 All right, so welcome everyone to the Schubert seminar before we start today's seminar. Let me remind you about the couple of announcements. The last seminar and there's announced that we are looking for nominations for either graduate students or for postdocs who are interested to give short talks. Short talks will happen probably at the end of November beginning of December and especially the early career people that are on the job market are encouraged to to nominate to be nominated. They can self nominate somebody else third party can nominate them and so on. So please, if you if you know anyone and that person is interested in giving a talk. Please send an email to any of us. And the next seminar is going to be next Monday at the same time with Dave Anderson. And now let's start today's seminar. So Chang Long Song from State University of New York in Albany is going to tell us about the elliptic periodic module and the 3D mirror symmetry. So please take it away. Yeah, thank you. First, I want to thank the organizers for inviting me to give this talk. Yeah, so this is some talk. It's a talk about my recent work with Christian Leonard and the guvans. So, yeah, here's the rough frame so I will talk about first mentioned some background about the equivalent elliptic homology and then I will say something about the elliptic homology equivalent elliptic homology of X of a variety and then I will talk about the construction of case theory periodic module. After that I will talk about the elliptic version, and then I will talk about the materialistic operators in the case and then the main result. So first, some background about elliptic homology. So the first part is but I mean there are many motivations many reasons of considering elliptic equivalent elliptic homology the first one comes from for me actually. The first one comes from the correspondences between three different objects. So the three big circles. Okay, the first one is about our matrices. So basically they are talking about solutions of classical Young-Baxter equations. So there are three types of solutions. For example, the other one, trigonometric, and the last one are called elliptic functions. So there are solutions of Young-Baxter equations and they can be used to construct different types of quantum algebras. The first type is called Youngian. The second one is called quantized universal European algebra. The last one is called content elliptic algebra. So basically there are algebras defined by using the generating relations using our matrices. That's my understanding of this. And the second category of objects consists of contains the one dimensional commutative formal groups. There are three types. They come from I mean the ones that come from outbreak groups. There are three types. One is the additive formal group. Second one is the multiplicity formal group. And the last one is collection or the elliptic formal groups. So that's the second collection of objects. And the last one is the generalized module series having some classes. So there are three types. The first one is the equivalent commutative. The second one is the equivalent k-theory. And the last one will be the equivalent elliptic commutative. And all the three collections, they correspond to each other. So for instance, equivalent commutative, they give you the additive formal groups. And they give the rational armature which gives the Youngian. Okay, for k-theory, you have the formal group will be the multiplicative formal groups. And then they will give you the universal UMMP algebra, the quantized version. And then also this quantum elliptic algebra, they correspond to the elliptic formal groups. And they are supposed to correspond to equivalent elliptic homology. Okay, so that's the relation between the three categories or three collection of objects. And that's the one of the motivations of considering this equivalent elliptic homology. So this is a theory. So the axioms are given by Gimburg, Kaplenov, and Vaselrod. They give some list of axioms, and that's in 1996, I think that's an archive paper never published. And the first construction was given by Gronowski. Basically, the two work appear in the same time almost in the middle of 90s. And later on, Ruri and many others have studied such theory, Ruri give a general construction of equivalent elliptic homology, and using this derived algebraic junction. So, and there's a lot of more people involved, so I couldn't give the complete list. Okay. So that's the first reason of considering this equivalent elliptic homology. So the second part comes from this so-called 3D mirror symmetry. So that's some theory from physics, theoretic physics, and the mathematical statements is made more concrete or precise and rigorous by recent work of many people. So there's a lot of logic, Akonkov, Romani, Smirnov, Vachenko, and so on, and there are many more others, mostly from Akonkov school. So the statement is, so it's more like correspondence, conjunctural correspondence between different objects. So here, X is a symplectic variety, symplectic resolution, and then it said there's a mirror of this variety, we call it X shriek, that becomes a standard notation. And then there's some correspondence between the two sides. The first one is that, so both of them will have torus action, so you have a torus acting on X, you have another torus acting on X shriek. And so then they satisfy this property, that is, so if you look at this diagram, so you have considered X cross X shriek, and now if you consider W from one of the t-fix points of X in X and W shriek, a t-fix point, a t-shriek fix point, then one can consider like this embedding, and also the other embedding, then what the 3D mirror symmetry statement says is the following. So there's some line bundle over this, so this equivalent elliptical model G of X cross X