 Thank you very much for the introduction. And also, thank you very much for inviting me to this nice conference and beautiful place. And I'm very happy to be here. So today I'm going to talk about Torek Mirosimitri in terms of tidal representation. So this is based on maybe one year ago paper. This is two years ago. And this one and the other one. And so this is about some Mirosimitri for Torek varieties. And Mirosimitri for Torek varieties is somehow well-studied. And many people know many things about this. And in this talk, I only talk about Gena 0 Mirosimitri. So in this conference, many people talked about higher Gena's chromatin variants and modularity or cost modularity or Jacoby forms. But today this talk is unfortunately only for Gena 0. But I'm considering some equivariant chromatin variants. So maybe I should start with what is equivariant chromatin variants? So this is maybe also a connection to the title of this conference. So equivariant chromatin variants are somehow related to, for example, gauged chromatin variants. So in that sense, my talk is related to gauge theory. And so let us consider a situation where T is a algebraic torus. Torus, just sister to something and acting on X. X is some quasi-projective. And so in this setting, maybe later we don't restrict to projective variety. So toric variety can be non-compact. For example, Toric-Clarviau manifold. And in this case, but this is completely general. So if you have a torus suction on X, then you have a torus suction on somehow, the modular space of stable maps. You have a torus suction on modular space of stable maps. And then you can define, from this you define equivariant chromatin variants. So this is a equivariant intersection theory on the modular space of stable maps. So it is, in general, of this form. So you have GMD. Maybe I don't go into the detail. So alpha i's are equivariant cohomology class of X. And D is the degree. And G is the genus of the curve. In this case, I only consider. In this talk, I only consider genus 0. But this, in general, so lies in the equivariant cohomological point. So the usual chromatin variant is a rational number. But now it is a function somehow. It is an equivariant cohomological point. This is a polynomial ring in the R variable, where R is a rank of the torus. Or more generally, fraction ring of this one. So when modular space is non-compact, then it may lie in the fraction ring. So this is an equivariant coherent invariance. And so from this, you can, for example, define equivariant quantum cohomology. The other thing, the other ingredient in my talk is a Seidel representation. And so this is a following map. So this is a map from home of sister to key. Maybe I'll review the construction later. But this is a map homomorphism from the group of co-characters to some invertible elements of quantum cohomology. So this is equivariant quantum cohomology. And this star means invertible elements. And so for each subgroup, maybe more generally, for each Hamiltonian cycle action on a symplectic manifold, you get some element, some invertible element in quantum cohomology. This is a Seidel representation introduced by Seidel in the early 90s. And so I'd like to understand mere symmetry in terms of this Seidel representation. So maybe I also want to explain some mere symmetry proposed by Given-Towell and also Horry and Baffa and many other people. So they claim that if you are given a funnel manifold or funnel-like manifold x, this should be mirror to low-long polynomial, polynomial f. So because I'm going to talk about not necessarily funnel-toric variety, so let me say funnel-like manifold. But this is somehow a conjecture that for each funnel-like manifold, we may have some mirror low-long polynomial and some of the symplectic geometry on x can be computed by some periods of f, periods or Gauss-Mann system of f. And so in the toric case, toric variety or obi-fold more generally or more generally, that corresponds to low-long polynomial f with generic coefficients. With generic coefficients. So if I have a toric variety, then that corresponds to low-long polynomial with somehow generic low-long polynomial. And if I take more interesting funnel manifold, that should correspond to some f with some more special coefficients. So for instance, if I'm interested in mirror for grass manian or partial-flag manifold, those are corresponds to f with some low-long polynomial but have some very special coefficients. So that is related to this mirror for toric variety by a toric degeneration, for example. And so in some sense, mirror for toric variety gives you kind of open dense subset of the space of all-long polynomials. And maybe if you go to some deep locus in the discriminant locus, then you may find more interesting funnel manifold. So this is a picture for mirror symmetry. But somehow, I'm not going to talk about this more interesting case, but just toric case. So but still, I have some new observation there. So I'm going to give a rough statement of the main result. So this is as follows. So let x be n dimensional. So dimension can be anything. And let mn be the space of all-long polynomials in n variables in maybe just say x1 to xn. So maybe this is some space. But actually, later, I think of this as kind of formal scheme. So this is just maybe you can think of this lively, some infinite dimensional space. Then the main statement says that there is a mirror map. Actually, this has some coordinate chart. Some mn has a coordinate chart given by the equivalent cohomology of x. So here, this star, I haven't spaced everybody's star, but the star is some base point. And certain mirror map between the space of