 So, first of all, it's a great pleasure to be here and I must say I was influenced by Maxime many times during my life. And what I will be talking about today is actually a joint work in progress with Maxime and Jan Soyberman. So let me start with writing the title and then somehow getting myself out of trouble with the title. Infrared, secondary, joint is joined to work with Maxime and Jan Soyberman. So, first of all, it may be a little unusual for mathematicians the term algebra of the infrared. It was borrowed from the title of the work of Gaiotte, Moore and Witten of this title. So, I will not start with this, I'll just say that this term comes from this work. I'll explain a little bit later and now let me do the most elementary logical exposition which will start with secondary polytops and their properties and then turns out that some of the properties are relevant for this physical point of view. So let me start maybe here, call it a reminder on secondary polytops, part one. So this is a concept that was introduced in my joint work with Gelfand and Zelivinski way back in the 90s with the goal of studying discriminants of polynomials and many variables. So I recall the context and then I recall the main setting and recall what was the old point of view on this setting and what somehow is the new source of examples for this setting. So we start with a finite set of points in Euclidean space. So in the old setting it was really a point, the set consisted of points with integer coordinates is the set of monomials, we interpret integer point as a Varan monomials and then we consider the space of polynomials over those monomials and look for those polynomials that define singular algebraic hyper surfaces. But this construction makes sense for any subset of points in Euclidean space integer or not. So we imagine something like this, so a bunch of points will take the convex hull, then some of the points are inside the convex hull, some points are vertices of the convex hull, let's call it Q, it's a convex hull of A, some convex polytop and we're looking at triangulations of this polytop into simplices with vertices in A, something like this, something maybe like this and maybe there are some other points of A which are not being used in the triangulations, so let me move it up. I'll just read here. We consider the totality of triangulations, we've called them T of Q into straight simplices with vertices in A. Now inside this set, it's a finite set because we fixed then the finite set of possible vertices, inside there are so-called regular triangulations, those which possess a piecewise convex function with respect to this triangulation, which is actually strictly convex in the sense that actually breaks down on every intermediate phase, regular triangulations. Does exist, strictly convex, T piecewise linear function. Do you allow it to be constant in one phase? Yes, it may be constant, it might be linear, so yes, so and the secondary polytop is a polytop in the space, or let me write it here, so sigma of A, this is the secondary polytop, it lies in the space of functions on A, which means R to the A, is a polytop, again convex of dimension equals cardinality of A minus D minus 1 whose vertices are in bijection with this type of triangulations, in general it is strict inclusion. So vertices, I call them, let's say phi T, phi T of regular triangulations. So it's not a definition, it's a reminder, yes, so it can be, so this is a certain particular function, so this is a function from A to R, I can write it, phi T is a function from A to R, and phi T of a point W is the sum of the volumes of the simplices of the star of the point W in the triangulation, equals volume of star, that's the definition and the theorem is that all those points are actually vertices of some convex polytop, which are priori they, it's not clear why they are. It's possible, but this has not been started, in particular it's an interesting direction is the hyperbolic case, in which case it's very close to the Penner's construction of triangulation. You define this convex set by the function, because you get the same notation for function of a convex set, and then you... No, so it's a function on the finite set, so it's A is a finite set, so we consider a function, so you can euclidean space of dimension equal to the cardinality of A. So you define it as a convex function of a convex set? No, no, I define the vector, a vector. Vertices of polyhedral, phi sub... Vertices of sigma of A. Elements of, functions of A. Yes, so the polytop is in the space of functions on A. So each vertex is an individual function on A. To each triangulation we can associate an individual function, which to every point associates the volume of the star of this point. It's a mysterious definition. It is mysterious, yes. Yes, so I'll say a little more