Stable Envelopes, Bow Varieties, 3d Mirror Symmetry
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Rimányi, Richárd
There are many bridges connecting geometry with representation theory. A key notion in one of these connections, defined by Maulik-Okounkov, Okounkov, Aganagic-Okounkov, is the "stable envelope (class)". The stable envelope fits into the story of characteristic classes of singularities as a 1-parameter deformation (ℏ) of the fundamental class of singularities. Special cases of the latter include Schubert classes on homogeneous spaces and Thom polynomials is singularity theory. While stable envelopes are traditionally defined for quiver varieties, we will present a larger pool of spaces called Cherkis bow varieties, and explore their geometry and combinatorics. There is a natural pairing among bow varieties called 3d mirror symmetry. One consequence is a ‘coincidence' between elliptic stable envelopes on 3d mirror dual bow varieties (a work in progress). We will also discuss the Legendre-transform extension of bow varieties (joint work with L. Rozansky), the geometric counterpart of passing from Yangian R-matrices of Lie algebras gl(n) to Lie superalgebras gl(n|m).
 Thank you very much. Thanks to the organizers. I'm very happy to be here and to give a talk. It's actually my first talk in person after one and a half years as for most of the speakers, I guess. So my goal in this lecture is to give very down to earth, very approachable tourist guide type of introduction to these three subjects that are in the title. So pretty many delicacies I will hide, but I want you to give some, to have some picture of these, I have some understanding of these after the talk. Let me acknowledge my quarters on different papers that are related to this story, Rosensky, Varchenko, Smirnov, Zhuh, Weber and Xu. This picture is the plan of the talk. So first focus on that black arrow on the top left part, which says 3D mirror symmetry for characteristic classes. So that the explanation of that double arrow will be the first half of the talk. So I try to explain what it means, what characteristic classes for some certain singularities in a space are, what they are and what does it mean that for two different spaces, in this case, in this example, I chose the Grasmagnian of two planes in C5 and another space, which is a Nakajima variety decorated by that, or defined by that red quiver. So these two are related in terms of characteristic classes of singularities. So that is the first part of the talk, explanation of that black double arrow. And the second half of the talk will be explanation of the rest of the picture. I want to talk about Cherkis' Bo varieties. It's a, it will turn out a very convenient rule of spaces, more general than Nakajima quiver varieties, and looking at spaces like Bo varieties instead of quiver varieties or homogeneous spaces, it sort of explains why, for example, those two spaces are mirror symmetric and also it will answer other questions. So this is the plan. Any questions? Then let me jump into, well, let me recall this notion that was already defined in the excellent talks of Joel Kamnitzer, the quiver variety. So if you see a picture like that on the left, so we call it a type A quiver. As a side remark, everything in this talk will be type A, just for simplicity. So this is a type A quiver, which means just a, you know, a combinatorial object which you see on the left. These W's and V's are non-negative integers. Associated to that picture, as we learned last week from Joel, there's a variety which I call script N of Q, the Nakajima quiver variety. So I'm not repeating, I'm not going to repeat the definition, but here are some examples. The examples on the left are familiar spaces. They are cottingen, total spaces of cottingen bundles over partial flag varieties. For example, the Grasmania, you get by just one portion of a quiver and here is an example of a partial flag variety, or cottingen bundle of the partial flag variety. Quiver varieties are of course more general than cottingen bundles of partial flag varieties. Here's an example. So this quiver variety is dimension two and it is the usual resolution of a Kline and Singularity, C2 over Z mod 3. More interesting than the actual definition, let's see some properties of quiver varieties. In this case, type A, they are smooth. They carry some holomorphic, carry a holomorphic symplectic form and there is a tolu section on it. So let me spend a little time on the tolu section, so at least naming what the tolu section is. At each of the framed vertices, the framed ones are the little squares, you imagine a tolu section of that dimension, of the dimension of W and the product of all these tori act on the quiver variety plus there is one more factor, one more C star, which comes from the fact that in the definition you considered the cottingen bundle of something before you cut it down by a group action and the cottingen bundle, in the cottingen direction you multiply by an extra C star, which we denote by C star H. So in this setting, the number of tolu fixed point is finite and we have some tautological bundles for each vertex on top of the quiver. So of rank V1, V2 and V capital N, there's a tautological, there's a bundle over the space. So this we are rather familiar with since last week at least. So eventually I want to talk about the cohomology ring, the torus-equivariant cohomology ring of the Nocajima quiveriates, look at the top left part, so that's what I want to explain how I will talk about an element of the equivariant cohomology ring. So the way we will name elements in the equivariant cohomology ring is we will name their images under the localization map. So consider this map on the top left called LOC lock. So that is just restricting a cohomology class to the torus-fixed points. It is just the most innocent mapping cohomology, the restriction map. It turns out that in very general situations, for example here, the localization map is injective. So if I name the image under the localization map of an element of the cohomology ring, then I name this. This is an injective map. And why is it a simple thing? Because we have finitely many fixed points, so the restriction to the torus-fixed points is the sum of the restriction for the cohomology ring of each fixed point, and the equivariant cohomology ring of a point is just a polynomial ring. So in this case U1, UN, H bar. So the picture on the right is of course an example. It is T star GR grass 1, 2, C4. It has six fixed points. These are the vertices of this graph. So to name an equivariant cohomology class in the cohomology of this space, I need to name a polynomial at each vertex. So to name a colloquial class is a tuple. At each vertex I name a polynomial in the U's and the H. I cannot name any tuple of polynomials. That's what I'm addressing in the bottom left. The image of the localization map is not the whole thing. There are constraints among the components. And this is what is explained or indicated by the edges of this picture. So there are invariant curves in the space, and those are the edges of the picture. And they come with some decoration. In this case the decorations say 1, 3 on the leftmost edge. It says that an element at top, okay, so I coordinate at that fixed point 2, 3, and I coordinate at that fixed point 1, 2. They cannot be independent. They have to satisfy constraints. And the constraint is that if you plug in U1 equals U3, then they have to be equal. So these kinds of constraints must be true for an image in the localization map. Here's an example. So you see everything that is in blue is a sixth tuple of polynomials. And because they satisfy these consistency conditions, they do represent an element in the equirincology of the Grasmanian. So let's check one constraint. So the bottom right constraint, the edge decorated by 2 and 4. So that means that the two polynomials associated to the two vertices, they have to be equal if you plug in U2 equals U4. And indeed you will see that there's a U4 minus U2 factor on the polynomial, so they are indeed equal. And you can check all the others. This particular sixth tuple is an element in the equirincology of the Grasmanian of 2, 4. This particular sixth tuple is actually something that will be called stable envelope. It's an example of a stable envelope later. This slide is just a warning that I said that the components are not independent. And I said that for every edge there is a constraint. Unfortunately, the edges actually are not always discrete. In this example, which is a Nakajima quiver rarity, the edges come in moduli. There is a one-parameter family, a pencil of curves you see in the middle, actually at two places as well. And in those cases, the constraints on the components are deeper than just coincidence under some linear form. The coincidence have to happen not only for the polynomials, but for some higher derivatives as well. So I'm not giving the concrete statement. I'm just saying that the constraints are more involved than just the coincidences. So now that we have a way of thinking about the equirincology ring of the Grasmanian or Nakajima quiver rarity, I want to name some special elements in them, which will be named stable envelopes or cohomological stable envelope classes. There will be one for each Taurus fixed point. So the notation is sub p. So that's what I want to define. The definition is not on the slide yet. So it will be on the next slide. So these definitions on this slide is just preparation for defining an element in the equirincology ring. So the preparation is the following. I'm going to fix a one parameter Taurus subgroup, which say, for example, this u maps to u to the first, u to the second, u to the third, and so on. For h bar, I plug in one. If that is fixed, then we can talk about this so-called bjarjnyski Biroula cells. For every fixed point, p, so I'm talking about the second pink line, for every fixed point, t, I can define the leaf itself, the collection of those points that under this one parameter, this one-dimensional Taurus, they flow into the point p. That's what the definition says. So the limit of sigma z times x is equal to p. So it's called usually bb cell, but we call it the leaf, the leaf of that point. Actually, I will come back to the rest of this slide, but I'm going to jump ahead by one slide and show you an example. So look at only the left. This is, of course, the moment graph. This is the skeleton of one of the Necajunic varieties. One of the fixed points is called 1,4. I don't know how much it's visible, but there is something shaded blue, or it looks like green for me here. All the points that flow into the fixed point 1,4. That's the leaf of 1,4. I'm going to go back. So if we have these notions of leaves, we also have a partial order by just taking that which fixed point is in the closure of another fixed point. So that way you get a partial order. And if you have a partial order, then we are ready to define the bottom line on this slide, the slope of a fixed point. You take the leaf of your fixed point, but also then you look at the points which are in the closure of that leaf, and then take the leaf of those, and then you look at the points which are in the closure of that leaf, and then those are fixed points in the leaf of those and so on. So it's not just the closure of the leaf. These leaves all have the same dimension, the Necajunic varieties. So if you are familiar with working with homogeneous spaces, think of the leaves as the conormal bundles of a Schubert cell. And conormal bundle of anything is always same dimensional. And so here you take the leaf is the conormal bundle of your Schubert cell, but then on the boundary there are some other Schubert cells, and take the conormal bundles of those as well, which have the same dimension, and then in iterative you do the same thing. So that's the slope. And it is illustrated on the right hand side of this picture. So the slope of 1,4 is the leaf of 1,4 of course, but in the boundary there is 1,5 or 2,4, and then take the leaves of so-do. So whatever is painted blueish here is the slope of 1,4. So this is the geometric part of a definition that is needed to define the stable envelope. This is the Maulik-Okunkov axioms definition of stable envelopes in cohomology. So the stab, the stable envelope associated to a fixed point P is a unique class that satisfies three conditions. The first one is the support. This must be supported on the slope of that fixed point. The second axiom is a normalization that the stable envelope of P restricted to P itself. It has to be some expected obvious thing, namely the Euler class of the normal bundle of the slope there. So the slope is not a smooth manifold, but at P it is smooth, so it has a normal bundle of it. It has a normal bundle in the ambient space, and you take the equivalent Euler class. And there is a boundary axiom that if you restrict the step of P to anything else but