commutative algebra with  a Poisson bracket and here it's quantization in the sense of deformation quantization so  it's meant to construct a one parameter family depending on veg bar which deforms your commutative  algebra into some algebra which is typically non-commutative and with non-commutativity  is controlled by the Poisson bracket.  So if you're saying it sounds like deformation so what does that mean?  In some sense here is just an example of deformation as associative algebra.  It's just a particular type of deformation as associative algebra.  So the point is the Poisson bracket and the deformation is constrained by this Poisson  bracket.  So this becomes on the last line.  Yes.  As a crossing point of the three lines corresponds to what object on the left hand side.  Sorry.  The crossing point on the three lines on the right hand side.  Here.  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 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 the C's ray and so this thing  corresponds to curve intersecting this side of the triangle at a single point with contact  order one.  And 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.  So the crossing point is has no corresponding to the left hand side.  You mean the point here?  Yes.  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 on pole 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 innovative 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 involves  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 innovative problems.  So the most interesting is the point which would sound slightly mysterious but it is  what more or less mirror symmetry tells you to do to use these numbers this kind of rational  curves you construct generating series out of them and more or less you use the generating  series to define gluing transformations between simple pieces.  So I will not give more details here is some rough picture 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 involves in that definition all these counter-fragile  curves I was describing previously.  Should we view this mirror variety as an actual family or is it a variety with a regular function?  Yes, you get some 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 invariance  is naturally to have a variable t to the beta keeping track of the curve class beta.  So all series of 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 family over  these t variables and in general there are convergence issues but for example I am describing  everything converges and you get some honest algebraic family.  My question was more like why do you get a family of a single variety?  Yes, you get a family because you have these variable t's so the specific value of t is  like fixing one particular fiber.  So the hodge number of the mu is related to y or the hodge number of y is related to  mu?  The hodge number of mu?  Yes, because it is non-compact it is certainly subtle to phrase.  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 here some of the mirror is itself a family of affine cubic surfaces.  So some of these kind of two dimensional situations are kind of two sides of the mirror at least  topologically in fact they are the same.  It is like a bit similar to the fact that in dimension 2 a mirror of a k3 is again a  k3.  That is some of the construction of this new family do you have to come across which  touch the corner?  Yes, I pray 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 is cleaner to say that you have a fixed divisor  and a fixed variety and you can't curve which possibly go into the corner.  Because here you somehow say you primitive and you can't do it to this thing.  But this is when three copies glue together then you have to come to it.  Yes, I pray you need to count all of them.  In fact in the example there are really things going in the corner in fact there are a lot  a lot of them going into the corner.  So the upshot of this slightly fake story is that starting from a pair YD surface with  an anticanonical divisor you count these genital zero 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 this mirror family naturally comes with a Poisson structure.  So 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 that's our Poisson structure.  In fact the output is really a mirror, it's really a family of non-compact Calabria.  So yeah like Fabriola of Dimension 2, it's a family of olomorphic 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 this construction  using animative geometry you can ask is there a way to produce a deformation quantization  by deforming the construction coming from your symmetry.  And we deform this game of counting genital zero curve which produces a mirror family,  can you in some way deform it to produce deformation quantization of the mirror family?  Maybe I'll skip that.  And the claim is that you can do it if you replace your count of genus zero curve by  count of iogenus 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, genus 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  edge bar to the power 2g minus 1 where g is a genus.  And here the fact to call it edge bar is not a coincidence, the claim will be this deformation  at the level of animative geometry replacing genus zero curve by iogenus curves is replacing  our previous generating series by new generating series depending on edge bar.  And the claim is that you can deform the mirror symmetry construction by now gluing simple  non-combative pieces, you can glue them together using this generating series of iogenus invariants  and you produce a non-combative algebra which is a deformation quantization of the mirror  family.  Is it associative?  Yeah that's right, associative, yeah so it's part of the quantization condition, you produce  some associative algebra, non-combative which is a deformation of the commutative algebra  formula function.  How do you translate associativity in that case into properties of the numerical deny?  