in the reminder. So an example of this, and... Again, the object is a polytop, right? The object is a polytop, yes. You define function, I'm going to show you. I define what would be a vertex of the polytop. I define vertices of polytop. Individual vertex is individual of this thing. So altogether they form the set of vertices. You see that all these functions we're pointing to also. All, yes, to all triangulations. All triangulations, yes. So you find this set, and then you say convex, convex, convex. Yes. But you don't know a priori it's convex or not? A priori, a priori, it's a theorem that all those points exactly are in the convex position. It's a theorem. So consider an example. Suppose A itself is a set in the plane. So D equals top. And this is in convex position, or A itself in the convex position. Means it's something like this. So all the points are here, right? We label them, say, 0, 1, 2, n. Then a triangulation is the same as a bracketing of symbols. So for instance, I write here, say, A1, I write A2. Here I write An. So here I write some bracketing. This will write A1, A2. For instance, this is 3. So this is 4. So this would be A3, A4. This would be the bracketing like this, A3, A4. And so on. So here it would be bracketing. So in this case, sigma of A is the so-called Stashev polytope, or Asashaya hydra. So vertices are bracketing of A1, An. And those polytopes are, of course, very classical, very well known. And what's important, let's call it polytope k, n plus 1, kn, that they form an operand, which means every phase of Asashaya hydra is a product of other Asashaya hydra. Can you say, why is this dimension, why is this dimension, why is it all dimension? So this is the space. This corresponds for global affine linear functions. No, maybe I just explained something. Let me show you these are formulas. You can see the functions on set A, on all real functions. And you get kind of low convex hull. And you can get different pieces in the composition. You get on space of linear function, you get some kind of dual polytope for this phase. That's it. So every phase, so we have maps like this, kn cross k A1 cross odd, cross k An is embedded as a phase into k A1 plus An. So this is a structure of an operand. So what happens in general is that the arbitrary secondary polytope also has this type of structure. So every phase is a product of other secondary polytopes, just like here. So let me call this the factorization property. Let's say that every phase of sigma of A is a product of other secondary polytopes, of other sigma of some AI. More precisely, one can describe what the phases are. So then, indeed, it's probably better to use the convex hull construction as Maxime pointed out. This goes, yes, yes, but I'm afraid that with the third one, I'll paint myself into a corner. I don't know what to do with that. No? OK. You can make any problem. Ah, OK. OK, maybe I'll use it next time. Yes. So what really happens is that phases correspond to triangulations that phases correspond to something in between, which is subdivision, but not necessarily into simplices. So phases Fp correspond to, again, regular, in this sense, regular polyhedral subdivisions. It's called p, which says they represent q as the union of some polytopes qi or something like this, for instance. So I suppose we have this point here, and this we subdivided our hexagon into three quadrilaterals. To be completely precise, we need this plus a choice of A subset ai in A containing the vertices, such that qi is the convex hull of ai. So to really speaking, there is some freedom here. So in the corresponding phase, Fp is simply a product of sigma of ai. There is a particular choice. So we can, for instance, ask, why don't we put ai to be qi intersected with a? We can do this, but in this way, we don't get all the phases. So let's call this the geometric phases. They're also interesting. So-called geometric phases. So anyway, we have this kind of self-referential factorizing structure that every phase of the polytope is a product of similar polytopes, which is something like the operatic structure, but really is a little more general. And one has to understand it in some other way. So now let me do an equally elementary reminder about Lie algebras and L infinity algebras. So if I recall it, we have a G-Lie algebra. Then to this, we associate this co-chain complex, which is the exterior algebra of G is the differential, which is a commutative dG algebra. So in more general, a Lie infinity algebra is sort of a derived analog of a Lie algebra. So it's something like a complex. Yes, yes, dole, dole, dole, yes, of course. Yes, yes. With the differential and with bracketing, which only satisfies Jacobi identity up to the next bracketing. So there is a differential d, which we consider as operation of order 1. We have bracketing from