 All right, so welcome everyone to the Schubert seminar. Today we're happy to have Chang-Zeng Li from Sun Yat-Zen University telling us about the Pluker coordinate mirror for partial flag varieties and quantum sugar calculus. So please take it away, Chang-Zeng. Okay, so maybe I just start. So first I would like to thank the organizers of the seminars for the invitation and thank Leonardo for the introduction. And this is my recent work with Connie Lych, Min Zi-Yang and Xu Zhang. So this part, this work somehow shows some interaction between mirror symmetry and quantum sugar calculus. In the first part, I will focus more on the mirror symmetry part or more precisely on the B side. And in the second part, I will focus more on quantum sugar calculus. So once we refer to mirror symmetry, we always say something called a side and something called a B side and the equivalence between the both sides. So usually on the A side, we say some information, geometric information from the simplicity geometry and on the B side something from the complexity geometry and some information equivalent to each other in the following sense. Here on the A side, if we stick to fundamental force, then the B side, so if A side is fundamental force, then B side is a function on some space. The whole thing, X, AF, called the Landau-Kinsberg model. And this is X is non-compact killer and this is a homework function. And the equivalence could be set at the various levels. For the first level, we would say that on the A side, we consider the quantum comm module ring, the small quantum comm module ring. And on the B side, we consider the Jacobi ring of the super potential F. And the Landau-Kinsberg ring, then these two are as more to each other. And this is something, something we find the statement of this isomorphism. Because if we have this isomorphism, then C1 is here. Then the image should be equal to F inside here. So then this says this. If the image of the first-gen class is equal to the class of the super potential, then image just says that the eigenvalues of the first-gen class are precisely the political value of F. Here eigenvalue just means that we consider QH to QH and the alpha mapped to C1 times alpha. So we have linear in the morphism, then we can consider eigenvalues in these things. Then we can ask isomorphism as a demodule of both sides. And the following model, we can ask isomorphism as for the business workforce. So anyway, there were various equivalents between both sides. And for me, I just, I mean, in the end, I want to understand the mismatchy for flag manifolds on the level of the business workforce. But today, okay. So we focus on the case when X check is of a flag manifold and we will talk about these two parts. So we talk about these two parts. Okay, so this may not be so good. So flag manifold is a question of a group by subgroup. And the laser classification. So once the G is fixed, then laser one to one in correspondence with a pair of roots of base and the subset of the base. So this is a subset of Delta. And this is some classification. And the notation says that, so the Bolloy subgroup corresponds to your empty set and the G corresponds to the data G equals to data L. This is a convention. In particular, in the case of type A, then the patch flag variety always parameterize the patch flag in CN plus one. And most especially if we consider the complete flag, then we can see that FL4 is V1 in V2 in V3 in C4. Alternatively, we can say that this is a point, this is a point, this is a line, this is a plane, this is in P3. So an element in FL4 is a pointer in a line P2 and then the whole thing looks like a flag. So this picture is from Sarah's PPT, which I like it very much. And another special case is that if Delta P is a maximum subset, then this is a graph meaning. So here are some examples of other lead types. We refer to this case just because, so for some asymmetry progress, which would involve some cases. In some cases, especially for minuscule graph maintenance. Now, we say the quantum cohomage. The classical cohomage is the intersection theory. So it tells us the number of intersection point of super varieties in general positions. And the small quantum cohomage tells us and small general information becomes the number of holomorphic sphere with fixed values that passing through the three given sub varieties. So in particular, if the value is equal to zero, this becomes a pointer. So in this sense, QH is a deformation of the cohomology and the number of this volume is given by Jonas Zellow from Witton invariance. Slightly precisely, the classical cohomage of flagority has a basis of superclasses. And WFP is a subset of the Y group. We don't need to worry about the subset at the moment. Just a comment only subset. And then the quantum product can be written down in this form with the structure constant defined rigorously by using the pullback of the superclasses to the multispace and take the integration over the multispace. And geometrically it counts the number of holomorphic maps with fixed degree. Such later the pointer F0, F1, F infinity belong to the super variety correspond to sigma u, sigma v, and the sigma w sharp in general position. So in particular, from here, we can see that this number is non-active integer. And so on the A side, we study the structure of quantum