all-long polynomials and the equivalent cohomology, t equivalent cohomology of x, t is a natural torus acting on x. So for any toric variety, we have n dimensional torus action on x. And such that the following. So this is maybe sort of a thought of as a coordinate chart on this infinite dimensional space. And such that this is infinite dimensional space. And this is maybe considered as flat coordinates. Maybe you can think of this as flat coordinates, such that if you consider the kyoji-saito structure, such that is isomorphic to equivariant quantum cohomology. So equivariant quantum cohomology is some family of rings. So this is a Frobenius-Manifold structure. This is family of rings and parameterized by tau. Tau itself is in the equivariant cohomology group. So this is infinite dimensional Frobenius-Manifold. For each point, you have a product structure on the tangent space. And on the other hand, there is some kyoji-saito structure, which appeared in the Togo beloved on Monday. And so that can be associated with family of low-lamp polynomials in this context. So this is, again, some another Frobenius-Manifold structure, and they coincide. And moreover, you have some group of, there is group action on this space, the change of variables, or change of variables, of variables x1, xn. So you're considering low-lamp polynomials of variables x1, xn. And on this space, you can consider a very big group that changes variables somehow. This is actually some formal group of change of variables, and that acts on this space, on this space. And that corresponds to the quotient by the change of variable that corresponds to non-equivariant limit, non-equivariant. So the quotient by the change of variables, that exactly corresponds to that non-equivariant limit on this side. And then, under somehow quotient by the change of variables, some of the cytostructure reduces to more usual cytostructure on a finite-dimensional base, and you get more somehow usual Frobenius-Manifold on the usual homology. So this is a rough statement. And so in some sense, this resulted that toric variety of the same dimension have the same mirror. Somehow, the difference arises in this base point. So for each toric variety have different base point. We will see it in the example soon. And for each toric variety have different base point. And somehow, near each base point, you have somehow flat coordinates, different flat coordinates. Those could be completely different. So this is somehow the picture. And so to make this rough statement more precise, I'm going to explain this in some example. And also, moreover, maybe I should mention that all these isomorphisms somehow mirror map and the isomorphism between cytostructure and equivariant quantum This is all these are constructed in terms of cytorepresentation. So let's consider the case x is p2. So this is maybe the most easy case. But so p2 has a fan. And according to Gibbenthal and Horibaba, we just write fan for p2 is fan for p2, one diagram. And I choose just one-dimensional generator for one-dimensional cones. And each generator corresponds to a term in the mirror law and polynomial. So this is mirror to fx1 plus x2 plus 1 on x1, x2. So the rule is simple. So I just assign one variable for each primitive generator for each ray. And so I assign multiplicatively. So this is later, yes. This is only classical case. And also, we also consider some Q variation, some deformation prime to Q. So this is, I think, some classical version of the mirror symmetry by Gibbenthal. And so in this case, a statement that is the following. So quantum cohomology of x, this small quantum cohomology of x, this is isomorphic to Jacobi ring of f. Jacobi ring is, by definition, this is just a quotient of this ring, roll and polynomial ring, by the idea generated by derivative of f. And under this isomorphism, some of Q corresponds to Kela parameter of x. And this is Gibbenthal result. And if I want to make this equivalent, then what I do is the following. So on P2, we have a torus sister square action on P2. And so we have two equivalent parameters, say lambda 1, lambda 2, the invariant parameters. So that the equivariant homologable point is a polynomial ring in lambda 1 and lambda 2. And then, if lambda, I introduce the following mirror. So this is always some general rule, but just consider. So I introduce this function. This is a multi-valued function. But still, we have the same statement. So small equivariant quantum homology of P2 is isomorphic to Jacobian ring of f. This is also due to Gibbenthal. But one can rewrite this in a different way. So Jacobian ring of this one can be computed by taking a derivative of this function. And it turns out it is easy to see that this is a coordinate ring of the following. So of lambda i is equal to df d log xi. So if I try to find the critical point of f lambda, then I just consider differentiation in log xi. And then you see this equation. So this is the equation for critical points. So equivariant quantum homology of x is a coordinate ring of this space. And from this presentation, maybe it is clear that this is a Lagrangian sub-manifold. This is some Lagrangian sub-manifold inside x lambda space. Because this is a graph of the differential of a function. And in fact, there is a proposal or some viewpoint proposed by Konstantin Telman. I learned his point of view after writing my paper. But this is, I think, a very clear way to understand that situation. So he said that actually, in order to determine this amount f, it suffices to know that how the spectrum of the equivariant quantum homology is embedded into the symplectic space, x lambda symplectic space. So x and lambda are