G tensor G to G, which consider operation of order 2. But Jacobi up to lambda 3 from G cube to G and so on. And this is encoded, again, as very familiar, by differential in the symmetric algebra of the dual space to G and shifted by 1. And some differential, let's call it d, satisfying that d squared equals 0. So one identity, d squared equals 0, it contains in itself, concerned the fact that this is 0, that the Jacobiator of this is the boundary of this minus 1. No, that actually I could never remember. Yes. So it goes, for instance, if you look into this, G star goes into, well, first of all, is G star of minus 1. That's the differential, that is lambda 1. It goes to S2 of G star of minus 1, which is in the more familiar notation. It's lambda 2 of G star shifted by minus 2. So that's the dual to the bracket. So then the next would be lambda 3 and so on. So from this point of view, this one condition is a sequence of this recursive conditions going forever. And it contains the Jacobi identity. Do you have a strong reason for setting lambda 0 to 0? What is lambda 0? Is the curvature? No, yeah, but also it's the constant term in d. In those applications, this so far was not necessary. But maybe later. That's the answer to that question. Yes, maybe later it will be. OK, so now if I do a permutation. Permutation, so this goes up and the two of those go down. This is a permutation. So now let me go to part 3 to combine those two observations and let me call it the chain complex. Chain complexes of the secondary polytope chain or let's say of secondary polytops in general. So I keep this notation. If I have any convex polytope whatsoever, let's consider the most naive cellular chain complex called c of p. So it's just the dimensions of the chains of the i-chains with the number of faces. I can write it like this, direct sum over f being faces. And here, I was strictly speaking, the corresponding sum is this orientation space. So let's say we work over a field k. So here, we have orientation space of f and simply shift by dimension of f. Because f capital N. Yes, over all faces. Yes, over all faces, the most naive thing that is studied in elementary courses of combinatorial topology. So this means in degree minus the dimension of f. So in here, of course, the differential squared equals 0. So now let's apply this to the secondary polytope and apply the information about the structure of faces. Apply to p equals sigma of a. So what really happens is that the differential takes a face into some of all of its subfaces of good dimension 1. But subfaces are products of other secondary polytopes. So we have some kind of multiplicative structure. So it looks like a differential in an algebra. And let's do this. So let's form a vector space v to be direct sum over all a prime of those subfaces v a prime. And this is simply v i prime. It's simply this. Or it is orientation of sigma of a prime shifted by dimension. Yes, otherwise it sort of collapses. Yes, cardinality of a prime. So this is a vector space. And let's form a symmetric algebra of this vector space. Form the symmetric algebra. So with this operation, call it dot. So my claim is that the factorization property of the secondary polytopes implies that this algebra acquires a differential, an algebra differential, satisfying d squared equals 0. So let me first state this claim. The factorization property gives a canonical algebra differential in symmetric algebra of v, satisfying differential d, satisfying d squared equals 0. So basically, it is unraveling of these definitions. Let me explain what happens on generators. You mean the duration? Yes, I mean, it means satisfying the lameness rule. Yes, yes. So as usual, such a derivation can be described on generators. So maybe I'll do it here. On generators say v a prime. And here I'm going to use the chain complex of the secondary polytope of a prime. So use chain complex of. Ah, OK. So sigma of a prime is the secondary polytope. v is just one dimensional space which is associated to this. The summand in the chain complex. So I want to combine them all together. So the chain complex of individual polytope is a finite dimensional thing. But because of the kind of regularity present in this factorization property, they can be combined together into differential and infinite dimensional algebra. So the algebra itself is infinite dimension. It will be finally generated. So we use the chain complex of the secondary polytope. So look at faces of sigma of a prime. They correspond to subdivisions, fp prime, subdivisions, into some, let's say, q second i, a second i. So what happens is the part of the differential of the chain complex of the secondary polytope goes into from v i, v a prime, to this symmetric product of v a second i. So in this, we take this, take differential on v a prime to be sum of those for