Yeah that's right, so 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 fit together in a way to produce some associative commutative  algebra and the way it happens is not obvious at all, it's something you need to prove and  it's part 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 it's even more true, the question why going to iogenus, 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 some other black  books.  Can you go back to the formula, formula you wrote for high-g scale?  Yes.  If you put g equal to zero for the disk, so it scales as h bar, scales as inverse power,  but previously is it the same?  Yeah, so this thing is like some normalization convention, so indeed here I'm starting the  learning term as being one over h bar and so this thing will 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?  Yeah, so roughly yes.  And it's roughly because taking classical limit even in the most simple quantum mechanical  example it's not as simple as just setting h by equal to zero, you need to take an appropriate  limit.  But if you do this correctly, yes.  So for example I could have for simplicity just for get minus one, then I set h by equal  to zero and I recover the previous formula.  If it scales as one by eight, yes that's right, but this thing is just a choice 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 still, that's the question.  So 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 set h by equal to zero naively, sometimes it's  a wrong thing to do.  But if you do this properly, everything works.  Okay, 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 second kilzibbert 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 geochirm of lines.  So you count your rational curves, you run the symmetry machinery and the output is exactly  this family.  And now running this quantum Iogenes version of the symmetry, running the machinery counting  Iogenes 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 of algebra of business, this low 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 the left hand side here we realize your geometry has a character variety and then  you use the fact that it is a character variety to run the scan algebra story to produce a  deformation quantization.  On the other side you just use the fact that it is a family of non-compact calabino surfaces  which appear as a family of something.  And then deforming this remark, the mirror symmetry construction you cook up a deformation  quantization.  And the theorem is that these two deformation quantizations 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.  So at the very end there was some physics slide that probably I will not really go into.  So the claim is that roughly seems a slightly surprising connection between low dimensional  topology on one side.  So on one side you are really studying links and knots.  So you are drawing curve on a four puncture sphere.  Whereas on the other side you are counting all the morphic 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  one morphic curve story is more natural.  So here is the side already put at the beginning.  It is a summary side or conclusive side.  So on the c side adds low dimensional topology.  On c side adds animative geometry.  And 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 deformation quantization.  And the two ways agree.  And something I do not stress in c-stalk is that once you know that then you can derive  non-trial application.  Because you know about all the morphic curves in cubic surfaces, somehow you can prove stuff  about curve on the four punctures field.  Which is a bit surprising but it is 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 the same equals to 2 is zero written theory.  So as far as I recall, zero written theory has a finite model space.  It means finite number of points.  How this is related to this plus, sorry, vector variety.  This one is finite.  It means finite number of points only count them to get an ingredient for the general  choice of matrix.  What do you mean by finite many points?  So you said zero written geometry.  Yeah, so.  So between geometry means it satisfies this zero written equations and it keeps your model  space which is finite number of points at the end.  Yeah, so here I am using cybeg written geometry in some kind of physics sense.  So this is not a model space of solutions with cybeg written equations.  It is a different terminology.  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 cybeg written that given some n equal to four dimensional  gauge theory.  There is a corresponding geometry attached to it which in physics language controls  the low energy effective theory.  So in fact, so cybeg written arrived to the mathematical cybeg written equations through  this kind of story but it's a bit long to explain the connection.  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 long story.  Can you do that and the formation continuation with the geometry for other surfaces?  Yeah, so in fact you could imagine realising the story.  So in fact there are other examples where the character variety is still a family of  surfaces and they are still this kind of delpato geometry complement of some divisor in the  delpato geometry.  And the ones which are very interesting are difficult.  So definitely you can run this construction, you can run this deformation quantisation  through innovative geometry but I don't approve that it's agree with a version of the sky and  algebra.  But forgetting the character variety side, so do them...  Yeah, so certainly the...  Do the pyrogene environments satisfy the right properties to give you...  Yeah, yeah, that's right.  They produce one deformation quantisation.  What is difficult is to compute the numbers explicitly to know what is explicitly the  locomotive algebra but yeah, so construction makes sense.  But in this case you computed explicitly on your...  Yeah, that's right.  In some sense the story I just compute explicitly.  You said that character varieties always emit a person structure and you gave us like the  formula for some specific examples.  Can you write down like a formula in general or I mean this is like a non-constructed proof?  Yeah, so I think maybe in the world of character variety maybe some key word, it may be God  band bracket, Poisson bracket.  