P, so I'm reading the bottom line, so anything else but P, then it will be divisible by H. The stable envelope will have a degree half of the dimension of the space, so all these restrictions are of the same degree. So it's divisible by H means that if for some person who doesn't see H, just put flux in H equals 1, then this means that the step restricted to Q is smaller degree than expected. So the fact that you like to call it a degree axiom, that the step of the number of restricted to anywhere else but P, it collapses a little bit. So it's a smallness condition, always you think of it as the restrictions are small. Actually maybe as I remarked now that I mentioned the H equals 1 substitution. So in special cases for example for G over P, this table number of were known before and they have a name called the Schwarz-McPherson classes. So in special cases it recovers some old notion with H equals 1. Okay, so in this picture I'm going to explain just further elaborate on the axioms of step 1, 4. So again step 1, 4 is a cohomology class, so it has component at every vertex. We know that these components cannot be arbitrary, they have to satisfy some consistencies. But let me see, beside these consistencies what other constraints we get from the axioms. For example the one in yellow, it says that the stable number of restricted to 1, 4, it has to be some explicit thing and then you see that, I hope the picture is rather intuitive, at 1, 4 there are two directions which point out of the slope and the product of those, the Euler classes of those has to be the restriction there. So the yellow is that the restriction has to be that thing, it's equal to that. We also have the support axiom and part of the support axiom means that restriction to 1, 2, 1, 3 and 2, 3 must be 0 because this class is supported on the slope. But more than that, for example the support axiom tells me something about the restriction at 1, 5 because you see at 1, 5 there is a direction which is normal to the slope, which is to the northwest direction at 1, 4. Since this is normal to the slope there and the stable envelope is supported on the slope, the stable envelope restricted here has to be divided by the weight of that direction, in this case u1 minus u3 plus h. So the support axiom is more than just a bunch of zeros outside, even in the boundary it requires some divisibilities. So I hope most of these should be clear. And of course divisibility by h is an explicit thing everywhere but at 1, 4 the restriction has to be divisible by 1, 4. So if you just look at this as it's presented on this slide, it looks like very combinatorial saying that there is only one tentapole of polynomials which satisfy the consistencies which I haven't really told you the derivative constraints. But anyway, so the consistencies, there is just one tentapole which satisfies the consistencies together with these axioms. But it's true and that is the stable envelope. Okay, let me check the time. Okay, I'm going to skip this part I think. Okay, so let's just regroup. I might come back to geometric representation theory connections but probably not. So far what we have is that we have a way of thinking about the equivalent conjurings of knackage and molecular varieties and we define the comological stable envelopes of it associated to every fixed point. The way of thinking of that, I would like to think of it as that is the class of this loop. So if there is a big ambient space and you have a sub-variate then it's tempting to think about that as a cycle and that represents the co-homology in the ambient space as soon as you have some kind of point-caridurative type of relation in the ambient space. So there's a cycle. So that is one way of thinking about the class, the class of this loop, but it's not really that class, it's really some kind of H-bar deformation of it. So if you take the coefficient of the highest H-power of this class, it has no H in it anymore, that is like a Schubert class type of object. So this is an H-bar deformation of like 19th century fundamental class type of calculations. So that's what we have and okay that's where we are so far. Okay now we are fast forwarding. I'm just telling you that the stable envelopes have a generalization in K theory and in elliptic co-homology. And there is no way I'm going to define those, not even the theories and even less the actual stable envelopes, but I want to give you some feelings about them. So first of all how do I think about the K theory element on a lecogenic curate or an elliptic homology element in the lecogenic curate. So the first time says that a stable envelope or anything as a co-homology element restricted to a point is a polynomial in the equivalent parameters, right? Everybody agrees, that's what I said. But then the K theory element is pretty much like that except the restriction is a Laurent polynomial, not a polynomial. And in the elliptic, an elliptic homology case it's not a Laurent polynomial, it's an elliptic function, it's a section over a product of elliptic curves. In the same variable is used. Okay, so that's one thing I want to say is that the way you want to think about for example an elliptic homology element on lecogenic curate is a tuple of some elliptic functions. But there's one more thing on this slide is that in the elliptic line I added more parameters, the V's. And in the next few slides I want to give an intuitive feeling that in why in elliptic co-homology when you want to talk about characteristic classes you are forced to have some new parameters. These parameters will be called dynamical or scalar parameters. I won't be able to give the precise mathematical statement but I hope I will be able to give some intuition why you are forced to have some new parameters. Okay, so first this is the look at the theta function. This is a section of a line bundle over the elliptic curve. The way I want to look at it, this is that it starts with x to one half minus x to the negative one half which is in logarithmic variables is really up to a constant, it's sine of x which would be just the k theory part of it. So the theta function is really just a q-decoration, some q-deformation of the sine function. So you can think of it as a q-deformation of k theory. And I will use this delta function, delta A B is just a, you cook up from the theta function another two parameter function delta A and B. Okay, this is just definitions because then the next slide will be the one which I hope will give an intuition, we are feeling why in characteristic class theory in elliptic co-homology you need extra parameters. There is something which is not on the slide so I just want to say that if you want to define some kind of characteristic classes for the stable envelopes, there are many others. There are many approaches but one approach is that you resolve your singular sub-variety, you define some obvious thing in the resolution and then you push it forward. If you do this, you have to show that what you invented in the resolution, it was invented in the right way that your class does not depend on the resolution. So this notion that you define the good class should depend on some identities. For example, the two nearby resolutions are, you get the same class downstairs. And these identities are really always boils down to one identity. In elliptic world, it boils down to the top identities called phi's three second identity. It says that if x1, x2, x3 is equal to y1, y2, y3 and they are both equal to one, then that the product of that, some of the products of that functions is zero. It's a good exercise for your graduate students. So that was defined in the earlier slide and then this is a, it turns out that about elliptic functions, this is the only identity. Everything else follows from it, although highly non-trivial. Anyway, this is the identity which is behind the fact that in elliptic characteristic, in elliptic world, you can define actually characteristic classes. Now let's do the following, that take that top identity and plug in q equals zero. So let's go to k theory. Then what you, what you get is the middle identity, which is you see a trigonometric identity. Now you can give it to your, to your calculus students that if x1 plus x2 plus x3 is equal to zero, y1 plus y2 plus y3 is equal to zero, then this identity holds. Ignore the purple part for a minute. So this is the q equals zero specialization of the top line. And you can further approximate sine of x with x, which you know, which we always do, then you get cohomology. And then the identity that you get is the bottom line. Okay, now look at the purple, purple decorations. Is that the identities for a, in k theory and in the rational limits, they are, they are easier because they, they tell you more. They say that if the x is at the top to zero, then the left hand side is equal to one. And if the y is at the top to zero, then the right hand side is equal to one. In some, so you see that the, the identity splits to x variables and y variables. So because of that in k theory and in cohomology, what we did in, in, you know, in the past 200 years of mathematicians, they were not forced to work with the other set of variables because these identities, they, they, the governing identity behind characteristic classes, they, it separates to x and y variables. So you can just take the left side of this slide and then, and then build up characteristic classes in cohomology and k theory. However, in elliptic cohomology, the top identity doesn't, doesn't split to an x part and the y part. You are forced to, to work with y. Okay, so this is actually something that elliptic cohomology thought that we should do cohomology and trigonometry as well with two sets of variables with the Kirchhoff parameters, but we need to go out of our ways to, to, to introduce them. Okay, so I, okay, this was the explanation of the bottom right corner of this space that stable envelopes are defined and, and, but they depend on new variables, which I call v's. Any questions? All right, then, then we, we come to this fact that, that, which I call 3D mirror symmetry for characteristic classes. It turns out that there are, there are pairs of Naka-Jima-Kivir varieties. Let's call them x, x-tree, that the pairing together comes with some, I, a fixed bijection between the totals, fixed points for which the, the stable envelopes on one are equal to the stable envelopes on the other one in the sense which is on the slide, that you take the stable envelope of p restricted to q, p and q are fixed points on one, and you take the stable envelope of q restricted to p on the other one, then you have a polynomials or Laurent polynomials or elliptic functions, and they claim that they will be equal if you switch equivariant and, and Kirchhoff parameters as well as invert h bar. So, so this is the, the 3D mirror duality for elliptic, for elliptic characteristic classes. Here's an example. So everything above the purple line is about one Naka-Jima-Kivir variety, that you see the, the cottingen bundle of p2. Sorry, yes. We have a question. Yes. How the elliptic stub transforms with respect to the modularity transformation? Yeah, I, there is something and I never looked at it, so I won't be able to say. Of course, you should, you should restrict it to a point so that you really have an elliptic function, but yeah. Oh, what should I, okay, yeah, maybe I shouldn't. Okay, so let me continue. So over the, over the purple line, it's all about the quiverity of cottingen bundle of p2. So you see that it has 3 fixed points, there is the skeleton of it is on the left side of the slide, it has 3 fixed points and there are, those are the constraints among the restrictions. And the table on the top is the elliptic stable envelopes. In, in the following sense that you take, take the rows of that table. So the first, the first row is the stable envelope of F1 and that means that that's the one of restricted to F1 is that product of data functions, restricted to F2 is 0, restricted to F3 is 0. And the middle line is the table envelope of F2 and so on. Now the, the 3D, 3D mirror dual of that, the quiverity is, is, is this one in the, the, the one below the purple line and that has a totally different looking moment graph. But I think that's what I wanted to convince you with that as soon as you see the moment graph, you can write down the stable envelopes. At least in cohomology it's easy, in K theory it's much less easy and in elliptic it's, it's a lot of work but, but for small ones you can do it. So as soon as you see that graph you can calculate the stable envelopes you will have the, this bottom table. Again the rows are the stable envelopes. And the fact is that if you stare at these two tables that they are the same after transposing and switching U and V variables and inverting gauge. Okay. So this is a, this is the baby example of 3D mirror symmetry for stable envelopes. Here are some, some other random examples. With