all faces of co-dimension one. Sum of this for all co-dimension one faces. So we sort of patch together for every generator. We consider its own secondary polytope and use a piece of its chain complex. And then because in the setting, it is clear that d squared equals 0 just because for every polytope, d squared equals 0. So now, equivalently, it means that just from nothing, just from a bunch of secondary polytopes, we produced the infinity algebra. So what we have, we have a differential in this symmetric algebra of a vector space, which means that appropriately dual and shifted vector space becomes an infinity algebra. So equivalently, let's call this vector space g. It is the dual space to my space of generators. Right there, v star shifted by 1. Again, it's a direct sum over subsets a prime in a, cardinality of a prime greater than d plus 1. Here I have the orientation space. It can be dualized, not dualized. It's identified with its dual. Here I write minus 1, minus dimension, sigma of a prime. So e is a linear infinity algebra. Minus 1. OK, so right here. If one, that was my minus 1. OK, OK, yes, yes, yes, you're right. Could never. So just from nothing, we had produced certain the infinity algebra. And we can understand, by looking at what happens here, what are phases of dimension 1, what is the nature of operations in this infinity algebra. So lambda n, the nth linear operation, is approximately operation of gluing together. Is the operation of gluing together and sub-polytops. Sub-polytops in a subdivision corresponding to a phase of co-dimension 1. In and let's call this in a coarse subdivision. So if triangulations are vertices and arbitrary phases are subdivisions, that phases of co-dimension 1 are sort of minimal subdivisions which you cannot do less than that. And let's call them coarse. So what are examples of this? So for instance, if we dissect our polytop into two parts by hyperplane into A and B, then it gives me the binary bracket, which simply says that A, B is the whole thing. What's in the blue cap lambda 1 as well? lambda 1 is the differential. So a priori in this setting, we would have it. It would correspond to a subdivision which are differential of this. It would be sum over inserting of all the internal points. Yes, a priori, there will be a differential here. Yes. Again, if we restrict to geometric phases where this freedom is removed, then the differential will be 0. But this is by no means the only operation. So for instance, it will be example number 1. Example number 2, if you subdivide something like this, if there is three parts, A, B, and C, then lambda 3 of A, B, C would be the whole thing. And in general, in higher dimensions, there are many more possibilities for doing this. So there exist more examples. So it's sort of geometric way of gluing together, geometric way of formulating gluing together polytops. Yes, algebraic, yes. Gluing together is geometric, and this is algebraic, yes. Right. Yes, maybe, yeah, no, I can say this too. So let me just say that this lean infinity algebra is nilpotent, because clearly there is a limit on how far we can go. It's a nilpotent lean infinity algebra. So let me now describe a version of this. Once we sort of get into the hang of this construction, then we can think, OK, so now we took chain complex. Let's take chain complex with coefficients. There are not only homology with constant coefficients. There are homology with coefficients in the sheaf. So how we construct something where coefficients will be present? And let me, this actually is an interesting question. So let me explain this in the most interesting for us is the case of two dimensions. Let me call this part versions with coefficients in two dimensions. So then a good sort of setting would be like this, that for every point of our configuration, there would be an algebra for every, let me call them, i point, or a dg algebra associative, let's call it si, differential graded. So for instance, if just a base field, it's OK. For every pair of indices, it may or may not be commutative, but we don't require this. A bi-module, nij. So bi-module over si, left over si, right over sj. Besides that, nji, so we assume this is a projective. Projective over the tensor product si tensor sj op. So just if you forget the differential. So it's something that we can always do this up to, just for simplicity. And I want to assume self-duality in the following form, nji equals nij star, but I have to explain what it is nij star to be home, well, we can say home over this dual, over this space, over si tensor sj op from nij to si tensor sj op. Then this would be a system of coefficients. Then for every sub-polygon, we have, say, Q prime. Q prime on vertices in our set. We can form, let's say, Q prime, A prime. A prime as the set of vertices. Well, and we can form the cyclic