So I think you can write a formula in term of...  For the torus I explained just how to do with intersection of homology classes.  In more general you can still I guess more or less write down a formula in term of intersection  of curves drawn in your surface but you need to be slightly careful about some kind of  geometric intersection numbers.  But 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...  Here you are considered non-oriented links and nodes.  What happens if you consider the oriented links and nodes?  Yeah so the story I'll be describing is related to this Kaufman bracket, Skyin module, which  somehow is formulated in terms of unoriented framed links.  Like for R3 it produces the Kaufman polynomial but up to some easy transformation it's more  than the same thing as the Jones polynomial and usually the Jones 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 something's Kaufman bracket scan algebra in which I think you consider oriented  links and they are just different objects.  So somehow the one which appears in this story is Kaufman bracket scan algebra but maybe  in some not so unrated 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 SGN surfaces?  So definitely in the example where I compute everything explicitly for these four puncture  spheres the answer is yes and somehow it's one consequence of the story I'm describing.  So for general 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 is it like very specific  case or the story proved it.  Is there no further questions?  Let's just finish it again.  If there are no further questions let's just finish it again.
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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 their 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 serialization 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 will be talking about knots, links, and related object. In a complex animative algebraic geometry we will be interested in counting configuration of geometric objects satisfying various constraints. And here is a picture of one of the most well-known examples 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 a 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 the fact that sometimes a given supersymmetric gauge series of different string serializations. The point is that string theory often give geometric realizations of gauge theory dynamics and when you get different string theory realizations of the same theory you get a prediction for relation between 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 the Schuyler algebra or the four-point sphere. So 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 interval, open interval 01. The enumerative algebraic geometry part of the talk will be about counting allomorphic curves in complex cubic surfaces. And finally the physics part of the talk is about four-dimensional n-egal-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 part of the talk I will talk about the enumerative 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. 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 cross-section cross interval and some of the abstract theories that you are able to prove such a thing by studying the apriori completely different topic of counting Riemann surface, counting all morphic curves in a complex cubic surface. In this talk I will not focus on the applications to the scan algebra, I 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. 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 innovative geometry, on the left there will be the low-dimensional topology and on the bottom there is the physics. And I will come back to this diagram at the very end of the talk. So the plan is first I will talk about the low-dimensional topology part of the story and so 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 3D-mersion topology via the scan algebra. Then I will talk about the innovative 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 Iogenius curve counting and the main point of the talk is a comparison between the quantization of these cubic surface geometry given by 3D-mersion topology by the scan algebra and the one given by Iogenius mirror symmetry, so the one given by innovative geometry. So Iogenius means you are talking about this current difference between this kind of Iogenius? Yes, Iogenius, let's say Iogenius compact curve. So let me start at the very beginning by saying very generally a few words about character varieties. So if you have gamma, finitely a generated group and g, a reductive algebraic group of a complex number like GLNC or SNNC, then you can construct an affine variety of group morphism from gamma to g. And there is a natural action of g on these affine varieties 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 an 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 we'll be particularly interested 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 G compact orientable surface. And so what is 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 this affine variety. So I will not say what this Poisson bracket is in general, I will just give you some example. So if you just take L equal to 0, 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 by. So ESC space is very 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 where 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 certainly more interesting example which will be the example in which we will be focused in this talk is a case g equal to 0, so you start with a topological some sphere, two dimensional sphere and a equal to 4, so we move 4 points from the sphere, so it is what this picture is supposed to represent, you have some sphere and you remove 4 points out of it and you take for a 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 character variety that I will denote by x, so it is a sl2c character variety for the 4 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 4 parameter family of affine cubic surfaces, so x has dimension 6, complex dimension 6 and it is a family of surfaces depending on 4 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 monogram around loops of the 4 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 you have a 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 monogram 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 7 loops and you will get 7 generators. So there is a1, a2, a3, a4 which are trace around the small loops around punctures, so in this picture there is an obvious loop, for each point to remove you can consider a small loop around this puncture and you get 4 of them because there is 4 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 4 puncture sphere into 2 pair of pints. 