green on the two sides I'm, I'm indicating the dimensions of these varieties. Yes, you see the different dimensional varieties have, have this amazing coincidence that characteristic classes of singularities in those varieties are equal in this sophisticated sense. Okay, the bottom line, the bottom line is that the left hand side is of course a macogymn variety and I claim that there is no macogymn variety which, which is 3D mirror dual to it. But we will fix that later. Because now I'm starting the part two of the lecture. So maybe it's a good point to ask questions if you have. Yes. Do you expect the stable envelope for communism in capitalism? Okay. I, I guess yes. But it, it, you know, having a group versus a formal group law has some advantages. An algebraic group. So these three varieties that, three, co-homology theories that I named, co-homology, k-theorem, elliptic homology. These are the, the, the, the, the theories that corresponds to one-dimensional algebraic groups which is, which are parts of the, the formal group laws. So there are, there are, there are advantages of working with formulas. So indeed maybe there is a formula for, for the, the most general co-homology theory. But I just, I just want, I'm a formula person. So, so I, I certainly want to look at these three. Do we have a question in what sense are all theta function identities derivable from the tricycant identity? Okay. So that is a sophisticated sense and I won't be able to, I can find the reference paper which I looked at and, and I don't remember the details, but I just remember the, the intuitive statement. Yeah. So just at the very beginning we fixed this homomorphism from C star to T. So is that important? Yeah. So in, in the slides which I skipped, it is important. So if you, if you start playing with that, what one parameter subgroup you choose, then you recover representation theory. So after a while I will, I might comment on those. The stable envelopes of course depend on that. The stable envelopes do depend on it. It's not infinite, it's just, they depend on some, some chambers of choices. So there will be finitely many. And changing them you recover young, young matrices and so on and so forth. So that's where representation theory starts to be built up. Okay. So from now on I want, I want to, to define, so, so, so give a feeling about what bovarietes are. And actually I want to advertise them. I think we should look at bovarietes instead of queries. They have some advantages. Okay. So these will be associated to some combinatorial pictures, combinatorial data. The combinatorial data will be called the brain diagram. Here's a brain diagram. So the brain diagram combinatorial is just a collection of horizontal segments called D3 brains. They, they come with some non-negative integers, the dimension vector. But then the consecutive ones are separated by either NS5 brains or D5 brains. So I drew a, a blue or a red skew line. And so that you don't have to memorize this. It's on, on this board out here just because this picture will go away after a while. For future purposes I will also decorate the D5 brains with equivalent variables U sub I and the NS5 brains with the scalar parameters V sub I. So this is a combinatorial object for us. We can discuss some, some, of course some super-syntheory after. Okay. Okay. So what is the, what is the Cherkis-Bau variety? Start doing the same thing as you would do for quiver varieties, but only do them for NS5 brains. So look at the left side of the picture. If you see an NS5 brain, it's a red brain. Then you just do the same thing as you would do for a quiver variety. Take HOMC and to CM, where, where you know N and M are the numbers, the, the decorations on the two sides. Take the cot engine bundle and, and that, that of course has an action of GL and cross GLM. So you do the, almost the same thing for the other type of brain, five brains, but it will be a different space, not just the cot engine bundle of HOMC and CM. That is the left hand side I call the, the arrow, the arrow edge or, okay, arrow brain and then this is the, the bow brain. Actually this whole thing should be called a quiver, which you can put an arrow and, and both an arrow and the, a bow into a quiver. All right. So, so for, for the other type of brains, you put some of the, I, I indicated roughly what that space is, but of course a lot of things are skipped in this, what acts how, but it's also just a, you know, a Hamiltonian reduction or GIT quotient of, of, of, of rather obvious spaces. The, the, the, the, the, the, the, the key difference is that on this other space, an extra group acts is C star, that's in the bottom right of the slide, that there, there is a C star acts. Because what do you, what do you do after that when you, you want to build, build up the necrogenous variety is that you take the, you take the product of all these, all these T HOMS and, and B N M's and then you do reduction by GLM, cross GLM, you do the reduction by all the GL's. You do that, but these C stars will, will survive. So on the space that you get, there will be a C star action for every D5 brain. You see this, this is realized that every D5 brain will be decorated by an equivalent variable. So if you do this story, then, then what will this, the Cherkis variety C, script C of D will be the name of the Cherkis Boo variety. It will be smooth. It will have a holomorphic symplatic structure on it. It comes with tautological bundles coming from the D3 brains. So those numbers are the ranks of the bundles. It, later I will show you that it has finitely many, finitely many fixed points and it has a toro section and the toro section comes from the D3 brains plus there is a C star action from the fact that there were lots of T stars in the earlier slide. So everything that we liked about, like about necrogenous variety is sort of true for this one. Everything is very combinatorial and there is an advantage which will come very soon I hope. You should have asked a question. So what is the framing that is usually, it's replaced by this. The D5 brains, correct. The D5 brains has come, okay. Some of the, they collapse together to be the framing. Okay. Yep. Yep. So there is a dimension formula. Of course you don't have to memorize. You just imagine that if you see the brain diagram with those numbers, the dimension vector, then from that you calculate the sum and that is the dimension of the, of the Cherkis-Bovarietzi. For example, if you see this, this brain diagram that it's on the left bottom of the slide, then you plug in the numbers and you will get four. It's not a surprise. This will be T star P2, the Cherkis-Bovarietzi associated with this brain diagram. So you might say that things are getting more complicated because the quiver name of T star P2 was just one dot and one square and now it's somewhat longer. But there will be things that we've been at the end. Okay. So this dimension formula. Oh yeah. And now I think I'm going to answer your question now. How are quiver related special cases? So what I need to give you now is a combinatorial recipe that if you see a quiver, how do you build up a brain diagram? So the quiver has parts, this K-n part, look at the top left part and whenever you see this K-n part, just just through a segment ending with NS5 brains, the red ones, and put n blue brains in between, where n is the framing of the, or the dimension vector of the framing and decorate the D3 brains with Ks. So this is just, and then glue together these segments. For example, look at the bottom example. And whatever is decorated by yellow, so that part will be just, just go to the brain, the brain diagram on the right, which is decorated, which is decorated, you know, shaded with yellow as well. But it, the quiver has another part and just glue that part to the right of it. So you glue these segments together and you would have a brain diagram. Yes. And this works only for Taipei quiver, Nagajima quiver right here or for any? Okay. I only know Taipei. But what about loops? Yeah. Yeah, okay. I think the next slide. Okay. Okay, so I won't be able to do, you know, all kinds of loops, but one loop is fine. So A tilde. Yeah, so yeah, right. But otherwise, this is right. But make, make an observation here that the brain diagrams that we get on the right, they are special. They, they have this cobalanced condition, which is in the bottom, very bottom line of the slide, is that on the two sides of a D5, D5 brain, you will always have the same numbers. Okay. But here's the advantage that we do not have for quiver varieties. There is an extra, actually there will be two extra operations. This operation is 3D mirror symmetry. This is just the most innocent symmetry operation on these brain diagrams. Just reflect it down, reflect it by the horizontal axis. Or in other words, change red to blue and blue to red and so on. So let's call it 3D mirror symmetry for both varieties. Let's see an example. Let's find the 3D mirror dual of T star P2. So the top line is the, the brain name of T star P2. And we just formally create the 3D mirror dual of it. Okay. I can calculate this dimension, this dimension too. But then here we go a little depressed because this is not cobalanced. So I cannot recover it as a, as a knuckajumma quiver variety. However, I will be in a minute. I will be able in a minute. So just wait. Because I'm going to show another operation which exists on Boo varieties. And it's called Hanani Witten transition. Think of it as like a Reidermeister move. So you can locally rebuild your brain diagram without changing the, the space. So the, the, the rebuilding is such that you can, if you have a consecutive D5 and NS5 brain, you can switch them. The price you pay is that, that the, the dimension vector in the middle changes and it changes the way which is on the, on the right. And the theorem is that if you carry out such a change on your combinatorial model, then the associated Boo variety doesn't change its, its same. Actually the Taurus, Taurus parametrization, the Taurus action reparameterizes a little bit. But, but yeah, for the purpose of this talk, it's just an isomorphism. Okay. Then let's continue this example that we saw a, a few slides up. That the first two lines are just, we found the, the, the 3D mirror dual of, of T star P2. But now I'm going to play the game of carrying out Hanani Witten transitions. For example, first I carry it out at the yellow part. You see it. And then, then, then I hope you can just carry out this decision. And then, then after that I carry it out at the green part. I hope it's visible. And I will get to the brain diagram that I'm, I'm pointing at, or in the middle of the right column of the slide. And this one is cobalanced. Okay. So I was lucky enough to be able to carry out Hanani Witten transitions to make my brain diagram cobalanced. And if it's cobalanced, then I can recover it as a, as a Nakajima variety. So then we recover this thing, this example, the baby example of 3D mirror symmetry, or between two Nakajima quiver varieties. Any questions? Oh, okay. So I want to give you an FNA type. So, so look at the, on the left of the picture, there are quiver varieties. On the right, we have the same varieties, but in their bone names. And the transition between them, everything follows from earlier slides. If you see a quiver, then there is a way of drawing a brain diagram. The 3D mirror dual is just switching D5 and NS5 brains. And then in this one, it's such a simple thing, it's, it's accidentally already cobalanced. So I, I can, I can rewrite it as a quiver variety. This is of course a very well known example, right? Of Hilbert's Schemes and his dual. But you can play the game with more, more complicated type A or FI type, type A quiver varieties. Okay, any question? Yes. Is this the correspondence in Unicom? Unicom, so does it have an automorphism? Certainly this is the natural one. So, so the, oh, so your question is that are there quiver varieties which are isomorphic to each other? So different combinatorial codes isomorphic to each other. Yeah, that, that I doubt. Yeah, I doubt. Oh, yeah, yeah, I'm sorry. I can, of course, I can, I can present the one point space many different ways. So, so, so maybe I have to be, we'll back on that. I'm not sure. Okay, so in, in, so now I think I explained everything which is on this slide. This is just a repetition of the first slide that to find the mirror dual of, of Grassman 2C5, you just have to write its Boo picture. Then formally take its 3D mirror and then carry out an animated and moves if you can. And if you're lucky, in this case you are. And then you will get the, you will get a quiver variety. Right. What I want to talk about is some other very important structure that, that quiver, bovarietes come with. One of them is brain charge. So it will be a number associated, an integer associated to every five brain. So an NS5 brain, it is, it is just the difference of the two numbers on