tensor product. And Q prime, let me first write, is cyclic tensor product, which means that we have vertices over the polygon going around the circle, say, i0, i1, i2, in. So we tensor multiply this by model with this by model over this ring, and this by model with this by model over this ring. And we can go all the way together, so they're sort of holding hands together. And the result would be just a vector space. So let me draw this picture here. Cyclic tensor product. If they were intonation, yeah, then it could be used as a duality, right? Yes. Yes, so this is somehow set up in a symmetric way. Yes. So we just write n i0, i1, tensor over si1, n i1, i2, will be tensor over si2, n i2, i3. And in principle, we can go all the way around. So then, if a polygon is subdivided into a bunch of other polygons, then maybe I'll draw this here. So if a subpolyt of q prime is subdivided into some q second nu, let me just consider a case of subdivision into maybe something like this. So then we read this one counterclockwise. And we read this one counterclockwise. So on this edge, here, for instance, this is i and j. So here in the tensor product, it will be n ij. And here it will be nji. And we pair them together using this pairing appropriately. So we get a map gamma from tensor product of n q prime nu to n q prime. And this is associative in the sense that if we subdivide further, it will be composable. So in this way, what we get, we get a system of sheaves, which on each secondary polytope, we have a cellular sheave, a sheave which is constant on every face. But some vector spaces, those vector spaces are associated to faces. And the system of the sheaves will be factorizable, again, in that sense. So let me maybe do this here. This picture forces that you want to imagine, too? Well, in order to get to higher dimensions, we should speak not about Lie algebras or associative algebras, but we should speak about appropriate structure of E and E d algebras. So this is a natural context here. Maybe at the end, I'll say a few words. So how much time? Oh, I still have. So we have those maps, those generalization maps. So the gammas give rise to a factorizing system of sheaves on the sigma of a prime, which means that if I have a face of this secondary polytope, it's a product. And the restriction of the corresponding sheave to this face would be the external tensor product of those sheaves. And this, again, by the same formalism, give rise to g. An algebra which now will depend on this data, let me call this data just by 1n, called gn, will be this direct sum over a prime. Here I simply say n a prime tensor or sigma of a prime. So this is a complex. Minus d sigma of a prime, minus 1. Again, is an important l infinity algebra. So now, let me explain the motivation for considering this, which uses those words of algebra of the infrared. Yes, I should probably say this too. Yes, yes. So this is a l algebra, this l infinity algebra. And so inside will be gn geometric. This will be direct sum, not over those sets, but over sub-polygons. And here I write n of a intersected with q prime. This is the way to simplify this algebra, to remove the freedom of considering several choices of points inside one polygon. So it's a sub-l infinity algebra. Yes, it has trivial differential. Not exactly because if there's differential here, then it sort of percolates. Motivation will be in the algebra, infrared. So first of all, what is, so why infrared in the first place? So from general physical understanding, the infrared analysis correspond to study the vacuum, or different vacua. So infrared sort of equals to collection of vacua, plus instantons to tunneling between different vacua, and plus sort of perturbation in this direction, instantons between them. So in particular, so this is a kind of general physical principles. We are interested in what's called this Landau-Ginsburg model, which simply means for us that we have a scalar manifold x, and we have a holomorphic function. On this, call it w. To the manifold, it's not compact. In the holomorphic, let's assume it's a Morse function. Holomorphic. So then it's a scalar manifold. It has a scalar form omega 11, which is as well known as the Riemannian metric, plus i times simplexic metric. So and the set A would be the set of w of xA is the set of critical values. So it's a subset of complex numbers. So we mark it on the plane. So some of them lie inside the convex hull of others. So w is a Morse function, a holomorphic Morse function. So the real part of w equals a real Morse function. But you don't assume that being proper in some sense, compact fiber or not? Not necessarily, we can assume it, yeah, but it's not. So it can be just just as the. The real category you usually assume here. What? The real category you assume here. Yes, yes, but we consider sort of more, in a