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 pints decomposition, pair of pints decomposition. Can I ask a question, 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 singularities and so in this case we can study differential equations on p1 or your 4 singularities and study the monodromy of these differential equations they will produce, I mean the monodromy will be an element of these best acts. So I just said that there are 7 generators corresponding to 7 particular loops and in fact these 7 generators satisfy a single equation which is written explicitly here. So this thing was already known to 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 3 variables gamma v1, gamma v2, gamma v3 and this 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 define you some affine cubic surface in like affine 3 dimensional space. But previously you said that it was a 4 parameter family of cubic surfaces but then it is 8 dimensional right and then here you have it is the 6 dimensional thing. Yeah I say it is at it was a 4 parameter family of surfaces so something 6 dimensional, 4 dimensional, 4 dimensional, 4 parameters in surfaces. So here the 4 parameter a1, a2, a3, a4 and my surface is defined by 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 is an example you can write explicitly 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 were just a product of the corresponding trace function. The point is that in this example it's like true at the leading order of the Poisson bracket of gamma v1, gamma v2 is a product of gamma v1, gamma v2 but then there is a correction term involving gamma v3. So this problem in three variables you cannot split it into problem involving only two variables which will be what happens in the case of the torus. 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 this geometry appear in many different context. So first of all it's a character variety and as any character variety by Raymond Hilbert there will be some analytic asomorphism with a modu space of flag connection with regular singularities so which is related to my answer to Kiosk question. It has to do with second order linear differential equations. And then by the non-nabian odd correspondence C space in fact homomorphic to a modu space of parabolic X mandals and then you have some etching elliptic vibration story and in a physics language this thing will be the sub-adjoint return geometry for this particular SU2 engagement theory. Something very much related is that the smooth fibers of this family of surface, so this affine cubic surfaces admits complete hypercalametrics and something which is maybe a bit more specific to the four puncture sphere, this space is the 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 the 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 as goes through three dimensional topology and the second is not well known and it involves some Iogenous version of mere symmetry. And the non-trial result of the talk will be that these two ways to quantize this geometry will agree. So I guess this 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 natural questions like does such a thing exist is it unique and the kind of thing we will 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 or the character variety and in particular for the very specific example which is of interest for us in this talk X the character variety SL2 character variety on the four punctures sphere. So here is an example of deformation quantization going back to the simple case where I have no puncture and where my group is simply GL1 simply C star. In this case I explained that the character variety is simply a complex algebraic torus simply C star to 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 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 this 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 become rational in q. So H is real or complex? So in this talk everything 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 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 the coupled 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 algebra and which is coming from three dimensional topology and so because I will only consider this SL2C story I will only describe this scan algebra story for SL2C. So here I'm describing some the way I've presented things I introduce is character variety SL4SL2C for the four puncture sphere it's an example of Poisson variety you can ask and then you can ask the algebraic problem or 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 Kaufman bracket scan module of any oriented stream manifold it's a module of a ring Z bracket A plus minus so long 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 of the 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 follow up shape so so each time locally your frame links look like that you can replace it by C's validation 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 previous size of framing just a knot with a nowhere vanishing section of its normal bundle. More concretely it's one way to view your knot as a boundary component of some annulus. So this Kaufman bracket scan module is one way starting from a three manifold you take all possible frame links you impose this relation you get some module. And so for example if your three manifold is R3 you can show that this module is simply the base ring of Laurent Poinomial 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 would be an element in the base ring and it is a so-called Kaufman bracket polynomial of this frame link which is essentially equivalent to the data of the joints polynomial of your link in R3. Okay so the point is that if you take R3 this Kaufman bracket scan module is just the base ring and it is just where essentially the joints or Kaufman bracket polynomial is leaving. So some more says Kaufman bracket scan module it's come out historically from trying to generalize the joints polynomial story to more general three manifold. And so in this talk we'll focus on three 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 and 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 things 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. So here's the question. 