cohomology of flagority. And this we usually refer to quantum super calculus. And for this, we would like to ask a link presentation of the quantum cohomology in terms of polynomial in question by idea. So the question is a description of the idea. And also we would like to ask the superclasses inside qx in the question. And because the structure constant is a non-active integer, we would like to ask a positive combinatorial formula of this structure constant, and maybe a bit more, which are some context in the quantum super calculus. And now for the b side. And as we can see from the statement, we wanna say the quantum x is as much to the Jacobi ring of F. So before each, we should ask what is, so once this is given, so we would like to ask what is the pale? So therefore, the first question for the massimetry of flagority is the construction of the super potential. And historically, when the super potential for torque mainly for the one known earlier, slightly earlier, and the super potential are usually referred to Hollywood formula super potential. And in the case of flagority, so around the same time, Gucci, Holly, Sean give a constructor the Miller-LG model for grassmaning, and they give a constructor the super potential for complete flag. Let alone BCFKS, let construct the super potential for partial flagority. And Nithyna, Nohara, and Weta also construct a super potential for this with a different approach. In the end, in summary, all the construction turned out to give a same super potential in the same. We will describe it really soon. And so the list we call it as F tolic, because it's closely related with some super potential for some total variety. Then in around 2008, and Rich provided Miller-LG model for flagority for all flagority in general type, and her construction was in this theoretical way. And more recently, Matthew and Rich they interpret Rich's super potential in terms of preco-coordinate, and this is quite simple. And our joint work with Rich and Minzi and Chizan is to generalize this to all partial flagority in type A. There were also some other super construction. Number one, this is also in terms of preco-coordinate by Karachi-Nikov. And this is by using a billion or billionization. So we will see some examples. So let me explain the super potential as follows. So for the total one, that is, so we can consider the total flagority and consider the total degeneration to the get one certain total variety. And once we have total variety, then we just use the super potential of the total variety and pretend that this is the super potential of the flagority. And so if you know a little bit about the total degeneration, we can just easily write down it from the graph. That is once we are, for example, we are given consider the partial flagority. So this is four by four, this is seven by seven, and this is nine, and then this is 12. So this is the way we draw the box and the unit box. And you can see that the number of unit boxes is equal to the dimension of the partial flagority. So this is the pure graph. And what we have here, we give orientation and then in this way. So once we have this, this is X, Y, then this error just give us Y over X. And once we have X, Z, then this error just give us 10, Z over X. This is the way we write down the super potential for total variety. And in a special case, so GR24, so you can see the dual diagram is in this form, is of this form, this is one. This is like a D11, D21, D22, D12, this is Q. See, this is the way we write down it in this way. So we can write out a function and define the only powers. Okay, and this mirror, so as a first glance, as a first glance, it's not so complicated. So it is an intersection of the lower triangular matrices. This is the B minus. And the U plus is one star, like this. And this is some longest permutation in the Y subgroup. And this is longest permutation in SN. And this is WP invariant tolerance. So the level of factorization of this and then U1, U1 is like 1110, U12, and Un minus one, Un, and the star. And this just means U12 plus U23 plus Un minus one. And then similarly to U2, but I need to be careful that the factorization is not a unique, not a unique. This is well-defined. It's independent of the choice of the pair. In practice, this is complicated because it's difficult to interpret it in terms of the way we write it. So we have a lot of local coordinates of the matrices somehow. Okay. And in the case of Grassmanian, so we have a canonical embedding, we just map to this element. And then in this case, we have an integral divisor of the Grassmanian defined by using this. So this is cyclic. You can see 12k and 22k plus one until here. And then, so this is the module N. So in this case, like j2, 4. So then it is 12234 and 45. 