shriek, you can think of this as a variety. Okay, so over this variety, you have a line bundle. Then you can consider some section, they exist on section M, so that if you pull it back, you get a section over this variety, equivalent elliptical model G of X. You can also pull back to here, to the X shriek side, you also get a rational section. And then the statement says that, first of all, the t-fix points of the two sides should be bijective to each other. So it will map W to top shriek, that's how you identify the bijection between the t-fix points. After doing that, then you can look at the pullback of the regression sections. And then they will give you the so-called elliptic stable envelope defined by Aghanagic and Okonkov. And lastly is this identity between stable envelope, you can restrict to a t-fix point V. And for the other side, you can look at stable envelope for X shriek at V shriek, and then you can restrict to W shriek. Now notice that then you have this identity between the two sides, okay. But this identity is an identity up to some normalization. So in other words, you have to normalize the stable envelope by some factors, okay. So this three gives the 3D mirror symmetry statement of Okonkov and Aghanagic. So of course one can notice that this W and this W shriek, V shriek, they are switched, okay. That's one of the properties. And also some comments about this 3D mirror symmetry statement. So it's conjectured that this X and X shriek correspond to the so-called Higgs branch and the Coulomb branch, okay. There are two concepts from physics and I believe that Higgs branch is already constructed and is relatively easy to understand in math. It's some kind of hypothetical question, I think. Sorry, I'm not familiar with that. And the other one, Coulomb branch is constructed recently by this, by three mathematicians, Riveman, Fingerberg, and the last one is Nakajima. So one yard of space. So that's just some recent work. So there are a lot of things to be studied in this area. And the other thing is, the other remark is that this 3D mirror symmetry is also related to this symplectic duality, okay. That's also some mathematical statements concerning this category O and this cultural duality. Also, I didn't mention that the stable envelope. So we're going to define it later, but for now you can think of it as some rational section over this variety, okay. And it's indexed by the fixed points. So give you any fixed point, you will have a stable, a rational section that's for the stable envelope. That's the elliptic version. There's a similar version for case theory and the chronology that's well known. Okay. So that's the second motivation that I think one should consider you can run elliptic analogy. So, before I continue with any questions. Okay. Now the third motivation that one wants to consider elliptic homology equivalent elliptic homology comes from this study of elliptic genus. This is some work done by Boris off the global and later on it was used in this super calculus by the money at Weber. So basically, one wants to consider elliptic classes for a singular variety and then it turns out that you have to consider pairs instead of just one thing singular variety. So you have to consider pairs with a real divisor. So, what is off the global, they constructed this elliptic classes for the pairs, as well as this status. So this pair is a KLT pair KLT links for power model log terminal pair. Okay. Then after you if the pair satisfy this KLT condition then you can define this elliptic classes using you can use resolutions and they prove that this definition of these classes does not depend on the choice of this resolution. So if you want to study this construction for super variety, then there's a problem because this super variety together with boundary is not KLT so you cannot just apply this Boris off the global construction. Okay. And then it was realized by the money at Weber that instead of considering the boundary of the super variety. You can choose it or shifted it by the line bundle determined by this character lender. Okay. And as long as this lender is regular. This pair will be a KLT pair, which means this Boris off the global construction can be applied. Okay, so that's why in this case you can define these classes. Of course, in this case, then you have an actual parameter coming up that is this lender. Okay, so this lender in this case is called this dynamical parameter. Actually, some people I mean or cook off and some people also call it Kayla parameter. Okay, so that's the reason why the dynamical parameter comes up. Okay, so then the elliptic case, people can quite elliptic homology with the elliptic heck algebra or elliptic the materialistic operator with dynamical parameters. Okay. And the definition of this, of course, is resolutions and the canonical resolution is this for the semester resolution that exists for any shoeboard writers. So that's how this is defined. And because for the semester resolution is constructed recursively. So then this, you know, operator, the materialistic operators are defined. Okay, so this elliptic version of the materialistic operators are defined. And because of this body softly go goes work, such the classes defined in this way does not depend on the choice of the resolution. So automatically, it will tell you that is the operator satisfy the bridge relation. Okay, and of course there's one more relation that is the, what if you take the square. So if you take the offer the square is going to be equal to one. So it behaves like a while group. Okay, any questions. So, let's continue. So that's the background about this three reasons that I think people should consider equivalent elliptic homology. So, for us, we will not use the construction here is that we use the logic modules. But before that, I should say something a