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 some more agree with the story described. 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. I don't know the pictures. Can you please show the pictures? Stack one after another you say. I cannot imagine just by looking at the thing. It can even draw something so that I can learn. I don't think I can draw anything but I can try. So here is my surface. Here is my interval and at some level I put some link. 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 L1. And by definition the product of L1 L2 is simply the union of these two nodes. So just a link in your three manifold and this thing is a product. And why it's not commutative? Sorry, I just still don't understand. Yes, so here 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. You have to pass. You have some underlying assumption. So I don't know if it's related but all the things are considered up to isotopy, up to continuous deformations, embedded continuous deformations. And the claim, so in some cases in fact you can move L1 L2 without crossing but it's like the non-generic situation. Generically you will have to cross and then it will be because of this crossing you will get non-commutative. 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 dish-joint 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 so 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 3. And you can put some obvious framing coming from the vertical third direction. And the claim, Soci Serem of Pristiki tell you that other topic classes of multi-curves in fact form a basis of a 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 this scan 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 bar over 4, then this thing is a deformation quantization of the algebra of regular function on the SL2C character variety of your surface. And essentially the isomorphism when you specialize A to minus 1 is obtained by sending a multi-curve, so which as I say are basis of this kind of algebra. So this collection of curves drawn on the surface, the corresponding function of the character variety is just a trace along the various components of your curves. So most of these multi-curve in the scan algebra are some kind of lift of the 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 this multi-curve to trace function, the scan relation will reproduce this kind of identities. Okay? That's the classical limit of h bar goes to 0. Yes, yes. Okay, so the previous theorem is like general for any SL2C character variety of your surface, the theorem tells you that the scan algebra produce 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 escape some detail and essentially you just compute the scan algebra and people doing not have completed the scan algebra, the four puncture sphere 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 cubic relation. And now this cubic relation in the scan algebra in the non-combative level, this cubic relation is deformed in this cubic relation. So there are also this non-commutative deformation of these cubic surfaces or final surfaces more generally. Can you, does that match in any way? What do you mean? I don't. I think if you have like P2 for example, you can have non-commutative deformation and more generally also final surfaces. Can you view this 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 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. So the point is I see things are just deformation of the previous equation except now everything depends on the variable big A. When we said big A equal to 1, you recover the ordinary equation of the cubic surface. And what was not obvious was where to put the correct powers of A. For example in the original equation I wrote, there was a minus 4 and the claim is that in the quantization this 4 should be replaced by minus 2 times A to the 4 plus A to the minus 4. And so some of the Scandinabian story tell you that it's correct to replace 4 by C's polynomial in A and not something else. And so this non-commutative surface appear in many, many different places from many, many different point of view. Is there any question about this story? So there is this character right here of the four-point sphere. I want to summarize it. There is this general Scandinabian story which works for any SL2C character right here. You just apply it for this geometry and you get some explicit answer, some explicit presentation but you have relations of this kind of algebra which produce some non-commutative version of the cubic surface. If there is no question, I will somehow change topic and start talking about the different way to produce a deformation quantization of the cubic surface and just do with an automotive geometry. Maybe first it has to do with murosimetry. So very generally speaking murosimetry is some connection between let's say two Calabi varieties. It has 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 Calabi varieties. Or precisely the case of log Calabi varieties which are 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 Calabi variety. And so for this talk essentially I will use nothing about murosimetry except that it's a way, essentially it's a, it's predicts that you should be able to construct algebraic variety. If you have a variety, its mirror should exist and people studying murosimetry think about ways to construct this mirror. And in particular