 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 Toric mirror symmetry in terms of tidal representation so this is based on Maybe one one year go paper This is two years ago and this one and And So this is about some mirror symmetry for Toric varieties and mirror symmetry for Toric varieties is somehow Well-studied and many people knows many things about this and In this talk, I only talk about genus zero mirror symmetry. So So in this conference many people talk talked about higher genus gravity invariance and modularity and or cross modularity or Jacoby forms on the phone, but today I so this talk is unfortunately only for genus zero and But I'm considering some equivalent gravity invariance. So so maybe I should start with What is the equivalent gravity invariance? so So this is a maybe also a connection to the title of this conference so equivalent gravity invariance are somehow related to for example gauged gravity invariance, so So in some in that sense my talk is related to gauge theory and So Let us consider a situation where keys are as you like torus Torus just sister to something and acting on X X is some cause project here and So So in this setting so Maybe later later. We don't restrict to projective variety. So So Toric variety can be non-compact for example Toric Calabria meaningful and In this case are so but this is completely general So if you have a torus action on X then then you have then then you have a torus action on somehow the modular space Space of stable maps you have a Torus action on modular space of stable math and then then you can define from this you define Equivariant grohiting variance. 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 Some maybe I don't go into the detail, but so alpha i's are Equivariant cohomology class of X and D is a degree and G is a G is a genius of the curve in this case I only consider in this talk I only consider zero, but this in general So lies in the equivalent cohomological point So in the usual grohiting variant is a rational number, but now now it is a function somehow It is a covariant cohomological point. This is a polynomial ring in our variable where R is a rank rank of the torus or Or more generally fraction ring of fraction ring of this one. So when Modular space is non-compact then it is made maybe may lie in the fraction ring So this is a covariant grohiting variance and So from this you can for example define Equivariant quantum cohomology The the other thing the other ingredient in my talk is a cydo representation presentation and So this is a following map. So this is a map from home of sister to T To maybe I review the construction later, but This is a map homomorphism of from this the group of co-characters to some invertible elements of quantum cohomology, so this is Equivariant quantum cohomology and this star means a invariant invertible elements and so for each For each subgroup Some of for each maybe more more generally for each cycle action Hamiltonian cycle action on a Simplactic manifold you you get some element Some invertible element in quantum cohomology. This is a cydo representation introduced by cydo in early 90s and So I'd like to understand mirror symmetry in terms of this so So maybe I also want to explain some mirror symmetry first proposed by given town and also Horry and Bafa and And many other people so they they claim that If you are given the funnel manifold or funnel like manifold X They should be mirror to low-long polynomial polynomial F so So because I'm going to talk about not necessarily funnel toric variety. So I 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 simplect geometry on X can be Can be computed by some periods of periods or gas money system of F and So in the toric case Toric variety Or all be fold more generally or more generally or full 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 so more interesting funnel manifold That should corresponds to some F with some more special coefficients. So So for instance if I am interested in mirror manifold for 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 Toric degeneration, for example, and so in some sense a Mirror for Toric variety is gives you kind of open dense subset of the space of all all on polynomials and Maybe if you go to some Deep locus in the discriminant locus, then you you may find more interesting funnel manifold But so this is a picture for mirror symmetry, but somehow I don't go 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 So this is as follows. So let X be a n dimensional so dimension can be anything and Mn let Mn be The space of volume polynomials in n variables in maybe just say x1 to xn. So this is some maybe this is some Some space but Actually later I think of this as kind of formal scheme, but So this is just maybe you can think of this nightly some infinite dimensional space then the main statement says that this Mirror map actually this has some coordinate chart some Mn has a coordinate chart given by the equivalent cohomology of X so So here this star I'm not I Haven't space everybody star but sign some base point I and certain mirror map Between the space of all of polynomials and the equivalent cohomology T equivalent homology of X T is a natural torus acting on X. So for any Toric variety we have a n-dimensional torus action on X and Such that the following so this is maybe this may be sort of a thought of us coordinate chart on some on this infinite measure of space and such that This is the infinite measure of space and this is maybe Consider the flat coordinates. Maybe you can think of this as flat coordinates such that If you consider the Kyoji cycle structure Such that is isomorphic to Equivariant quantum Quantum so if you variant quantum cohomology is some family of rings So this is a Frobenius manifold structure. This is family of rings and parameterized by tau How tau itself is in the cohomology ring? Equivariant So this is a for being 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 cycle structure which appeared in the local beloved on Monday and So that can be associated with the family of low lamp polynomials in this context. So So this is again some another for means main for structure and they they coincide and more over more over you have Some group of there is group action on on this space the change of variables or change of variables variables x1 xn so so So you're considering we are considering low lamp