some canonical conjugates. And lambda is equivariant parameter. So we only need to know what is x. So Telman claims that xi is a side element. That means the side element is the image of the side representation. So the image of the side representation is some invertible element in quantum homology. And I call it side elements. So because torus is two-dimensional, I have two side elements. And spectrum is more equivariant quantum homology is a Lagrangian sub-variety in the x lambda space. So this is somehow how you can see the mirror tautologically. So this ring contains x and lambda, x and lambda i. So therefore, the spectrum of this can be identified with some sub-variety, affine sub-variety in x lambda space. And that is Lagrangian. And therefore, you can find some generating function f. So f is a generating function of this Lagrangian sub-variety. So in the toric case, this is just some generating function is a single valid function. And in general, it is not single valid. f is not single valid. But this fact is general fact. So Lagrangian sub-manifold in x lambda space. This is for general space. But somehow, the connection to mirror lambda gives model is only for toric case. But for other cases, maybe f may be multi-valid. So in some sense, my story is somehow extending this to big quantum homology. So we consider the following universal function. So let me stick to the example of p2. And in the case of p2, I consider the following function, large f, x, with parameter y. So this is a sum over all lattice points, k in z2. And x to the k, q to the beta k. So this is some unfolding of the previous function. So maybe I need to explain. So this is sum over all lattice points in the fan. So instead of all, originally, I only considered the sum of three lattice points. But I, yeah, so we also had this constant term. And also other points, any other points in the lattice. So in this case, maybe I didn't explain what is beta k. But this can be somehow explained in a diagram easily. So if this corresponds to x1, this corresponds to x2, then this corresponds to x1, x2. And this corresponds to x1 squared, x2 squared, and so on. So x1, x2 squared. And so this point corresponds to q on x1, x2. And therefore, so for each cone, I just write lattice point in that maximal cone as a linear combination of the edge vectors. So this vector is a sum of this vector plus this vector. So I think I assign the monomial x2 on q, a q on x2 on this. And similarly, somehow I assign x1 on q, and so on. So that explains how I put this beta k. And this is general rule for general case. In some sense, some of these q variables is somehow redundant because we can absorb these q variables into y variable. But it is somehow technically important because somehow q specifies the direction of the large radius limit point. That corresponds to the base point I talked about before. And the statement that somehow we have of some mirror map. That is a map from y, space of all y's, to tau of y. This is in the given homologer of x. This is some formal map. But isomorphism, formal isomorphism, such that, for instance, we have similar statement as before. So such that Jacobian ring of f, large f lambda. So large f lambda is f minus lambda log x, just as before. This is isomorphic to big quantum kohomomigio. So under this mirror map. So maybe I should say that this yk is a parameter. This is a parameter for the b model. And maybe first statement is on the level of ring. Somehow, if you consider the Jacobian ring of f lambda, then that is isomorphic to the equilibrium quantum kohomomigio x. So you don't understand why somehow this space of parameters, infinite dimensional space of parameters, correspond to equilibrium kohomomigio x. This is somehow intuitively not so difficult to see. So it is easy to see that the equilibrium kohomomigio x has a c basis parameterized by lattice points. So this is somehow a basic observation in Toric geometry. So although this mirror map is not just a linear map, but linear approximation gives you somehow this statement. So more precisely, if I take just, for example, in this p2 case, this is some fact. And p2 case, if I take d1, d2, d3 as Toric divisors, then t equivalent kohomomigio p2 is generated by d1, d2, d3. With only relation d1, d2, d3, the product 3 is 0. So in this diagram, for example, x1 corresponds to d1, x2 corresponds to d2, and this point corresponds to d1, d2. And this point corresponds to 1, d3, and so on. So because the product of 3 is 0, so you cannot multiply elements in that different call. That just gives you 0, so 1, d3, and so on. So these elements give spawns. These elements are linearly independent and spawns this infinite dimensional space. So it somehow explains why this correspondence is not so strange. So actually, I want to somehow, this statement is only at the level of rings. But I want to lift this statement to the statement on a d module. So and this is somehow important in the proof. So what I want to do is, as I said before, I want to show the cycle structure of F lambda. Is this more quick to quantum d module? Equivariant quantum d module, thanks. Maybe this is the so-called quantum d module. And this side is some Gauss-Manning system. So roughly speaking, I don't have time to explain in detail what the cycle structure. But roughly speaking, this consists of oscillating forms. So I need some oscillating parameter z, a new parameter z. This plays an important role. And omega, this consists of some oscillating differential forms. And omega is a differential form, omega 2 on sister square, which is in z, and also maybe lambda. Also everything contains from the y, which I just omitted and also