the two sides of the NS5 brain, L minus K, plus the number of D5 brains on the left of it. So here you see that this is not just type A, not F find type A. In F find type A there is something local charge, but anyway, so this is, let it be just finite type A. And for D5 brain a very similar integer associated. So K minus L plus a number of different type of five brains to the right of it. Here's an example. Is everything visible? Not much. Take the left most NS5 brain. His brain charge is 2 minus 0, because these are the numbers on the two sides. Plus, plus the number of D5 brains to the left of it is nothing. So that is the top red 2 on, on this, on this diagram. So for some reason I will collect these charges on, on the top and on, on to the left of an empty table. Okay. For, for the time being this is just a decoration. So I collect the charges of NS5 brains left of this empty matrix, the charges of D5 brains on top of the empty matrix. It's actually a, a, a, an easy theorem that the two charge vectors, the red and the blue charge vectors is a complete invariant of the Hanani-Britain class of, of, of the brain diagrams. Hanani-Britain class is the isomorphism. Is that you, if you, you switch to consecutive different type of brains, you have a different diagram. But the brain, the charges will not change. Yeah. And vice versa. So, so if I, if I didn't want to define your Bouvaritis, but Bouvaritis up to Hanani-Britain, then I would have just told you that they are associated to a, a pair of vectors. This pair of vectors have this one extra perfect, the sum of the red numbers is the same as the sum of the blue numbers. Okay. So again, so up to Hanani-Britain transitions. Bouvaritis are, are, when you, next to this empty matrix, you put arbitrary numbers on top and on the left. So the, the sums add up together. And among these, the ones which are, at least in one of the representatives is a quiver variety. These are the ones which were in the top, the top vector is a, is a partition. Those numbers are weakly decreasing. And if you put numbers there fully one, then these are just the, the topics of Schubert calculus. So why, why do we care? Because actually if you are, if you're coming from representation theory, then you don't, then, then you think that you can, okay, so I didn't say enough to support this, but it's fact that in geometric representation theory, you, you, you permit yourself, you allow yourself to permute those numbers on top. But not here. So the representation theory is the same, but the underlying space is different. So if you want to look at the space, then, then the both varieties are more general than, than quiver varieties. And why, why do we like that both vectors are arbitrary? Because it comes from one extra operation, trans, transpose, which is essentially three-dimensional duality. Which, so this was not complete for quiver varieties, as you can see. Oh, okay, so, okay, since I keep mentioning geometric representation theory, I might say that the following, that, so if you see these two vectors, then the height of this, this matrix is a number, say, n. And then you say, take the young end of g l n. That's, that's a quantum group. Then those numbers on top, read them as, as, as, if the numbers are a, b, then take lambda a of the, of the vector representation, tens or lambda b of the tens of representation and so on. So you take the, the fundamental representations and multiply them together. That's a representation of a, of a quantum group. And now you read these numbers on the left and take that weight space of that representation. And it turns out that that weight space is, is very naturally identified with the cohomology of the associated Boole variety. Okay. So again, this is the, why, this is why these spaces are important in geometric representation theory. On the accurate conjoinings of these varieties, quantum groups act. So one Boole variety is one, one weight space of such a quantum group. And this is true in, in k theory and elliptic. Is this the analog of this Moldik and Kuhnkov Young end? Yeah, yeah, yeah, this is exactly. So if this was, if these numbers on the top were just partitioned, this is the Moldik and Kuhnkov Young end. Yeah. And now, okay, okay, that was just, okay. I want to show you the, the beautiful combinatorics of fixed, torus fixed points. So if you ever looked at the combinatorics of torus fixed points on the Kuhnkov varieties, it was a couple of partitions and, and was somewhat messy. It is a different picture here, of course, equivalent and of course more general because for Boole varieties. And I find it fascinating. So the claim is that fixed points are in bijection with tie diagrams. The tie diagram is on the picture. It consists of ties. A tie must connect a blue brain with a red brain. And you know, my picture of drawing them in these skew lines, it tells you, yeah, so it's a natural way of connecting them. And, and of course each D, not of course, but it's true that every D3 brain has to be covered by ties as many times as this, as its multiplicity. So just imagine that you only see the brain diagram and the numbers and it's your task to put ties there. There are lots of choices. For this particular brain diagram there are 123 brain diagrams. This is one of them. So this associated Boole variety has 123 fixed points. Here are the fixed points of, of grass mints you see four. And you see there's a natural bijection between four choose two, just like in any other names. This, this tie diagrams beautifully transform under Hananividen transition, which looks like a Reidemesser III move. And they beautifully transform with 3D mirror symmetry, just take the, the, you know, the reflection of the image. So that means that not just that we have a bijection between Hananividen equivalent Boole varieties, which should be because they are isomorphic, but there is a bijection, natural bijection between, between 3D mirror Boole varieties, the bottom line. That was needed for the 3D mirror symmetry statement. Let me also mention this other combinatorial gadget that this, this matrix used to be empty, but now put, put zeros and try to put zeros and ones there, so that the row sums are the red numbers and the column sums are the, are the green numbers. If you are, if you manage to put zeros and ones in such a way, then you call it a binary contingency tables, the binary contingency table. You know these contingency tables come from statistics, but here it's only zeros and ones are permitted. It's also a, a, these theorem that fixed points are in bijection with binary contingency tables. Maybe, maybe I want to draw a picture is that, that if you come from Schubert calculus, then, then you learn to work with full, for example, the full flag variety and everything about the full flag varieties is, is parameterized by, by, by permutations. This is a permutation. In this language, what I'm saying is that quiver varieties, in quiver, in the, in the geometric quiver varieties, everything is parameterized by, oh no, first, first what I want to say is that a partial flag variety would be parameterized by an order set of subsets, which can be identified with these, these kind of bipartite graphs where the degree doesn't have to be one on the left. So it's like a permutation, but I permit coincidences on the left. This is a partial flag variety, everything Schubert's, all those fixed points, everything is parameterized by these things. And with both varieties, all we are doing is that we are permitting higher, higher degrees on the right as well. So bipartite graphs, which are the same thing essentially as binary contingency tables, are the, are the objects that, that parameterize all the, you know, the cells or torus fixed points or, or stable envelopes. So there will be a stable envelope for each of them. So on the right, on the right side, there are, there are the BCT codes of torus fixed points on graph 2C4. Okay, so I think I, at the beginning of this talk, I wanted to convince you that, that, that to find characteristics from a space, you have to walk through the following path. From the space, first you find the torus fixed points. You also find the, the invariant curves, which I skipped. There's combinatorics behind that as well. But then you have the moment graph. And from the moment graph, the axioms give you the, the, the stable envelopes. So I gave you half of the story. I gave you the combinatorics of the, of the torus fixed points. Oops. Anyway, so, so, so if you see what's on the top of this, of this slide, then you can create the torus fixed points. You will have the vertices on the thing on the left. Yeah. And you do some more combinatorics. You will find the edges. You will find what's on there and, and below the decorations. And then the, the stable envelopes are defined. From this one, you can calculate. Okay. The same, same for this other one. I might, I might mention at this point that, of course, that's how you define stable envelopes, but they are hopeless to calculate using the definition. So, so you don't really do that. So as we are understanding better and better, so there is some kind of co-homological whole algebra type of structure among stable envelopes. You want to, to, to, to, to, to, to, to, by convolution multiply two of them together third one. And there are some, some formulas as well in certain special cases. Okay. So then, then I'm just walking through something that I already showed you is that if you start with those two diagrams, then you play the game, you will get the torus fixed point, you will get the invariant curves, you will have these two graphs. And then you can calculate the stable envelopes and you recover this slide that you already see. You already saw. So all I want to say is that everything follows from just the brain diagrams, nothing else. Okay. So this statement that the stable envelopes match for, for a 3D mirror dual-bubu varieties is proved in certain special cases. For the Grasmanian and its dual, it's proved by the paper with what, it's a Smirnov, Varchenko's, for the full flag variety in type A being self-dual is, is, is proved in a different paper by the same authors and in general type with, with, on J Weber. The hyperb toric being 3D mirror dual in terms of stable envelopes with this dual is proved by Smirnov and Juh. And, and we calculated finitely many other cases. Maybe I should emphasize many. So it's, it's, it's quite a well-established conjecture that it should happen. Of course, everybody think it should happen. This should, I mean, you might remember the, the Higgs branch, Coulomb branch interpretation of this in, in, in the Commits are stuck. The, they are, oh, that's, okay. So may, It may not be that this is, I don't know. Yeah, I don't know. So I'm not sure that that is a special case of what I'm presenting. Certainly the middle of the general G over L and when it's longer on the word, it's not a quiviriety or booviriety of type A. So that's not a special case of what I'm saying, that in these are the cases for which the 3D mirror symmetry for characteristic classes is established or proved. So here is a summary and actually I'm on time. The summary of what we learned is that if there are certain nice spaces, for example, varieties of the total section, there is a characteristic classes. They have relations to the numerative geometry. They are relations to representation theory of quantum groups. Okay, they are related to some very important Q difference equations and actually two sets of Q difference equations. So they are there, yeah, I advise everybody to study them. They are very important notions. We also learned that they come in pairs, such that this table, I mean, the space is coming pairs and there is such that this table envelopes on the two are related and the natural pool of detecting it is booviriatives, which is close for 3D mirror symmetry and easy combinatorics govern their 3D mirror symmetry. The end. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Any questions? What's the relation, if any, between the boviriatives and the construction of cooling branches we saw last week? So that's Nakajima Takayama paper that they prove it. That these boviriatives are the cooling branches. Taipei, probably more general, but I understand the Taipei. There was a question in the Q&A, how the stub changes under the, and any width and the mover. Is this clear? Yeah, so that's an isomorphism, so it shouldn't change. But the way I set it up, you have to make some choices. The way I set it up, the tourist gets reparameterized, but that just means that in one of the variables, one of the equilibrium parameters, instead of u1, you write u1 times h. But so it's an isomorphism, so there's no change. There was another one. The boviriative description of a given quiverite includes some additional carrier parameters that were not visible on the quiverivariative side. Do you know how this should be dealt with when a computer stable emerald stops boviriatives? They should, I didn't tell this in the last sentence, but before that I want to say. Do you know how this should be dealt with when a computer stable emerald stops boviriatives? So maybe my first remark is that those Vs are also there for the quiver picture. They are at these top vertices. So there is here, of course there's not enough. So this is V1 over V2, and this is V2 over V3. So at this bottom, the top vertices are decorated by the carrier parameters. They come from the PICAR groups. So there's a line bundle, the determinant bundle here and here and here, and these are the carrier parameters. Okay, the question, how to deal with them? Yeah, I mean, in the definition of elliptic stable envelope, those display a role, and they are just at the right place. And also, it's quite, I don't know how you appreciate it or not, but the 3D mirror symmetry, equivariant parameters and carrier parameters just switched beautifully, so as they should. So that's just for me, just another incarnation of the fact that this is a right way, this is a good way of looking at these varieties. So here the equivariant parameters and carrier parameters is not clear how they switch. I mean, but in the bov picture it's rather nice. In the recent paper, this is a result of you mentioned that there is some super-efficient design. So I only did the GLN part, that it is extended to GLNM with Leprosensky and myself. What happens is that each of these five brains you can define, for some of them you put a star, and then you apply a, in a sophisticated sense, you apply a Lejeune transform there, and there's a different associated space. And in the cohomology of that, not the Yangians of GLN, but the Yangians of GLNM relaxed. In Kuhamal. Can you explain what are the results? No, it's rather sophisticated. So first of all, we don't do it in the GIT way, because first of all, we have to give up holomorphic symplactic structure. It might be twice, but certainly we have to give up. And the omega. So we rephrase the definition to, some Lagrangian intersection definition, and then, this is quite involved for me. And then, before, each of them will be a Lagrangian, generalized Lagrangian variety, and the original would be their intersection in some sense. But then, for, there is a star, first you apply some Lejeune transform, and then you intersect them. So that will be the new variety. So that's actually on the archive already, this paper. Okay, I should put here, many of the things that I learned about physics and everything is from Lejeun's case, so he certainly deserves his name. Any further, yeah? You did pass one to use the combinatories of these diagrams to calculate the stable I will open. In some sense, they define it, but it's just very complicated. So actually, there is a student, Thomas Obotta, who has a great way of doing this in cohomology, for in the quiver settings, and it's some kind of cohomological whole algebra multiplication. I have a paper from 10 years ago, but that's only in the Schubert calculus settings, that if you, you take some cohomological whole algebra that Marcus will define tomorrow, so I'm sorry to take it for granted, and then you take some natural elements one, there are many natural elements called one in it, and if you multiply them in the right way, then it will be a, it will be a cohomology class, and that is the stable envelope, but that is in some Schubert calculus settings, when you take quotient bundles of partial flag right. But Marcus will convince you that this cohomological whole algebra multiplication is non-trivial, it's beautiful, but it's not just multiplication, it's non-concutitive, for example. The computation that you are mentioning, it does after localization, is it some shuffle algebra way? Actually, that's correct, but it's, yeah, so I'm cheating a little bit, so indeed that's a local calculation, right, so it's only for polynomials. So it's a shuffle algebra. But for elliptic, we are trying to set it up here, there are lots of technical difficulties. Any further question? Well, I must say that Thomas Bode does it for elliptic, as well, but for quiver varieties. Maybe then you can point me out, please. Except... OK, let's thank each other again.
 Thank you very much. Thanks to the organizers. I'm very happy to be here and to give a talk.  It's actually my first talk in person after one and a half years as for most of the speakers,  I guess. So my goal in this lecture is to give very down to earth, very approachable  tourist guide type of introduction to these three subjects that are in the title. So pretty  many delicacies I will hide, but I want you to give some, to have some picture of these,  I have some understanding of these after the talk. Let me acknowledge my quarters on different  papers that are related to this story, Rosensky, Varchenko, Smirnov, Zhuh, Weber and Xu. This  picture is the plan of the talk. So first focus on that black arrow on the top left  part, which says 3D mirror symmetry for characteristic classes. So that the explanation  of that double arrow will be the first half of the talk. So I try to explain what it means,  what characteristic classes for some certain singularities in a space are, what they are  and what does it mean that for two different spaces, in this case, in this example, I chose the  Grasmagnian of two planes in C5 and another space, which is a Nakajima variety decorated  by that, or defined by that red quiver. So these two are related in terms of characteristic  classes of singularities. So that is the first part of the talk, explanation of that black  double arrow. And the second half of the talk will be explanation of the rest of the picture.  I want to talk about Cherkis' Bo varieties. It's a, it will turn out a very convenient  rule of spaces, more general than Nakajima quiver varieties, and looking at spaces like  Bo varieties instead of quiver varieties or homogeneous spaces, it sort of explains why,  for example, those two spaces are mirror symmetric and also it will answer other questions. So  this is the plan. Any questions? Then let me jump into, well, let me recall this notion  that was already defined in the excellent talks of Joel Kamnitzer, the quiver variety.  So if you see a picture like that on the left, so we call it a type A quiver. As a side  remark, everything in this talk will be type A, just for simplicity. So this is a type  A quiver, which means just a, you know, a combinatorial object which you see on the left. These W's  and V's are non-negative integers. Associated to that picture, as we learned last week from  Joel, there's a variety which I call script N of Q, the Nakajima quiver variety. So I'm  not repeating, I'm not going to repeat the definition, but here are some examples. The  examples on the left are familiar spaces. They are cottingen, total spaces of cottingen bundles  over partial flag varieties. For example, the Grasmania, you get by just one portion of a quiver  and here is an example of a partial flag variety, or cottingen bundle of the partial flag variety.  Quiver varieties are of course more general than cottingen bundles of partial flag varieties.  Here's an example. So this quiver variety is dimension two and it is the usual resolution  of a Kline and Singularity, C2 over Z mod 3. More interesting than the actual definition,  let's see some properties of quiver varieties. In this case, type A, they are smooth. They  carry some holomorphic, carry a holomorphic symplectic form and there is a tolu section  on it. So let me spend a little time on the tolu section, so at least naming what the  tolu section is. At each of the framed vertices, the framed ones are the little squares, you  imagine a tolu section of that dimension, of the dimension of W and the product of all  these tori act on the quiver variety plus there is one more factor, one more C star,  which comes from the fact that in the definition you considered the cottingen bundle of something  before you cut it down by a group action and the cottingen bundle, in the cottingen direction  you multiply by an extra C star, which we denote by C star H. So in this setting, the  number of tolu fixed point is finite and we have some tautological bundles for each vertex  on top of the quiver. So of rank V1, V2 and V capital N, there's a tautological, there's  a bundle over the space. So this we are rather familiar with since last week at least. So  eventually I want to talk about the cohomology ring, the torus-equivariant cohomology ring  of the Nocajima quiveriates, look at the top left part, so that's what I want to explain  how I will talk about an element of the equivariant cohomology ring. So the way we will name elements  in the equivariant cohomology ring is we will name their images under the localization map.  So consider this map on the top left called LOC lock. So that is just restricting a cohomology  class to the torus-fixed points. It is just the most innocent mapping cohomology, the  restriction map. It turns out that in very general situations, for example here, the  localization map is injective. So if I name the image under the localization map of an  element of the cohomology ring, then I name this. This is an injective map. And why is  it a simple thing? Because we have finitely many fixed points, so the restriction to the  torus-fixed points is the sum of the restriction for the cohomology ring of each fixed point,  and the equivariant cohomology ring of a point is just a polynomial ring. So in this case  U1, UN, H bar. So the picture on the right is of course an example. It is T star GR  grass 1, 2, C4. It has six fixed points. These are the vertices of this graph. So to name  an equivariant cohomology class in the cohomology of this space, I need to name a polynomial  at each vertex. So to name a colloquial class is a tuple. At each vertex I name a polynomial  in the U's and the H. I cannot name any tuple of polynomials. That's what I'm addressing  in the bottom left. The image of the localization map is not the whole thing. There are constraints  among the components. And this is what is explained or indicated by the edges of this  picture. So there are invariant curves in the space, and those are the edges of the  picture. And they come with some decoration. In this case the decorations say 1, 3 on the  leftmost edge. It says that an element at top, okay, so I coordinate at that fixed point  2, 3, and I coordinate at that fixed point 1, 2. They cannot be independent. They have  to satisfy constraints. And the constraint is that if you plug in U1 equals U3, then  they have to be equal. So these kinds of constraints must be true for an image in the localization  map. Here's an example. So you see everything that is in blue is a sixth tuple of polynomials.  And because they satisfy these consistency conditions, they do represent an element in  the equirincology of the Grasmanian. So let's check one constraint. So the bottom right  constraint, the edge decorated by 2 and 4. So that means that the two polynomials associated  to the two vertices, they have to be equal if you plug in U2 equals U4. And indeed you  will see that there's a U4 minus U2 factor on the polynomial, so they are indeed equal.  And you can check all the others. This particular sixth tuple is an element in the equirincology  of the Grasmanian of 2, 4. This particular sixth tuple is actually something that will  be called stable envelope. It's an example of a stable envelope later. This slide is  just a warning that I said that the components are not independent. And I said that for every  edge there is a constraint. Unfortunately, the edges actually are not always discrete.  In this example, which is a Nakajima quiver rarity, the edges come in moduli. There is  a one-parameter family, a pencil of curves you see in the middle, actually at two places  as well. And in those cases, the constraints on the components are deeper than just coincidence  under some linear form. The coincidence have to happen not only for the polynomials, but  for some higher derivatives as well. So I'm not giving the concrete statement. I'm just  saying that the constraints are more involved than just the coincidences.  So now that we have a way of thinking about the equirincology ring of the Grasmanian or  Nakajima quiver rarity, I want to name some special elements in them, which will be named  stable envelopes or cohomological stable envelope classes. There will be one for each Taurus  fixed point. So the notation is sub p. So that's what I want to define. The definition  is not on the slide yet. So it will be on the next slide. So these definitions on this  slide is just preparation for defining an element in the equirincology ring. So the preparation  is the following. I'm going to fix a one parameter Taurus subgroup, which say, for example, this  u maps to u to the first, u to the second, u to the third, and so on. For h bar, I plug  in one. If that is fixed, then we can talk about this so-called bjarjnyski Biroula cells.  