little bit about equivalent elliptic homology. So, for that one, once you start with the point and that's the easiest case. So if you consider equivalent homology of a point, we know this is equal to the symmetric algebra of the group of chapters. And you can think of elements here as functions polynomial functions over this T's lowest star tensor C. Okay. So similarly, if you consider the case theory, it will be just the symmetric at the group ring. And again, you can consider that as functions over these colors. So here, of course, he lowest size, just free of being group. So this gives you a product of C star. So that's why if you take the spectrum, you will get a fine space. And if you take spectrum of the case theory, you will get the torus. Okay, so that's why if you consider the elliptic version, you will be looking at functions over. And there's a typo, as you put it here is an elliptic is the elliptic curve that you fix at the beginning. Okay. So, and then, if you take the spectrum, you're going to get T low star tensor E. Okay, this is the variety that we call it a. So it has morphic to E to the m is the length of the torus. So actually, I put this way, but in the remaining part, I will also always consider this spectrum. This a, as equivalent commodity of a point. So yeah, what's from now on, if I say equivalent commodity of a point or equivalent commodity of a variety. I'm talking about some IBM writing. So in the point case is just this IBM writing for general X, it will be a variety over this EM. Okay, so that's why we define this a to be this T low star tensor E, and then the dual of this ability is T up a star the group of chapters tensor with E. So the tensor is over C, okay, over the ring of integers. And that's also isomorphic to the degree zero pick up group of a. Okay. So that's why that's why it's called the dual ability. So I made another mistake. So I will just think of elliptic homology of a point as a itself. Okay. And so when we talk about elliptic classes, we are talking about some rational sections of certain line bundles over this variety and now if you consider the T fixed points, you can embed it into here and then you have pullback of elliptic classes, you can restrict to the T fixed points V. Okay, then in this case, this becomes a product of elliptic functions over the over a. So this becomes functions over this a this ability right. Yeah, also I should say that these elliptic classes of the money waiver, they also corresponds to the stable envelope of agonistic. Okay, this is stated in the money and waivers original work and I think for some cases, this identification is not written down yet. Okay. Okay. So that's the setup of this equivalent elliptic homology of varieties. So there are varieties. And so to continue, I would like to first review some part from case theory that's a classical story about case theory of Cotanium Mando. So, we know that this alpha and hack algebra. This is as a morphic to this case theory equivalent case theory of the fiber this standard variety. So this is already noted by the standard variety. And this work I should refer to rustic. Some classical work of the tool. So basically, one can identify this hack after hack algebra with this case theory of alpha of the standard variety which is a singular variety, and then it acts on case area of Cotanium Mando by convolution. Okay. And of course, you can look at the T fixed points of this Cotanium Mando, which will be projected to the wild group. So then you can go back to T fixed points case area of T fixed points. So this is called the restriction. So then inside this alpha hack algebra, that's one special element that's called the materialistic operator for case theory, usually people denote it by T alpha, alpha is a simple root. So this, the presentation of this operator can be written as this fraction plus this one plus S alpha, S alpha is the wild group action. Okay. And then you can let this, this is an element. This is a class or class in this case theory. So it will act on the case of case theory of the Cotanium Mando. And then, for instance, you can take a special element, the class corresponding to the identity points. Okay. So, then this T alpha by definition is, it belongs to this key product. Okay, that's the group ring of wild group. And this is the field of fractions of the case area of a point of course we know that this is basically this group ring. And they add this actual parameter, because of this C star action. Okay. So this work actually refer to custom kuma that's not classical work. Of course also realistic. Sorry. So, T alpha you can think of it as an element in this product of this field with this group ring, and they set by this quadratic relation T alpha square is equal to q minus one T alpha plus q. So now if you, they set vibrator region so we can define T w. Okay, after defining T w, then you can let the T w, which comes from the affine heck algebra, you can let them act on this fixed point. Sorry, this, this class that you fixed at the beginning. And then you will get some classes indexed by the by the world group. So this is the case theory stable basis for w. Okay. So case here stable basis was constructed by Molyk and our concoff and of course this case theory stable basis depends on many several parameters depends on a chamber polarization and slope. So this identity is just for one of them you have to fix some special chamber polarization slope. For example, this identity was proved by my work with Sun Jian Su and Goofang Zhao. And also, there's another story related to this that is, it's also equal to this motivation class of super cells. And construction coming from electric geometry. So that's some work of a Luffy mihasher Schumer and Sue, and also for type A, type A flag right is, that's also studied by Lemanian wiper. Sorry, Lemanian