q. And this consists of some oscillating forms of this one. So for example, omega is just like dx1, dx2, some volume form. And on this one is this one also, this is some lift of the quantum cohomology to d module. And this is something appears in Rabini's manifold for in general. So it is a tangent bundle of cohomology group. This is infinite dimension space, but infinite dimensional vector bundle over infinite dimension space and equipped with the probing connection. So I'm sorry, I forgot to say. On this side, you have Gauss-Manning connection. On this side, you have Gauss-Manning connection. So you have somehow a parameter y. So there is a flat connection on the y parameter space. And similarly, you have somehow tangent shift and equipped with some flat connection. And the probing connection maybe I just briefly recall. If I use some basis and coordinate, this is plus 1 on z. This is some connection on the tangent shift. And again, we have some parameter z. So we have some oscillating parameter z here. And that corresponds to this parameter 1 on z here. It may sit in a smaller group because it should preserve metric. But yeah, it is somehow contained in a given total group. But it has to be simplected. But yeah, maybe even contained in a much smaller group. So in general, some monotomy of the quantum cohomology should somehow, it is a conjecture, but it should arise from derived equivalence of x. So it should be contained in a much smaller group. So yeah, so this is the probing connection. And I want to actually some lift-sider representation to quantum D module. And that is called shift operators. So side of representation, quantum D module. So this is called shift operators. So shift operators introduced by Brabherman appeared in many people's work. So Brabherman, Molek, Okonko, and Panharipande. So I think originally Okonko and Panharipande used it in the context of quantum cohomology of Hilbert's scheme of points on C2. And then Brabherman, Molek, Okonko somehow formulated the more general context. And so this is somehow side of representation on quantum D module. So maybe I want to review the construction briefly. So again, somehow x is a T variety. And I pick any sister subgroup in T. Then I have a following, so-called side-o space. So ek is x cross C2 minus 0. This is some subspace associated with sister action on x. And where this sister action is defined by as follows. So s acting on x, v1, v2, as s to the k acting on x, s inverse v1, s inverse v2. So this is the action. And so for instance, if k is a trivial homomorphism, then this is just a product of x cross p1 because we don't have action here. But this is twisted by sister action. So in general, this has a structure of an x bundle over p1. There is a projection to the second factor. So sister, this is p1. You have a projection to the second factor. But you don't have a projection to the first factor. So this is x bundle. The fiber is x over p1. And this is somehow important things, introduced by side-o. And we have the following action. We consider the following t cross sister action. So maybe let me just draw a picture. So this is a ek, picture of ek, the fibering over p1. And I have 0 and infinity as a. And I have a fiber over 0, x0, and fiber over infinity. So then actually, I want to consider some group action on ek. So that is given by t cross sister action. That is a t cross sister action on ek. This is just in formula, I just give the definition. So this is basically something like x cross p1. And on x factor, original t acts. And on p1 factor, sister acts. So this can be done in this twisted bundle. So this is the action. And maybe the crucial point in the definition of shift operator, so this is their work, is the following. So this fiber 0 and fiber infinity, so x0 and x infinity, they are invariant under t cross sister action. And the action, so t cross sister action on x0 and x infinity will be something like this. So on x0, x infinity will be, so t u acts on x0 as t times x0. And t u times x infinity will be t uk to the x infinity. So on this fiber, maybe by definition, it is easy to see on this fiber, this is a 1 0 fiber. It is easy to see that somehow t u, just t cross sister just act by projection to t. But on this factor, there's some twist. So there's some twist. And this twist is somehow here. So it just not act by projection to t, but there's some mixing. So mixing between t and sister. And so this is somehow the key point of the shift operator. So we consider, as Seidel did somehow, we consider the following correspondence. So we consider the modularized space, polynomial fixed sections. So we count, instead of just counting a map to x, but somehow we consider a map to this twisted bundle, section of this twisted bundle. And we consider the modularized space, and then take certain compactification. So this is, for example, you can just take the stable map compactification. And then you have a correspondence between x0 and x infinity. You have a validation map of 0 and infinity. So this gives you an operator, which I write sk from t cross sister equivalent cohomology of x0 to t cross sister equivalent cohomology of x0. And actually, so this t cross sister equivalent cohomology of x0, this is just because sister factor act trivially on x0, this t equivalent cohomology of x with one variable z. So polynomial ring in z, answered. So z is the same parameter as we see before. So this sister equivalent parameter becomes somehow a slightly parameter on the Seidel side. And on this side, this is a little bit subtle, but you can identify, you can do some identification with x infinity and x0. Somehow, this is more because x0 and x