For every fixed point, p, so I'm talking about the second pink line, for every fixed point,  t, I can define the leaf itself, the collection of those points that under this one parameter,  this one-dimensional Taurus, they flow into the point p. That's what the definition says.  So the limit of sigma z times x is equal to p. So it's called usually bb cell, but we  call it the leaf, the leaf of that point. Actually, I will come back to the rest of  this slide, but I'm going to jump ahead by one slide and show you an example. So look  at only the left. This is, of course, the moment graph. This is the skeleton of one  of the Necajunic varieties. One of the fixed points is called 1,4. I don't know how much  it's visible, but there is something shaded blue, or it looks like green for me here.  All the points that flow into the fixed point 1,4. That's the leaf of 1,4. I'm going to  go back. So if we have these notions of leaves, we also have a partial order by just taking  that which fixed point is in the closure of another fixed point. So that way you get  a partial order. And if you have a partial order, then we are ready to define the bottom  line on this slide, the slope of a fixed point. You take the leaf of your fixed point, but  also then you look at the points which are in the closure of that leaf, and then take  the leaf of those, and then you look at the points which are in the closure of that leaf,  and then those are fixed points in the leaf of those and so on. So it's not just the closure  of the leaf. These leaves all have the same dimension, the Necajunic varieties. So if  you are familiar with working with homogeneous spaces, think of the leaves as the conormal  bundles of a Schubert cell. And conormal bundle of anything is always same dimensional.  And so here you take the leaf is the conormal bundle of your Schubert cell, but then on  the boundary there are some other Schubert cells, and take the conormal bundles of those  as well, which have the same dimension, and then in iterative you do the same thing. So  that's the slope. And it is illustrated on the right hand side of this picture. So the  slope of 1,4 is the leaf of 1,4 of course, but in the boundary there is 1,5 or 2,4, and  then take the leaves of so-do. So whatever is painted blueish here is the slope of 1,4.  So this is the geometric part of a definition that is needed to define the stable envelope.  This is the Maulik-Okunkov axioms definition of stable envelopes in cohomology. So the  stab, the stable envelope associated to a fixed point P is a unique class that satisfies  three conditions. The first one is the support. This must be supported on the slope of that  fixed point. The second axiom is a normalization that the stable envelope of P restricted to  P itself. It has to be some expected obvious thing, namely the Euler class of the normal  bundle of the slope there. So the slope is not a smooth manifold, but at P it is smooth,  so it has a normal bundle of it. It has a normal bundle in the ambient space, and you  take the equivalent Euler class. And there is a boundary axiom that if you restrict  the step of P to anything else but P, so I'm reading the bottom line, so anything else  but P, then it will be divisible by H. The stable envelope will have a degree half of  the dimension of the space, so all these restrictions are of the same degree. So it's divisible  by H means that if for some person who doesn't see H, just put flux in H equals 1, then this  means that the step restricted to Q is smaller degree than expected. So the fact that you  like to call it a degree axiom, that the step of the number of restricted to anywhere else  but P, it collapses a little bit. So it's a smallness condition, always you think of  it as the restrictions are small. Actually maybe as I remarked now that I mentioned the  H equals 1 substitution. So in special cases for example for G over P, this table number  of were known before and they have a name called the Schwarz-McPherson classes. So in  special cases it recovers some old notion with H equals 1. Okay, so in this picture I'm  going to explain just further elaborate on the axioms of step 1, 4. So again step 1,  4 is a cohomology class, so it has component at every vertex. We know that these components  cannot be arbitrary, they have to satisfy some consistencies. But let me see, beside  these consistencies what other constraints we get from the axioms. For example the one  in yellow, it says that the stable number of restricted to 1, 4, it has to be some  explicit thing and then you see that, I hope the picture is rather intuitive, at 1, 4 there  are two directions which point out of the slope and the product of those, the Euler classes  of those has to be the restriction there. So the yellow is that the restriction has to  be that thing, it's equal to that. We also have the support axiom and part of the support  axiom means that restriction to 1, 2, 1, 3 and 2, 3 must be 0 because this class is  supported on the slope. But more than that, for example the support axiom tells me something  about the restriction at 1, 5 because you see at 1, 5 there is a direction which is  normal to the slope, which is to the northwest direction at 1, 4. Since this is normal to  the slope there and the stable envelope is supported on the slope, the stable envelope  restricted here has to be divided by the weight of that direction, in this case u1 minus u3  plus h. So the support axiom is more than just a bunch of zeros outside, even in the  boundary it requires some divisibilities. So I hope most of these should be clear. And  of course divisibility by h is an explicit thing everywhere but at 1, 4 the restriction  has to be divisible by 1, 4. So if you just look at this as it's presented on this slide,  it looks like very combinatorial saying that there is only one tentapole of polynomials  which satisfy the consistencies which I haven't really told you the derivative constraints.  But anyway, so the consistencies, there is just one tentapole which satisfies the consistencies  together with these axioms. But it's true and that is the stable envelope. Okay, let  me check the time. Okay, I'm going to skip this part I think. Okay, so let's just regroup.  I might come back to geometric representation theory connections but probably not. So far  what we have is that we have a way of thinking about the equivalent conjurings of knackage  and molecular varieties and we define the comological stable envelopes of it associated  to every fixed point. The way of thinking of that, I would like to think of it as that  is the class of this loop. So if there is a big ambient space and you have a sub-variate  then it's tempting to think about that as a cycle and that represents the co-homology  in the ambient space as soon as you have some kind of point-caridurative type of relation  in the ambient space. So there's a cycle. So that is one way of thinking about the class,  the class of this loop, but it's not really that class, it's really some kind of H-bar  deformation of it. So if you take the coefficient of the highest H-power of this class, it has  no H in it anymore, that is like a Schubert class type of object. So this is an H-bar  deformation of like 19th century fundamental class type of calculations. So that's what  we have and okay that's where we are so far. Okay now we are fast forwarding. I'm just  telling you that the stable envelopes have a generalization in K theory and in elliptic  co-homology. And there is no way I'm going to define those, not even the theories and  even less the actual stable envelopes, but I want to give you some feelings about them.  So first of all how do I think about the K theory element on a lecogenic curate or an  elliptic homology element in the lecogenic curate. So the first time says that a stable  envelope or anything as a co-homology element restricted to a point is a polynomial in the  equivalent parameters, right? Everybody agrees, that's what I said. But then the K theory  element is pretty much like that except the restriction is a Laurent polynomial, not  a polynomial. And in the elliptic, an elliptic homology case it's not a Laurent polynomial,  it's an elliptic function, it's a section over a product of elliptic curves. In the  same variable is used. Okay, so that's one thing I want to say is that the way you want  to think about for example an elliptic homology element on lecogenic curate is a tuple of  some elliptic functions. But there's one more thing on this slide is that in the elliptic  line I added more parameters, the V's. And in the next few slides I want to give an intuitive  feeling that in why in elliptic co-homology when you want to talk about characteristic  classes you are forced to have some new parameters. These parameters will be called dynamical  or scalar parameters. I won't be able to give the precise mathematical statement but I hope  I will be able to give some intuition why you are forced to have some new parameters.  Okay, so first this is the look at the theta function. This is a section of a line bundle  over the elliptic curve. The way I want to look at it, this is that it starts with x  to one half minus x to the negative one half which is in logarithmic variables is really  up to a constant, it's sine of x which would be just the k theory part of it. So the theta  function is really just a q-decoration, some q-deformation of the sine function. So you  can think of it as a q-deformation of k theory. And I will use this delta function, delta  A B is just a, you cook up from the theta function another two parameter function delta  A and B. Okay, this is just definitions because then the next slide will be the one which  I hope will give an intuition, we are feeling why in characteristic class theory in elliptic  co-homology you need extra parameters. There is something which is not on the slide so  I just want to say that if you want to define some kind of characteristic classes for the  stable envelopes, there are many others. There are many approaches but one approach is that  you resolve your singular sub-variety, you define some obvious thing in the resolution  and then you push it forward. If you do this, you have to show that what you invented in  the resolution, it was invented in the right way that your class does not depend on the  resolution. So this notion that you define the good class should depend on some identities.  For example, the two nearby resolutions are, you get the same class downstairs. And these  identities are really always boils down to one identity. In elliptic world, it boils  down to the top identities called phi's three second identity. It says that if x1, x2, x3  is equal to y1, y2, y3 and they are both equal to one, then that the product of that,  some of the products of that functions is zero. It's a good exercise for your graduate  students. So that was defined in the earlier slide and then this is a, it turns out that  about elliptic functions, this is the only identity. Everything else follows from it,  although highly non-trivial. Anyway, this is the identity which is behind the fact that  in elliptic characteristic, in elliptic world, you can define actually characteristic classes.  Now let's do the following, that take that top identity and plug in q equals zero. So  let's go to k theory. Then what you, what you get is the middle identity, which is  you see a trigonometric identity. Now you can give it to your, to your calculus students  that if x1 plus x2 plus x3 is equal to zero, y1 plus y2 plus y3 is equal to zero, then  this identity holds. Ignore the purple part for a minute. So this is the q equals zero  specialization of the top line. And you can further approximate sine of x with x, which  you know, which we always do, then you get cohomology. And then the identity that you  get is the bottom line. Okay, now look at the purple, purple decorations. Is that the  identities for a, in k theory and in the rational limits, they are, they are easier  because they, they tell you more. They say that if the x is at the top to zero, then  the left hand side is equal to one. And if the y is at the top to zero, then the right  hand side is equal to one. In some, so you see that the, the identity splits to x variables  and y variables. So because of that in k theory and in cohomology, what we did in, in, you  know, in the past 200 years of mathematicians, they were not forced to work with the other  set of variables because these identities, they, they, the governing identity behind characteristic  classes, they, it separates to x and y variables. So you can just take the left side of this  slide and then, and then build up characteristic classes in cohomology and k theory. However,  in elliptic cohomology, the top identity doesn't, doesn't split to an x part and the y part.  