we will use the fact that in dimension two, this kind of, this kind of thing we came about in dimension two, there is a mirror symmetry construction for log Calabi surfaces due to growth like in Cal and here this 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 murosimetry 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 Calabi surface YD by some appropriate counts of rational curves in this pair, then it's possible to construct its mirror of family. And its mirror of family here I wrote it curly V goes to spec C bracket N of Y. So N of Y is a cone of effective curves on Y and C bracket is a corresponding monoid algebra. So essentially the mirror of construction gives you a mirror of 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 symplactic or color parameter or the original thing which has to do with essentially H2 homology. And so the claim is that it is possible to recover the particular geometry we care about since X, this family of affine cubic surfaces over A4 can be obtained as an output of such mirror of symmetry construction. And this mirror of family is obtained by applying the mirror of 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 of configurations. And you take that for D. So here there is something which is possibly 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 of family but the place where the curve counting would be happening would be the mirror cubic surface. So shall we complain that some of the cubic surfaces H2 equal to 7? Yeah, you should complain. And so essentially there is a way if you do things properly you get a 7 dimensional family but there is a natural action of a 3 dimensional torus coming from the fact that this D has 3 components and when you quotient by this 3 dimensional torus you get this 4 dimensional family. Okay, so here is 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 an anti-canonical cycle and we assume a fractional curves. And then if you fix a curve class on your surface there is 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 are the property to intersect the anti-canonical device or D in a single point. So maybe there should be a picture somewhere. So here is a picture, let's say on my cubic surface with 3 line D1, D2, D3 on it and then I am looking in Y to rational curves which are the property to intersect this triangle at a single point. And this 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 there is this 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 device order. If you take some integral point which is on one of these ray here and these 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 the 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 are 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. So roughly there is some kind of space of integral points which parameterize all possible ways your curve can intersect the triangles at a single point. You are counting all morphic or algebraic curves so when I say effective class, it is some class which is represented by some all morphic or algebraic curve. I mean it is in play in the fact that maybe at this point it is not fully clear what we are doing. So the point is that we are counting these curves so for every beta curve class you will solve this innovative problem. Counting this curve which is given class, which is given contact order, meeting the boundary in one point and then you will get numbers. So this thing is just innovative geometry on the surface and you get a lot of numbers out of solving these innovative problems. And maybe before answering your question I just give an example in our concrete geometry where I was a smooth projective cubic surface in P3, d a triangle of lines on the cubic surface and I said at the beginning there is 27 lines on such cubic so already 3 of them form the boundary d so there is 24 of them remaining and the claim is that each of the 24 lines not contained in d intersect d in a single point. So you have this kind of picture where for each side of the triangle you have like 8 lines which will intersect this side in exactly one point and nothing else. So these lines on this cubic surface are examples of curves we are counting in this problem. Can you say again what does that mean by quantization? Quantization is whether you have a Poisson algebra, so it's a commutative algebra with a Poisson bracket and here it's quantization in the sense of deformation quantization so it's meant to construct a one parameter family depending on veg bar which deforms your commutative algebra into some algebra which is typically non-commutative and with non-commutativity is controlled by the Poisson bracket. So if you're saying it sounds like deformation so what does that mean? In some sense here is just an example of deformation as associative algebra. It's just a particular type of deformation as associative algebra. So the point is the Poisson bracket and the deformation is constrained by this Poisson bracket. So this becomes on the last line. Yes. As a crossing point of the three lines corresponds to what object on the left hand side. Sorry. The crossing point on the three lines on the right hand side. Here. 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 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 the C's ray and so this thing corresponds to curve intersecting this side of the triangle at a single point with contact order one. And 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. So the crossing point is has no corresponding to the left hand side. You mean the point here? Yes. 