way, more formal study of this. And we look at gradient flow. Gradient flow. And this is well known as the same as Hamiltonian flow. Hamiltonian flow for imaginary part of w. And the Hamiltonian is preserved. So it means that the images of the gradient flow lines would be half lines like this. So typically, if they all have different imaginary parts, so they could never intersect. But if you rotate it so that they become on the same horizontal line, there may be some non-trivial intersection here. So let's write it like this. So for every i and j, so i and j would be points here. So look at the rotated function e to the theta ijw. It would be this angle. And we'll look at nij to be the number of gradient trajectories of a real part of this, which goes from i to j. So they will project into straight intervals. So gradient trajectory of real. Yes, yes, appropriate number. Yes, no, actually, no, no, no, I just want let's say the number and then make a vector space out of them. Re-e to the theta ijw from critical point xi to critical point xj. And let n, big nij equal space spent by them with appropriate grading and floor type differential. Space spent by them. Yes, the angle, yeah. Yeah, so then this gives rise to a system of coefficients n, which corresponds to all algebras being the field. And those nij, those nij. So in principle, one can imagine a more general situation when this is not necessarily a Morse function. So singularities are not necessarily Morse, not necessarily isolated. In this case, those would correspond to the Foucailles ideal categories of those local singularities. And this would be certain by models, which also, in principle, one can imagine such a construction. I'm not, probably, this construction has not yet been fully developed in simplexic topology. But anyway, so algebraically, what can imagine this setting? This is the motivation, and we want to study this Lie algebra, so Lie algebra corresponding to this setting is particularly interesting from physical point of view. So corresponding, corresponding g is important. But g will be in the infinity algebra. Yes, it will be in the infinity algebra. So now it's actually important in this case that in two dimensions, actually in any number of dimensions, we can consider the relative setting, which means one point at infinity. And this actually putting one point of infinity is exactly the setting when Foucailles ideal category is defined in simplexic topology, which is put. So this move of everything into one direction. But we can consider formally the setting. We can consider that our set A has one extra point, which can be either actually at the projective infinity or simply far away, something like this. So we can have a new set, say A tilde, which is all set A, together with one point infinity, which is outside of everything. Then in this case, we can form the corresponding Lie infinity algebra called g tilde. We can do it with coefficients. We can do it without coefficients. And it has, so first of all, it has an ideal, in the obvious sense, ideal called g infinity, which consists of infinite polygons. Polygons, which one vertex is at infinity? It will span off. Well, I use this notation, vA for the generators. I can say it again, so vQ prime for Q prime contain infinity. And the subalgebra, subly infinity algebra, gF, which is simply the algebra corresponding to this A, so g. And this is a direct sum of both of them. So which means there is a canonical map, say alpha, from gF to, so when we have an algebra, Lie algebra g, g hat, which is a direct sum of a subalgebra and an ideal, it means that the subalgebra acts on the ideal. So in the L infinity situation, we will have sort of derived action. I write derived function of derivations of g infinity. So this is Lie algebra. The differential, have I got this right? The differential preserves gF, but it doesn't preserve g infinity. The g infinity is an ideal, so it preserves even more in a bigger sense. Well, yeah, but I don't see that, because I could imagine having convex subsets that don't include infinity. Because maybe I've lost this track. If you have something which is finite and something is the infinite, the result will be infinite. Yeah, it's a illusion. Yeah, so in this way, yeah. So it's derived derivations. In particular, a Maurer-Cartan element here, so Maurer-Cartan, give deformations of g infinity. So but in dimension two, so this can be done in any number of dimensions. But in dimension two, this infinite part is really near infinity. The situation is, somehow, has one dimension less. And everything is ordered. Oops, I'm sorry. And everything is ordered. So in d equals 2, situation near infinity is one dimension. So everything is ordered. So g infinity can be lifted from a homotopy l-algebra to a homotopy associative algebra. Let's call it a infinity