You are forced to, to work with y. Okay, so this is actually something that elliptic  cohomology thought that we should do cohomology and trigonometry as well with two sets of  variables with the Kirchhoff parameters, but we need to go out of our ways to, to, to  introduce them. Okay, so I, okay, this was the explanation of the bottom right corner  of this space that stable envelopes are defined and, and, but they depend on new variables,  which I call v's. Any questions? All right, then, then we, we come to this fact that,  that, which I call 3D mirror symmetry for characteristic classes. It turns out that there are, there are pairs  of Naka-Jima-Kivir varieties. Let's call them x, x-tree, that the pairing together comes  with some, I, a fixed bijection between the totals, fixed points for which the, the stable  envelopes on one are equal to the stable envelopes on the other one in the sense which is on  the slide, that you take the stable envelope of p restricted to q, p and q are fixed points  on one, and you take the stable envelope of q restricted to p on the other one, then you  have a polynomials or Laurent polynomials or elliptic functions, and they claim that  they will be equal if you switch equivariant and, and Kirchhoff parameters as well as invert  h bar. So, so this is the, the 3D mirror duality for elliptic, for elliptic characteristic  classes. Here's an example. So everything above the purple line is about one Naka-Jima-Kivir  variety, that you see the, the cottingen bundle of p2. Sorry, yes.  We have a question. Yes. How the elliptic stub transforms with respect to the modularity  transformation? Yeah, I, there is something and I never looked at it, so I won't be able  to say. Of course, you should, you should restrict it to a point so that you really have an elliptic  function, but yeah. Oh, what should I, okay, yeah, maybe I shouldn't. Okay, so let me continue.  So over the, over the purple line, it's all about the quiverity of cottingen bundle of  p2. So you see that it has 3 fixed points, there is the skeleton of it is on the left  side of the slide, it has 3 fixed points and there are, those are the constraints among  the restrictions. And the table on the top is the elliptic stable envelopes. In, in the  following sense that you take, take the rows of that table. So the first, the first row  is the stable envelope of F1 and that means that that's the one of restricted to F1 is  that product of data functions, restricted to F2 is 0, restricted to F3 is 0. And the  middle line is the table envelope of F2 and so on. Now the, the 3D, 3D mirror dual of  that, the quiverity is, is, is this one in the, the, the one below the purple line and  that has a totally different looking moment graph. But I think that's what I wanted to  convince you with that as soon as you see the moment graph, you can write down the stable  envelopes. At least in cohomology it's easy, in K theory it's much less easy and in elliptic  it's, it's a lot of work but, but for small ones you can do it. So as soon as you see  that graph you can calculate the stable envelopes you will have the, this bottom table. Again  the rows are the stable envelopes. And the fact is that if you stare at these two tables  that they are the same after transposing and switching U and V variables and inverting  gauge. Okay. So this is a, this is the baby example of 3D mirror symmetry for stable envelopes.  Here are some, some other random examples. With green on the two sides I'm, I'm indicating  the dimensions of these varieties. Yes, you see the different dimensional varieties have,  have this amazing coincidence that characteristic classes of singularities in those varieties  are equal in this sophisticated sense. Okay, the bottom line, the bottom line is that the  left hand side is of course a macogymn variety and I claim that there is no macogymn variety  which, which is 3D mirror dual to it. But we will fix that later. Because now I'm starting  the part two of the lecture. So maybe it's a good point to ask questions if you have.  Yes.  Do you expect the stable envelope for communism in capitalism?  Okay. I, I guess yes. But it, it, you know, having a group versus a formal group law has  some advantages. An algebraic group. So these three varieties that, three, co-homology theories  that I named, co-homology, k-theorem, elliptic homology. These are the, the, the, the, the  theories that corresponds to one-dimensional algebraic groups which is, which are parts  of the, the formal group laws. So there are, there are, there are advantages of working  with formulas. So indeed maybe there is a formula for, for the, the most general co-homology  theory. But I just, I just want, I'm a formula person. So, so I, I certainly want to look  at these three.  Do we have a question in what sense are all theta function identities derivable from the  tricycant identity?  Okay. So that is a sophisticated sense and I won't be able to, I can find the reference  paper which I looked at and, and I don't remember the details, but I just remember the, the  intuitive statement. Yeah.  So just at the very beginning we fixed this homomorphism from C star to T. So is that important?  Yeah. So in, in the slides which I skipped, it is important. So if you, if you start playing  with that, what one parameter subgroup you choose, then you recover representation theory.  So after a while I will, I might comment on those.  The stable envelopes of course depend on that.  The stable envelopes do depend on it. It's not infinite, it's just, they depend on some,  some chambers of choices. So there will be finitely many.  And changing them you recover young, young matrices and so on and so forth. So that's  where representation theory starts to be built up.  Okay. So from now on I want, I want to, to define, so, so, so give a feeling about what  bovarietes are. And actually I want to advertise them. I think we should look at bovarietes  instead of queries. They have some advantages.  Okay. So these will be associated to some combinatorial pictures, combinatorial data.  The combinatorial data will be called the brain diagram. Here's a brain diagram. So  the brain diagram combinatorial is just a collection of horizontal segments called D3  brains. They, they come with some non-negative integers, the dimension vector. But then the  consecutive ones are separated by either NS5 brains or D5 brains. So I drew a, a blue  or a red skew line. And so that you don't have to memorize this. It's on, on this board  out here just because this picture will go away after a while.  For future purposes I will also decorate the D5 brains with equivalent variables U sub  I and the NS5 brains with the scalar parameters V sub I. So this is a combinatorial object  for us. We can discuss some, some, of course some super-syntheory after. Okay. Okay. So  what is the, what is the Cherkis-Bau variety? Start doing the same thing as you would do  for quiver varieties, but only do them for NS5 brains. So look at the left side of the  picture. If you see an NS5 brain, it's a red brain. Then you just do the same thing as  you would do for a quiver variety. Take HOMC and to CM, where, where you know N and M are  the numbers, the, the decorations on the two sides. Take the cot engine bundle and, and  that, that of course has an action of GL and cross GLM. So you do the, almost the same thing  for the other type of brain, five brains, but it will be a different space, not just  the cot engine bundle of HOMC and CM. That is the left hand side I call the, the arrow,  the arrow edge or, okay, arrow brain and then this is the, the bow brain. Actually this  whole thing should be called a quiver, which you can put an arrow and, and both an arrow  and the, a bow into a quiver. All right. So, so for, for the other type of brains, you  put some of the, I, I indicated roughly what that space is, but of course a lot of things  are skipped in this, what acts how, but it's also just a, you know, a Hamiltonian reduction  or GIT quotient of, of, of, of rather obvious spaces. The, the, the, the, the, the, the,  the key difference is that on this other space, an extra group acts is C star, that's in the  bottom right of the slide, that there, there is a C star acts. Because what do you, what  do you do after that when you, you want to build, build up the necrogenous variety is  that you take the, you take the product of all these, all these T HOMS and, and B N M's  and then you do reduction by GLM, cross GLM, you do the reduction by all the GL's. You  do that, but these C stars will, will survive. So on the space that you get, there will  be a C star action for every D5 brain. You see this, this is realized that every D5 brain  will be decorated by an equivalent variable. So if you do this story, then, then what will  this, the Cherkis variety C, script C of D will be the name of the Cherkis Boo variety.  It will be smooth. It will have a holomorphic symplatic structure on it. It comes with tautological  bundles coming from the D3 brains. So those numbers are the ranks of the bundles. It,  later I will show you that it has finitely many, finitely many fixed points and it has  a toro section and the toro section comes from the D3 brains plus there is a C star  action from the fact that there were lots of T stars in the earlier slide. So everything  that we liked about, like about necrogenous variety is sort of true for this one. Everything  is very combinatorial and there is an advantage which will come very soon I hope.  You should have asked a question. So what is the framing that is usually, it's replaced  by this.  The D5 brains, correct.  The D5 brains has come, okay.  Some of the, they collapse together to be the framing.  Okay.  Yep. Yep. So there is a dimension formula. Of course you don't have to memorize. You  just imagine that if you see the brain diagram with those numbers, the dimension vector,  then from that you calculate the sum and that is the dimension of the, of the Cherkis-Bovarietzi.  For example, if you see this, this brain diagram that it's on the left bottom of the slide,  then you plug in the numbers and you will get four. It's not a surprise. This will be  T star P2, the Cherkis-Bovarietzi associated with this brain diagram. So you might say  that things are getting more complicated because the quiver name of T star P2 was just one  dot and one square and now it's somewhat longer. But there will be things that we've  been at the end.  Okay. So this dimension formula. Oh yeah. And now I think I'm going to answer your question  now.  How are quiver related special cases? So what I need to give you now is a combinatorial  recipe that if you see a quiver, how do you build up a brain diagram? So the quiver has  parts, this K-n part, look at the top left part and whenever you see this K-n part, just  just through a segment ending with NS5 brains, the red ones, and put n blue brains in between,  where n is the framing of the, or the dimension vector of the framing and decorate the D3  brains with Ks. So this is just, and then glue together these segments. For example,  look at the bottom example. And whatever is decorated by yellow, so that part will be  just, just go to the brain, the brain diagram on the right, which is decorated, which is  decorated, you know, shaded with yellow as well. But it, the quiver has another part  and just glue that part to the right of it. So you glue these segments together and you  would have a brain diagram. Yes. And this works only for Taipei quiver, Nagajima quiver  right here or for any? Okay. I only know Taipei. But what about loops? Yeah. Yeah, okay. I  think the next slide. Okay. Okay, so I won't be able to do, you know, all kinds of loops,  but one loop is fine. So A tilde. Yeah, so yeah, right. But otherwise, this is right.  But make, make an observation here that the brain diagrams that we get on the right, they  are special. They, they have this cobalanced condition, which is in the bottom, very bottom  line of the slide, is that on the two sides of a D5, D5 brain, you will always have the  same numbers. Okay. But here's the advantage that we do not have for quiver varieties.  There is an extra, actually there will be two extra operations. This operation is 3D  mirror symmetry. This is just the most innocent symmetry operation on these brain diagrams.  Just reflect it down, reflect it by the horizontal axis. Or in other words, change red to blue  and blue to red and so on. So let's call it 3D mirror symmetry for both varieties. Let's  see an example. Let's find the 3D mirror dual of T star P2. So the top line is the, the  brain name of T star P2. And we just formally create the 3D mirror dual of it. Okay. I can  calculate this dimension, this dimension too. But then here we go a little depressed because  this is not cobalanced. So I cannot recover it as a, as a knuckajumma quiver variety.  