45, which means one four if the zero. And the middle space is the complement of Grassmanian of this integral divisor that is C4 removing the three head surfaces. And in this case, the super potential is defined as very simple. This is just like this. And in terms of box, in terms of box, this is just like empty one box. This is like one box to one box in terms of box. Just me and the partition just corresponds to a canonical pre-coordinate in this way. So this part is actually p this box and p one box in the queue. So you can see the pre-coordinate is simple. And for FL124, then we can compare which the super potential currently cross-unit curve and the power ratio looks like this. So you can see that this looks simple. But it's not only a simple expression, but something more, this is enough. This is right mirror. This is weak. This is weak. Weak is in a sense. So if it's a right mirror, at least we should have the dimension of the flag should be equal to the number of grid points of F. So the chronicle emoji of X corresponds to a jacqueline. And naturally, we should have the measure of dimensions. But 1q1 equal to q2 equal to 1. So the number of grid points are not enough. In this case, this has 11, this has 11, this is 12. So that means something, some critical points will go to infinity. So in essence, this is not enough. And we can also see the Goudtrap mirror. So even for Guassmanian, it contains a lot of valuables. So in that case, it's complicated. And this is the Peterson framework, which is closely related to mass metric in this way. So Peterson variety is a sub variety in the nonland fuel flag variety. And in this case, A side, we can see the B check. And there's a six direction on the Peterson variety. It has a finite fixed point, which leads to strata decomposition of the Peterson variety, yp plus and yp minus. Peterson claims that the chronicle emoji of flag variety is as much to the regular function of yp plus. And so this is about yp minus. So this is one statement. And also Rich showed that the grid points of yp plus is grid points. The regular function of yp plus is as much to the Jacobi ring of the super potential. So if we combine these two, then we obtain this. This is the mass metric. This is the A side. This is the B side. OK. And this is some story on some higher level. So on a level D module, two weeks ago, I mean last month. So Chihong Chiu gave another talk on the module mass metric for flag variety. You may track the records that you can see later. It's about the algorithm of quantum D module, which compatible with the quantum connection on the A side and the quantum connection on the B side. This is the definition of here. So usually we should, as a mechanism, should compatible with all directions, Q i and also Z directions. So in the case of Grass Manian, mass and the rich proved this to be an injecting morphism. And Thomas Lan and temporary proved this for minus Q Manian as a morphism. But in this case, this is precise and this isomorphism is less precise. So Chihong Chiu also showed the isomorphism for general GOP, but in his case, so Z direction was not considered yet. And here we can see something. So we will compare this. We can see this. This roughly tells us that C1 should be related with F. Now indeed, this is our main aim of this work. And this is on the level of the business manual that there are fewer progress as far as I know, only known for Grass Manian indirectly and it was also known for project absorption. And this is for some homologous mismatches and this is something related to that. This is also mismatches, but this is open string. And the previous one, this is closed string mismatches. And our results just sets following. So for all patch flag variety in type A, we can express rich super potential in terms of pre-coordinate function. And moreover, under the isomorphism of FL, isomorphism to Jacobi. So this is equal to Jacobi. So under this isomorphism, we can say that the image of first class is really the image of the function. And previously, this was only known for total final manifolds. And we can take first glance of our super potential. So it is always, our super potential is of this form. So there are some key points of our expression. First, the number of turns is equal to m minus 1 plus r. And this is equal to the number of components of the integral divisor. And this is a canonical integral divisor. This is an integral divisor by Newton, Thomas Land and the spare. So given by the sum of projected retraction hyper surface. And each turn is either a quotient of a monomial by monomial or some quadratic pp summation pp of this form. And each turn has exactly a simple pole along a unique elitical component of here. All these, these phenomena are somehow important from the viewpoint of a mere symmetry. And as an example, we can see that if we can see the FL1 minus 1n, in this case, then the super potential is of this form. So here, PI, if we take this is, this is P1, P2. And this is like a P, this one is like a PN check. So this is your form. So as a summary, we just selected. So you will consider a mere symmetry for flag variety. You can see it from the well beginning. So here we can see the j over P. So the first part, I just emphasize the construction of F. And we want to select F minus is something good, the super potential. I mean, it is, I mean, it is some good expression of the super potential. Yeah, that's what I want to emphasize. And because of mere symmetry, we can, because of this isomorphism, we can see later. So information of the quantum core logic will be capitalized from the ring structure of the jacquie ring of AFL in principle. So we may ask if we could really say something from the B side. Yeah, that's what I want to want to say in the second part. Okay. All right. So I'm done. Thank you. Thank you. So then we're going to take a small break, maybe four minutes until five or five.