algebra. So algebra from which, by the analog of the formula that the bracket is a b minus b a. And let's call this algebra r. And it has an upper triangular structure. Because what happens here is just we can combine them in the order. So it has upper triangular. r is upper triangular. Or using the more derived category of language, the category of r models has a semi-arthogonal decomposition. And we have a map. We can extend this to the homomorphism psi from g to the partial complex of r, which, again, is well-known governs deformations. Gf. Gf. Yes, yes, Gf. So in fact, if we work with coefficients, then we'll get the image, not the entire Hochschild complex, but a sub-complex which governs deformations that preserve this structure, that preserve semi-arthogonal decomposition. So let me just do one more permutation here. I'm almost done. So this psi takes values. Let's call the or in the sub-complex. Let's call Hochschild with an arrow, which is, let's write something like this. It's the most natural thing that writes itself. It would be something like this over i0 less than i1 less than in. And less than means in the sense of that order upstairs. So we have hom over si0 star tensor sin. Here I have r i0 i1. So this refers to this triangular structure, tensor si1 r i1 i2 tensor r in minus 1 in over sin minus 1. And then in this complex, and then what we can prove is a sort of universality theorem that this algebra gf is the corresponding morphism is a quasi-isomorphism called the universality theorem. Oh, oh yeah, home to, yes, yes, to r i0 in. So we think of this now as a sort of category of a semi-arthogonal decomposition. Yes, and we try to deform it in a way that would preserve this thing, so that's what we write. Universality theorem that the L infinity morphism, that will be an L infinity morphism, L infinity morphism called psi from gf to ordered Hochschild complex of r is a quasi-isomorphism. It means that every deformation of any semi-arthogonal deformation comes from a Maurer-Cartan element. i.e., any deformation preserving semi-arthogonal structure comes from a Maurer-Cartan element in gfinet. And then the somehow physical motivation of Gaiota-Morwin-Witton was to give particular examples of such elements which would recover the Foucaille-Seidel category. So let's try the Gaiota-Morwin-Witton goal. So here in principle, there may be several approaches. It's still thinking on other approaches to this problem to find the Maurer-Cartan element, let's say gamma, find gamma recovering the Foucaille-Seidel category of W. So anyway, so this gives a certain class of Lie algebraic structures, just coming from basically from nothing, from secondary polytops. And there are interesting relations with Hochschild complexes and in higher number of dimensions. This should be understood in terms of E d and E d minus 1 algebras. I think it's Swiss cheese of glass. Yes, Swiss cheese of glass. So this is a sort of by now a classical area of thinking, so how we think of the formations of E n algebras. So this is a version of the lean conjecture. But anyway, so the structure itself is completely elementary, and this is what I wanted to present today. Thank you very much. Is this supposed to work in the case when we have a one-dog input model with one isolated single line? Well, so this setting is sort of designed to work with this whenever the symplectic people will construct corresponding categories. So how do you set a symbol in this language? Well, a symbol is exactly, so those lines are intersections of symbols. So a gradient line from one critical point to another. So what if they are not as red? No, that I don't know. So in your construction of G, you neglected those sets of vertices which didn't span. Yes. Is there any way that any of them might have a role, do you think, to give some negative? In principle, yes, but I don't know. Yeah, I asked physicists about that. So there may be. So it may. Yes, I agree it's a little annoying that we disregard them. It would be better to not have to disregard them, but to do something. So we all have the same reaction, but without knowing what we're meant to be. What physics we're meant to learn. OK. What question? In the case where on a face, you could factorize the polynomial. So typically in singularity, we have a face which is a line. And in the middle of this face, some element, such that the whole face polynomial becomes square. Then you run and travel. So in the case of middle number, it goes drastically up. Here I'm talking about the secondary polythorpe, but not of the set of monomials at all. Yeah, but I'm still allowed to think that alphas are integral values or integers. Yes. So then I can take, normally, these two piece-up pairs. No, but we don't do this at all, because those w's are not monomials. They're critical values. Yes, they're complex numbers.