polynomials of variables x1 texan and On this space you can consider a big 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 Noniquivariant limit So the quotient by the change of variables that exactly corresponds to that noniquivariant And then and there are some our quotient quotient by this change of variables some of cytostructure reduces to more usual cytostructure on a finite dimensional base and you get more Somehow usual for being useful on on the usual So this is a rough statement and So in some sense this resulted that Toric variety of the same dimension have the same era somehow The difference arises in this base point so So for each Toric variety have different base point. We were we will see it in the example soon and for each Toric variety have different base point and somehow near the Near each base point you have somehow flat coordinates different flat Those could be completely different. So this is somehow the picture and So to make more to make this rough statement more precise. I I'm going to explain this in a Some example and also moreover. Maybe I should mention that all these isomorphism somehow mirror map and Isomorphism between cytostructure and equilibrium to quantum commas. This is all these are constructed in terms of cytos representation so Let's consider the case x is p2. So So this is a maybe the most easy case but So p2 has a fan and according to give entail and Horibafa. We just write fan for p2 Diagram and I choose just one-dimensional generator Sorry generator for one-dimensional cones and Each generator corresponds to a term in the mirror alone for you. So this is mirror to f x1 plus x2 that's one on x1 x2 So so The rule is simple. So I just Assign one variable for each primitive generator for each ray and but So I assign multiplicatively so this later. Yes, this is only classical case and And also we We somehow we also consider some q variations some deformation prime to q So this is I think some classical version of the mirror symmetry by give entail and So in this case a statement that is the following. So quantum cohomology of x This is small quantum cohomology. I mean we have small s small quantum cohomology of x This is isomorphic to Jacobi ring of F the Jacobi ring is by definition, this is just Quotient of this ring roll and polynomial ring by the idea of generated by derivative of that and Under this isomorphism some of q corresponds to keller parameter of x keller parameter of x and Yeah, and this is a given tall result and And if I want to make this equivalent then what I do is the following. So So on p2 we have a thoracic star square action on p2 and So we have two equivalent parameters say lambda 1 lambda 2 Be very in front is so that the take a very incomparable 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 cohomology P2 is as mothic to Jacobian ring of This is also due to give and tell and but One can rewrite this in a different way. So So Jacobian ring of this one can be computed by taking a derivative of this function and this is it turns out It is easy to see that this is coordinate ring coordinate ring of following so Lambda i is equal to df d log x i so if I try to find a critical point of f lambda then I just Construct differentiation in log x i and Said and then you see this equation. So this is a question for critical points so Equivariant quantum cohomology of x is a coordinate ring of this this space and From this presentation, maybe it is clear that this is a Lagrangian sub manifold in 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 Constant entailment Actually, I learned I learned his point of view after writing my paper But this is I think a very clear way to understand the situation. So So he said that Actually, so in order to determine us somehow F It suffices to know that how the spectrum of the equivalent quantum cohomology is embedded into the Simplactic space x lambda symplectic space. So x and lambda are some of cannon co conjugates and lambda is Known some of lambda is equivariant parameter. So we only need to know what is x So we claim that so tell them claim that x i Is a side of element that means side of element is a image of the Side of representation. So the image of the side of representation is some invertible element in quantum Tohomology, and I call it side of elements. So because the torus is two-dimensional. I have two side of elements and And the and spectrum it's more equivalent quantum cohomology is Is a Lagrangian sub variety variety in the X lambda space so So this is somehow How you can see the mirror tautologically? so 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 you can find some generating function f So f is a generating function generating function of this Lagrangian sub variety and So in the toric case, this is just Lagrangian some a generating function is a single valid function and in general it is not single valid f is not single valid But but this fact is general fact. So Lagrangian some manifold in X lambda space. This is for for general space and But somehow the connection to mirror lambda gives model is only for a toric case, but But because some for other cases, maybe f may be multi-valid so Okay, so Maybe yeah, so in some sense my story is somehow extending this to big quantum homology so So we consider the following universal function so Let me stick to the example 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 next to the k Q to the beta Q. So, so this is some unfolding of the previous function so Maybe I need to explain so this is some over all lattice points in the fan So instead of all so originally I only consider the sum of three lattice points but I Yeah, so we also at this constant turn and Also other points any other points in the lattice so So in this case, maybe I didn't explain what is beta k But this can be somehow explaining the diagram easily So if this corresponds to x1 this corresponds to x2 then this corresponds takes one x2 and this corresponds to x1 square x2 square and so 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 maximum cone as a linear combination of the edge vectors So so this vector is a sum of this vector plus this vector So I 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 so so that that explains how I put this beta k and this is general rule so for general case In some sense some of this Q Q variables is somehow redundant because we can absorb these Q variables into y