infinity are the same other space. So you can identify in some way. But this identification is not linear over equivalent cohomology of a point. So in this way, you get the map here from equivalent cohomology of x to itself. And this is a shift operator. And there are several properties. And also, maybe I don't mention it here, but you can also extend this definition to big case. So you can also insert parameter tau. You can define with some bulk deformation parameter. And then what you do is, what you see from the construction, in particular, this shift of the t cross sister action is the following properties. First, this sk shift the action of lambda. So sk lambda i is equal to lambda i minus kiz. So there is some Heisenberg group type commutation relation between equivalent parameter. Lambda i was a t equivalent parameter. And recall that this z is a c straight equivalent parameter. And this property 1 just follows from somehow the difference of the action x0 and x infinity. So they differ by just somehow k factor. And the second property, this is somehow the original side-o representation. sk is tau sl is proportional to, there is some factor. This is some side-o representation. So I'm some side-o representation. And also, this sk also commutes with the robin connection. It's the robin connection. And so you have some rich structure on quantum d module, equilibrium quantum d module. And maybe, yeah, I should first mention that these first relations somehow, they are somehow, they satisfy some kind of Heisenberg relation, commutation relation, canonical commutation relation. This is a response to the fact that the lambda and s are the side-o variable are conjugate. So recall that we have somehow lambda variable and x variable are conjugate to each other. And x variable corresponds to this side-o element. So this explains why lambda and s are conjugate variable in equilibrium quantum cohomology. In some sense, quantum d module is a quantization of quantum cohomology. So this is a canonical quantization commutation relation. And the second remark is that originally some side-o element, side-o element, is just a limit of this shift operator. So sk is a limit of z goes to 0. This c-strike invariant goes to 0 of sk applied to the unit. So this is a side-o element. This was the original side-o element. So by this somehow shift operators and side-o elements, we can totologically, in some sense, totologically construct mirror map and mirror isomorphisms. So that's what I'm going to explain now. So what we want to do is to construct the isomorphism between side-o structure of lambda and quantum d module, equilibrium quantum d module. And such that in such a way that you have Gauss-Mannin connection that corresponds to the Broglie connection. And also, you have an action of xk. Actually, I need this extra term. But this is not very important. Because in some sense, side-o representation is ambiguous up to powers of q. And it's not so important. This corresponds to sk, the shift operator. And so actually, somehow, I claim that if you want to find an isomorphism satisfying these two properties, then isomorphism can be uniquely fixed. So for instance, if I take an element here, if I take this oscillating form, so px1, just assume I'm just still working on p2. And take this element. This is a section of this module. And what this goes to. So this goes to some element in quantum, equivariant quantum cohomology. This is just a section of the tangent shift. Section of the tangent shift of the equivariant cohomology. And I want to know what is this section. And this can be determined uniquely, somehow, almost uniquely. Because this element satisfies the following differential equation. So z, d, dy, d, dy, k. If I differentiate this in lambda, x, y, and z, d, x, and x. There's some cohomology class of this. By very simple differentiation, you just put some big one terms. So this is xk, q to the k, by the very definition of f lambda. e to the f lambda, x, y, and z. So you have this differential equation. And by this requirement, so this is Gauss-Mannian connection. Maybe I didn't say explicitly, but this is Gauss-Mannian connection. So therefore, we should have the Roving connection, d, dy. Actually, I need to differentiate tau because this is somehow proved by the mirror map. Omega, this is equal to sk tau of omega, tau y of omega. So this is a differential equation satisfied by the image of this element. And you can somehow solve this. And moreover, if you take the z equals 0 limit, then you also find the following differential equation for the mirror map. So dy, dk should be determined by this differential equation. So this is somehow to logically determine the mirror map and the mirror isomorphisms. And somehow you can show the theorems. And maybe I also mentioned that, finally, I just want to mention one thing. So of course, some of these differential equations are very abstract, and that doesn't seem to be. Of course, you can show abstractly that this differential equation has a solution. But on the other hand, you can also determine tau as a mirror map and this omega by somehow i function. And so this is a more traditional method of given tau, so given tau i function. But actually, given tau i function can be also solved in this way. So recover, which is certain, maybe I don't say what is given tau function. It's certain hyper geometric function by this kind of differential equation. So this is some infinite dimensional i function. For some sk, some explicit difference of right and constant difference of right. And somehow from this, you can define i function and also solve these equations explicitly. So thank you very much.