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 on pole 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 innovative 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 involves 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 innovative problems. So the most interesting is the point which would sound slightly mysterious but it is what more or less mirror symmetry tells you to do to use these numbers this kind of rational curves you construct generating series out of them and more or less you use the generating series to define gluing transformations between simple pieces. So I will not give more details here is some rough picture 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 involves in that definition all these counter-fragile curves I was describing previously. Should we view this mirror variety as an actual family or is it a variety with a regular function? Yes, you get some 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 invariance is naturally to have a variable t to the beta keeping track of the curve class beta. So all series of 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 family over these t variables and in general there are convergence issues but for example I am describing everything converges and you get some honest algebraic family. My question was more like why do you get a family of a single variety? Yes, you get a family because you have these variable t's so the specific value of t is like fixing one particular fiber. So the hodge number of the mu is related to y or the hodge number of y is related to mu? The hodge number of mu? Yes, because it is non-compact it is certainly subtle to phrase. 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 here some of the mirror is itself a family of affine cubic surfaces. So some of these kind of two dimensional situations are kind of two sides of the mirror at least topologically in fact they are the same. It is like a bit similar to the fact that in dimension 2 a mirror of a k3 is again a k3. That is some of the construction of this new family do you have to come across which touch the corner? Yes, I pray 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 is cleaner to say that you have a fixed divisor and a fixed variety and you can't curve which possibly go into the corner. Because here you somehow say you primitive and you can't do it to this thing. But this is when three copies glue together then you have to come to it. Yes, I pray you need to count all of them. In fact in the example there are really things going in the corner in fact there are a lot a lot of them going into the corner. So the upshot of this slightly fake story is that starting from a pair YD surface with an anticanonical divisor you count these genital zero 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 this mirror family naturally comes with a Poisson structure. So 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 that's our Poisson structure. In fact the output is really a mirror, it's really a family of non-compact Calabria. So yeah like Fabriola of Dimension 2, it's a family of olomorphic 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 this construction using animative geometry you can ask is there a way to produce a deformation quantization by deforming the construction coming from your symmetry. And we deform this game of counting genital zero curve which produces a mirror family, can you in some way deform it to produce deformation quantization of the mirror family? Maybe I'll skip that. And the claim is that you can do it if you replace your count of genus zero curve by count of iogenus 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, genus 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 edge bar to the power 2g minus 1 where g is a genus. And here the fact to call it edge bar is not a coincidence, the claim will be this deformation at the level of animative geometry replacing genus zero curve by iogenus curves is replacing our previous generating series by new generating series depending on edge bar. And the claim is that you can deform the mirror symmetry construction by now gluing simple non-combative pieces, you can glue them together using this generating series of iogenus invariants and you produce a non-combative algebra which is a deformation quantization of the mirror family. Is it associative? Yeah that's right, associative, yeah so it's part of the quantization condition, you produce some associative algebra, non-combative which is a deformation of the commutative algebra formula function. How do you translate associativity in that case into properties of the numerical deny? Yeah that's right, so 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 fit together in a way to produce some associative commutative algebra and the way it happens is not obvious at all, it's something you need to prove and it's part 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 it's even more true, the question why going to iogenus, 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 some other black books. Can you go back to the formula, formula you wrote for high-g scale? Yes. If you put g equal to zero for the disk, so it scales as h bar, scales as inverse power, but previously is it the same? Yeah, so this thing is like some normalization convention, so indeed here I'm starting the learning term as being one over h bar and so this thing will 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? Yeah, so roughly yes. And it's roughly because taking classical limit even in the most simple quantum mechanical example it's not as simple as just setting h by equal to zero, you need to take an appropriate limit. But if you do this correctly, yes. So for example I could have for simplicity just for get minus one, then I set h by equal to zero and I recover the previous formula. If it scales as one by eight, yes that's right, but this thing is just a choice 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 still, that's the question. So 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 set h by equal to zero naively, sometimes it's a wrong thing to do. But if you do this properly, everything works. Okay, 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 second kilzibbert 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 geochirm of lines. So you count your rational curves, you run the symmetry machinery and the output is exactly this family. And now running this quantum Iogenes version of the symmetry, running the machinery counting Iogenes 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 of algebra of business, this low 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 the left hand side here we realize