variable But it is somehow technically important because somehow Q specify the direction of the large radius limit that that corresponds to the base point I talked about Before and Just taking that somehow we have of some mirror map. You're a map that is a map from y space of all y's to Tau y this is in given homologous x. This is some formal map but isomorphism formal isomorphism such that for instance, we have the similar statement as before so such that Jacobian ring of F large F lambda so large if lambda is Just This is isomorphic to big quantum So Under this mirror map. So, sorry, so maybe I should say that this yk is a parameter this is a parameter 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 commons of x So, you know to understand why? Why somehow this space of parameters infinite dimensional space of parameters correspond to equilibrium commons of x. This is somehow intuitively not so difficult to see so so It is easy to see that the equilibrium commons of x has a c basis parameterized by lattice points. So this is somehow basic observation in Toric geometry. So All those also this mirror map is not just a linear map, but this is but linear approximation gives you somehow this statement so 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 historic divisors divisors then Equivariant cohomology of t equivalent cohomology of p2 is generated by d1, d2, d3 with only a 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 So because because the product of 3 is 0 so you cannot multiply elements in the different call That just give you 0 so 1, d3 So these elements give spawns Some of these elements are linearly independent and spawns this infinite dimensional space. So It's in somehow explains why This correspondence is not so strange. So So actually I want to somehow this is only the level of this statement is only the level of at the level of rings But I want to lift this statement to the statement on a d-mojo so And this is somehow important in in the proof So what I want to do is as I said before I want to show Cycle structure of F lambda is isomorphic to quantum d-mojo Equivariant quantum d-mojo banks This is Quantum d-mojo and and this side is some Gauss-Mannin system So roughly speaking I I don't have time to explain in detail what side of structure, but roughly speaking this consists of Oscillating forms so So I need some oscillating parameter Z a new parameter Z plays an important role and the omega So this consists of some oscillating differential forms and omega is a differential form omega 2 on sister square with conditions in Z and also maybe lambda and Also everything contains from the Y which I just omit 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 this one is this one also This is some lift of the quantum homology to D-mojo and this is something Appears in rabbinis manifold for in general so it is a tangent tangent bundle of tangent bundle of Homology group. This is infinite dimension space but if you mention a vector bundle we over anything to measure space and equipped with With the probing connection the probing connection. So, okay. I'm sorry. I forgot to say on this side. You have a Gauss-Mannin connection on this side. You have Gauss-Mannin 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 tangent shift and equipped with some connection flat connection and The robin connection. Maybe I just Be free recall if I use some some basis and coordinate. This is plus one of Z this is some connection on the tangent shift and Again, we have some parameter Z Okay, so we have some oscillating parameter Z here and that corresponds to this parameter one on Z here. Yeah It may sit in a smaller group because it should preserve metric Yeah, but yeah It is somehow contained in a given to a group, but it has to be symphlectic But yeah, maybe even contained in a much smaller group So in general some monotony of the quantum homology should Somehow it is a conjecture, but it should arise is from derived equivalence of X. So So it should be contained in much smaller group. So yeah, so this is a developing connection and I Want to actually some a left-sided representation to quantum D module and that that was done that is called shift operators so Side of representation quantum D module So this is a called shift operators. So shift operators are introduced by Brabherman appeared in a Many people's work. So a brabherman Molek Okonkov and Pan-Hari-Pande so So I think originally Okonkov and Pan-Hari-Pande used it in the context of quantum homology of Hilbert scheme of points on C2 and then Molek Brabherman Molek Okonkov formulated the more general context and So this is somehow side of representation on quantum D module So 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 following the so-called side-dose space E k is X C2-0 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 P1 V2 as S to the K acting on X S inverse P1 So so this is the action and So for instance if K is a trivial 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 will be this has a structure of an X bundle over P1 that this is there is a projection to the second factor So sister You have a projection to the second factor But you don't have a projection to the first factor So this is a X bundle the fiber is X over P1 and this is somehow important things Introduced by side-dose and 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 E K picture of E K the fiber in over P1 And I have zero and infinity as a and I have a fiber over zero X zero and fiber over infinity So then actually I want to consider some group action on E K So that is given by T cross sister action That is a T cross sister action on E K This is Just in formula. I just give the definition So this is basically just this is basically something like X cross P1 and On X factor original T acts and on P1 factor sister acts So So this can be done in this twisted bundle so so this is the action and Maybe The crucial point in the definition of shift operator. So this is their work The following thing so up so So this fiber at zero and fiber of the infinity so X zero and X infinity they are invariant under T cross sister action and the actions such T cross sister action on T cross sister action on X zero and X infinity will be something like this. So on X zero X infinity will be So T U acts on X X zero as T times X zero and T U times X infinity will be T U K So So on this fiber maybe by definition it is