vachenko. Sorry. And I think I should also put the taros off. These three are for type A and then Luffy mihasher Schumer and Sue, they work for general Schumer right, general jima b, not just for type A. So that's the story for case theory. Okay, any questions. Okay, so as I said that's the case theory case and so basically this is called the periodic module. Okay. And for elliptic case we can start to do elliptic case and what we want to do is we want to mimic this action. And also this module and then this operator. Okay, so, so far everything I mentioned are not done by us. I mean, some of their classical and some of their recent. So now for the elliptic version, what do we do is the following. This is the concrete line bundle over this A cross a dual. So basically, this concrete line bundle will classify all the degree zero line bundles over a. Okay. So, and we look at some shift of P. This L is defined to be P tensor with all of H row. Sorry to H row tensor with all of negative two H row check. So here, H, it's an element in E that you fix at the beginning. This corresponds to the C star action so we have fixed an edge first. And this row, all of two H row. So, H row, row is an element, low is the sum of the positive rules and then divided by two. So to H row is a special element in a check. And then this all of two H row will give you a degree zero line bundle over a. And similarly this one will give you degree zero line bundle over this a check. So take the tensor and then tensor with P. That's the shift of this L. Okay. And then there are two, the well groups acts on a, because a is the T low star tensor E so well group acts on the group of collectors. So you have a well group action here which can define a well group action on a. And similarly, you have a well group action on a check. Okay, for the second one we call it w d is the initial for dynamical. We have two well group actions on a cross a check. And then we can define this as to be this w up star. V D star so D means dynamical so V D star really acts on a check w acts on a. So with these two actions you can pull back this L inverse and then you can tensor with L. This defines you a line bundle and then you can take the direct sum over all the W V in capital W. Okay, so of course, in case, in case if you repeat this construction, this bundle will be a some of it. So, because over C star in the case here case, you will be talking about over this torus and over this torus. You, this is our fine so you don't need to really worry about line bundles, but over a or over e elliptic curve that's projective variety so you cannot just. It's not our fine so you have to consider line bundles. So that's why we have this construction. So this is what we call the twisted group extra. The elliptic version of the one with dynamical parameters. Okay, so this is a line is a vector bundle over in this choreography. Shift of category of coherence shifts over a cross a check. That's a type of a cross a check. Okay, and it's also. So because of this piece, we can think of this back bundle as a algebra object in this by graded coherence shifts over across a check. So we can define a tensor product structure in this category and then this as becomes an algebra object. Okay. So this is the definition of this twisted group extra. And of course we can consider the quotient of a cross a check into the file into the wire group action. And this will give us a print an action of s on this L more precisely you have to push forward to this variety. So as to get this action. Okay, one other way to do that is use the tensor tensor structure over here. So that's what we call the, so this L is called polynomial representation. It corresponds to the, if you recall before, it corresponds to the action of this after I had captured on a T cross C star of a point. Okay, which is, which is the group ring. Sorry, it's already here. Just this one. Okay, that's called a polynomial representation. And then in the elliptic case, the polynomial representation is really the action on this line bundle L. Okay. So that's the definition of this polynomial representation and then we can talk about the periodic module. That is this one. Okay, this is to mimic the idea that for potential bundle. So what we can look at is T fixed points and that's just w. So that's why if you consider elliptic homology of these fixed points. You will be looking at elliptic homology of w which is basically direct sum of a lot of copies of a structure shift of a, but if you replace the structure shift by the line bundles. It will be this one. Okay. So the W is really to indicate the degree. So this becomes. So both of them. So this becomes a object in W graded coherence shift in a cross a check. Now for the dynamic version, you can do similar things. You can also apply the dynamic of our group action, which acts on the second component. So then this becomes an object in the coherence shift, the W graded, more precisely WD graded coherence shifts of O A cross A check. Okay. So that's the definition of this periodic module and we have an action of this S on L as I mentioned L and also we have action of S on M and action of S on M D. So the D will serve as the equivalent elliptic homology of T star G over B, where you have to replace G and B by the London's dual. Okay. Because the roles of the torus and the dual torus and the torus and the London's dual torus, they are switched. So that's the definition of this M D, the periodic module for the London's dual, and that's the periodic module for the loose system. And then with that we can start to define this, the materialistic operator in elliptic case. So for that, I'm sorry to interrupt so we usually take a few minutes break in the middle of the seminar I don't know if this is a good time to have a little break. Actually it is. Yeah, thank you for reminding me. Okay.