your geometry has a character variety and then you use the fact that it is a character variety to run the scan algebra story to produce a deformation quantization. On the other side you just use the fact that it is a family of non-compact calabino surfaces which appear as a family of something. And then deforming this remark, the mirror symmetry construction you cook up a deformation quantization. And the theorem is that these two deformation quantizations 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. So at the very end there was some physics slide that probably I will not really go into. So the claim is that roughly seems a slightly surprising connection between low dimensional topology on one side. So on one side you are really studying links and knots. So you are drawing curve on a four puncture sphere. Whereas on the other side you are counting all the morphic 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 one morphic curve story is more natural. So here is the side already put at the beginning. It is a summary side or conclusive side. So on the c side adds low dimensional topology. On c side adds animative geometry. And 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 deformation quantization. And the two ways agree. And something I do not stress in c-stalk is that once you know that then you can derive non-trial application. Because you know about all the morphic curves in cubic surfaces, somehow you can prove stuff about curve on the four punctures field. Which is a bit surprising but it is 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 the same equals to 2 is zero written theory. So as far as I recall, zero written theory has a finite model space. It means finite number of points. How this is related to this plus, sorry, vector variety. This one is finite. It means finite number of points only count them to get an ingredient for the general choice of matrix. What do you mean by finite many points? So you said zero written geometry. Yeah, so. So between geometry means it satisfies this zero written equations and it keeps your model space which is finite number of points at the end. Yeah, so here I am using cybeg written geometry in some kind of physics sense. So this is not a model space of solutions with cybeg written equations. It is a different terminology. 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 cybeg written that given some n equal to four dimensional gauge theory. There is a corresponding geometry attached to it which in physics language controls the low energy effective theory. So in fact, so cybeg written arrived to the mathematical cybeg written equations through this kind of story but it's a bit long to explain the connection. 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 long story. Can you do that and the formation continuation with the geometry for other surfaces? Yeah, so in fact you could imagine realising the story. So in fact there are other examples where the character variety is still a family of surfaces and they are still this kind of delpato geometry complement of some divisor in the delpato geometry. And the ones which are very interesting are difficult. So definitely you can run this construction, you can run this deformation quantisation through innovative geometry but I don't approve that it's agree with a version of the sky and algebra. But forgetting the character variety side, so do them... Yeah, so certainly the... Do the pyrogene environments satisfy the right properties to give you... Yeah, yeah, that's right. They produce one deformation quantisation. What is difficult is to compute the numbers explicitly to know what is explicitly the locomotive algebra but yeah, so construction makes sense. But in this case you computed explicitly on your... Yeah, that's right. In some sense the story I just compute explicitly. You said that character varieties always emit a person structure and you gave us like the formula for some specific examples. Can you write down like a formula in general or I mean this is like a non-constructed proof? Yeah, so I think maybe in the world of character variety maybe some key word, it may be God band bracket, Poisson bracket. So I think you can write a formula in term of... For the torus I explained just how to do with intersection of homology classes. In more general you can still I guess more or less write down a formula in term of intersection of curves drawn in your surface but you need to be slightly careful about some kind of geometric intersection numbers. But 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... Here you are considered non-oriented links and nodes. What happens if you consider the oriented links and nodes? Yeah so the story I'll be describing is related to this Kaufman bracket, Skyin module, which somehow is formulated in terms of unoriented framed links. Like for R3 it produces the Kaufman polynomial but up to some easy transformation it's more than the same thing as the Jones polynomial and usually the Jones 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 something's Kaufman bracket scan algebra in which I think you consider oriented links and they are just different objects. So somehow the one which appears in this story is Kaufman bracket scan algebra but maybe in some not so unrated 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 SGN surfaces? So definitely in the example where I compute everything explicitly for these four puncture spheres the answer is yes and somehow it's one consequence of the story I'm describing. So for general 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 is it like very specific case or the story proved it. Is there no further questions? Let's just finish it again. If there are no further questions let's just finish it again.
quantization of these cubic surface geometry denote by SGL which is a topological surface Poisson bracket on the algebra of regular puncture sphere and as I already repeated it is a quantization. And I will explain that there is algebra is a solution to the deformation the output of the mirror symmetry construction are the trace functions of SL2 character