easy to see on this five This is a one zero 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 mix mix mixing so mixing between T and sister and so this is somehow the key point of the Shift operator So we consider outside of this somehow we consider the following correspondence So we consider the modular space 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 modular 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 X zero and X infinity You have a violation map at zero and infinity. So this gives you a operator This gives you an operator s which I write sk from T cross sister covariant homology of X zero to T cross sister covariant homology of X infinity and Actually, so this T cross sister equivalent homology of X zero. This is just Because sister component sister factor of trivially on X zero T covariant homology of X With one variable Z So up polynomial ring in Z answered so Z is the same parameter as we see before so so this sister equivalent parameter becomes somehow a slightly parameter in on the site or site and On this side, this is a little bit subtle, but you can identify You can do some identification with X infinity and X zero Somehow these are small field there because X zero and X infinity are the same as a space So you can identify in some way. So but this identification is not linear over Equivariant homological point. So so in this way you get them up here from from Quivariant homology 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 To big case so you can also insert run to town You can define with some bulk deformation and then Then what you do is what you see from from the cost is construction in particular this Shift of the Degross Easter action in the following properties. So first first this SK Shift the action of lambda. So SK Lambda I is equal to lambda I minus Ki Z so this is some There's some Heisenberg group type commutation relation between Equivariant parameter lambda I was a T quivering parameters and Recall that this Z Z is a C striker very important and this is This property one just follows from the somehow the action to have action on the difference of the action X zero and X infinity. So they are they differ by Just some okay factor and the second property. This is somehow the original side-o representation SK As now as L is this is Proportional to there's some factor A plus L. This is some side-o representation Some some side-o representation and also this SK also commutes with the robbing connection is the robbing connection and So you have some rich structure on quantum demodule equally variant quantum demodule and Maybe yeah, I should first mention that these first Relations amount They are somehow They satisfy some some kind of Heisenberg relation commutation relation kind of canonical commutation relation This is a corresponds 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 each other and and X variable corresponds to this side-o element, so This explains why my lambda and X s are conjugate variable in Equivalent quantum In some sense quantum demodule is a quantization of a quantum common machine So this is a kind of a quantization commutation relation and And second remark is that originally some of side-o element side-o element Is just a limit of this? Shift operator, so SK is limit of Z goes to 0 this C-strike invariant point goes to 0 of SK Applied to the unit. So this is a side-o element This was original side-o element. So So by this somehow shift operators and side-o elements we can Totalologically in some sense, totalologically construct mirror map and a mirror isomorphisms. So that's what I'm going to explain now so so so what we want to do is to construct the isomorphism between side-o structure of a lambda and and Quantum demodule, equivariant quantum demodule 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 shift operator and so So you actually somehow I I claim that if you want to find an isomorphism Satisfying these two properties then isomorphism can be uniquely fixed. So So for instance if I take an element here say if I take take This oscillating form so ex1 just assume. I'm just still working on p2 and Take this element. This is an section of this module and What this goes to so this goes to some element in quantum equilibrium quantum cohomology. This is just a Section of the tangent sheaf section of the tangent sheaf of the equivariant one covariant cohomology and I want to know what did this section and But this can be determined uniquely somehow almost uniquely Because because this element satisfies the following differential equations. So Z ddy Ddyk if I differentiate this in lambda xy in Z And some homology class of this By very simple differentiation you just Somehow pick one term. So this is xk q to the meter k by the very definition of F lambda e to the F lambda x y on z So you have this differential equation and By this requirement So this is Gauss-Mannin connection. Maybe I didn't say explicitly, but this is Gauss-Mannin connection. So therefore So therefore we should have the rubbing connection ddy Actually, I need to differentiate Tau because because this is somehow pulled up by the mirror Omega this is equal to sk How? So 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 equal Z equal zero limit Then you also find the following differential equation for the mirror map so dydk should be should be determined by this differential equation so This is somehow to logically determine the mirror map and the mirror isomorphisms and And somehow you can show the theorems and maybe also mentioned that Finally, I just want to mention one thing. So So of course some of these differential equations are very abstract and it doesn't seem to be Of course, you can show abstractly that these Differential equation has a solution but on the other hand you can also determine Tau as some a mirror map and this omega by somehow I function and So this is a more traditional method of given also given the life function but actually given the life function can be also solved in this way so recover Which is certain? Maybe I don't say what it's given dark function with certain hyper geometric function By some this kind of differential equation So this is some infinite dimensional I function for for some SK some explicit difference of right and constant difference of greater and somehow from this you can Define I function and also solve these equations explicitly. So thank you very much