However, I will be in a minute. I will be able in a minute. So just wait. Because I'm  going to show another operation which exists on Boo varieties. And it's called Hanani  Witten transition. Think of it as like a Reidermeister move. So you can locally rebuild your  brain diagram without changing the, the space. So the, the, the rebuilding is such that you  can, if you have a consecutive D5 and NS5 brain, you can switch them. The price you  pay is that, that the, the dimension vector in the middle changes and it changes the way  which is on the, on the right. And the theorem is that if you carry out such a change on  your combinatorial model, then the associated Boo variety doesn't change its, its same.  Actually the Taurus, Taurus parametrization, the Taurus action reparameterizes a little  bit. But, but yeah, for the purpose of this talk, it's just an isomorphism. Okay. Then  let's continue this example that we saw a, a few slides up. That the first two lines  are just, we found the, the, the 3D mirror dual of, of T star P2. But now I'm going to  play the game of carrying out Hanani Witten transitions. For example, first I carry it  out at the yellow part. You see it. And then, then, then I hope you can just carry out this  decision. And then, then after that I carry it out at the green part. I hope it's visible.  And I will get to the brain diagram that I'm, I'm pointing at, or in the middle of the  right column of the slide. And this one is cobalanced. Okay. So I was lucky enough to  be able to carry out Hanani Witten transitions to make my brain diagram cobalanced. And if  it's cobalanced, then I can recover it as a, as a Nakajima variety. So then we recover  this thing, this example, the baby example of 3D mirror symmetry, or between two Nakajima  quiver varieties. Any questions? Oh, okay. So I want to give you an FNA type. So, so  look at the, on the left of the picture, there are quiver varieties. On the right, we have  the same varieties, but in their bone names. And the transition between them, everything  follows from earlier slides. If you see a quiver, then there is a way of drawing a brain diagram.  The 3D mirror dual is just switching D5 and NS5 brains. And then in this one, it's such  a simple thing, it's, it's accidentally already cobalanced. So I, I can, I can rewrite it  as a quiver variety. This is of course a very well known example, right? Of Hilbert's  Schemes and his dual. But you can play the game with more, more complicated type A or  FI type, type A quiver varieties. Okay, any question? Yes.  Is this the correspondence in Unicom?  Unicom, so does it have an automorphism? Certainly this is the natural one. So, so the, oh, so  your question is that are there quiver varieties which are isomorphic to each other? So different  combinatorial codes isomorphic to each other. Yeah, that, that I doubt. Yeah, I doubt. Oh,  yeah, yeah, I'm sorry. I can, of course, I can, I can present the one point space many  different ways. So, so, so maybe I have to be, we'll back on that. I'm not sure. Okay,  so in, in, so now I think I explained everything which is on this slide. This is just a repetition  of the first slide that to find the mirror dual of, of Grassman 2C5, you just have to  write its Boo picture. Then formally take its 3D mirror and then carry out an animated  and moves if you can. And if you're lucky, in this case you are. And then you will get  the, you will get a quiver variety. Right. What I want to talk about is some other very  important structure that, that quiver, bovarietes come with. One of them is brain charge. So  it will be a number associated, an integer associated to every five brain. So an NS5  brain, it is, it is just the difference of the two numbers on the two sides of the NS5  brain, L minus K, plus the number of D5 brains on the left of it. So here you see that this  is not just type A, not F find type A. In F find type A there is something local charge,  but anyway, so this is, let it be just finite type A. And for D5 brain a very similar integer  associated. So K minus L plus a number of different type of five brains to the right  of it. Here's an example. Is everything visible? Not much. Take the left most NS5 brain. His  brain charge is 2 minus 0, because these are the numbers on the two sides. Plus, plus  the number of D5 brains to the left of it is nothing. So that is the top red 2 on, on  this, on this diagram. So for some reason I will collect these charges on, on the top  and on, on to the left of an empty table. Okay. For, for the time being this is just  a decoration. So I collect the charges of NS5 brains left of this empty matrix, the  charges of D5 brains on top of the empty matrix. It's actually a, a, a, an easy theorem that  the two charge vectors, the red and the blue charge vectors is a complete invariant of  the Hanani-Britain class of, of, of the brain diagrams. Hanani-Britain class is the isomorphism.  Is that you, if you, you switch to consecutive different type of brains, you have a different  diagram. But the brain, the charges will not change. Yeah. And vice versa. So, so if  I, if I didn't want to define your Bouvaritis, but Bouvaritis up to Hanani-Britain, then  I would have just told you that they are associated to a, a pair of vectors. This pair of vectors  have this one extra perfect, the sum of the red numbers is the same as the sum of the  blue numbers. Okay. So again, so up to Hanani-Britain transitions. Bouvaritis are, are, when you,  next to this empty matrix, you put arbitrary numbers on top and on the left. So the, the  sums add up together. And among these, the ones which are, at least in one of the representatives  is a quiver variety. These are the ones which were in the top, the top vector is a, is a  partition. Those numbers are weakly decreasing. And if you put numbers there fully one, then  these are just the, the topics of Schubert calculus. So why, why do we care? Because  actually if you are, if you're coming from representation theory, then you don't, then,  then you think that you can, okay, so I didn't say enough to support this, but it's fact  that in geometric representation theory, you, you, you permit yourself, you allow yourself  to permute those numbers on top. But not here. So the representation theory is the same,  but the underlying space is different. So if you want to look at the space, then, then  the both varieties are more general than, than quiver varieties. And why, why do we like  that both vectors are arbitrary? Because it comes from one extra operation, trans, transpose,  which is essentially three-dimensional duality. Which, so this was not complete for quiver  varieties, as you can see. Oh, okay, so, okay, since I keep mentioning geometric representation  theory, I might say that the following, that, so if you see these two vectors, then the  height of this, this matrix is a number, say, n. And then you say, take the young end of  g l n. That's, that's a quantum group. Then those numbers on top, read them as, as, as,  if the numbers are a, b, then take lambda a of the, of the vector representation, tens  or lambda b of the tens of representation and so on. So you take the, the fundamental  representations and multiply them together. That's a representation of a, of a quantum  group. And now you read these numbers on the left and take that weight space of that representation.  And it turns out that that weight space is, is very naturally identified with the cohomology  of the associated Boole variety. Okay. So again, this is the, why, this is why these  spaces are important in geometric representation theory. On the accurate conjoinings of these  varieties, quantum groups act. So one Boole variety is one, one weight space of such a  quantum group. And this is true in, in k theory and elliptic.  Is this the analog of this Moldik and Kuhnkov Young end?  Yeah, yeah, yeah, this is exactly. So if this was, if these numbers on the top were just  partitioned, this is the Moldik and Kuhnkov Young end. Yeah. And now, okay, okay, that was just,  okay. I want to show you the, the beautiful combinatorics of fixed, torus fixed points. So  if you ever looked at the combinatorics of torus fixed points on the Kuhnkov varieties,  it was a couple of partitions and, and was somewhat messy. It is a different picture here,  of course, equivalent and of course more general because for Boole varieties. And I find it  fascinating. So the claim is that fixed points are in bijection with tie diagrams. The tie diagram  is on the picture. It consists of ties. A tie must connect a blue brain with a red brain.  And you know, my picture of drawing them in these skew lines, it tells you, yeah, so it's a natural  way of connecting them. And, and of course each D, not of course, but it's true that every D3  brain has to be covered by ties as many times as this, as its multiplicity. So just imagine that  you only see the brain diagram and the numbers and it's your task to put ties there. There are  lots of choices. For this particular brain diagram there are 123 brain diagrams. This is one of them.  So this associated Boole variety has 123 fixed points. Here are the fixed points of, of grass  mints you see four. And you see there's a natural bijection between four choose two, just like in  any other names. This, this tie diagrams beautifully transform under Hananividen transition,  which looks like a Reidemesser III move. And they beautifully transform with 3D mirror symmetry,  just take the, the, you know, the reflection of the image. So that means that not just that we  have a bijection between Hananividen equivalent Boole varieties, which should be because they are  isomorphic, but there is a bijection, natural bijection between, between 3D mirror Boole varieties,  the bottom line. That was needed for the 3D mirror symmetry statement. Let me also mention this  other combinatorial gadget that this, this matrix used to be empty, but now put, put zeros and try  to put zeros and ones there, so that the row sums are the red numbers and the column sums are the,  are the green numbers. If you are, if you manage to put zeros and ones in such a way,  then you call it a binary contingency tables, the binary contingency table. You know these  contingency tables come from statistics, but here it's only zeros and ones are permitted. It's also  a, a, these theorem that fixed points are in bijection with binary contingency tables. Maybe,  maybe I want to draw a picture is that, that if you come from Schubert calculus, then, then you  learn to work with full, for example, the full flag variety and everything about the full flag  varieties is, is parameterized by, by, by permutations. This is a permutation. In this language,  what I'm saying is that quiver varieties, in quiver, in the, in the geometric quiver varieties,  everything is parameterized by, oh no, first, first what I want to say is that a partial flag  variety would be parameterized by an order set of subsets, which can be identified with these,  these kind of bipartite graphs where the degree doesn't have to be one on the left. So it's  like a permutation, but I permit coincidences on the left. This is a partial flag variety,  everything Schubert's, all those fixed points, everything is parameterized by these things.  And with both varieties, all we are doing is that we are permitting higher, higher degrees on the  right as well. So bipartite graphs, which are the same thing essentially as binary contingency  tables, are the, are the objects that, that parameterize all the, you know, the cells or  torus fixed points or, or stable envelopes. So there will be a stable envelope for each of them.  So on the right, on the right side, there are, there are the BCT codes of torus fixed points  on graph 2C4. Okay, so I think I, at the beginning of this talk, I wanted to convince you that,  that, that to find characteristics from a space, you have to walk through the following path.  From the space, first you find the torus fixed points. You also find the, the invariant curves,  which I skipped. There's combinatorics behind that as well. But then you have the moment graph.  And from the moment graph, the axioms give you the, the, the stable envelopes. So I gave you  half of the story. I gave you the combinatorics of the, of the torus fixed points. Oops. Anyway,  so, so, so if you see what's on the top of this, of this slide, then you can create the  torus fixed points. You will have the vertices on the thing on the left. Yeah. And you do some  more combinatorics. You will find the edges. You will find what's on there and, and below the  decorations. And then the, the stable envelopes are defined. From this one, you can calculate.  Okay. The same, same for this other one. I might, I might mention at this point that, of course,  that's how you define stable envelopes, but they are hopeless to calculate using the definition.  So, so you don't really do that. So as we are understanding better and better, so there is  some kind of co-homological whole algebra type of structure among stable envelopes. You want to,  to, to, to, to, to, to, to, by convolution multiply two of them together third one. And there are  some, some formulas as well in certain special cases. Okay. So then, then I'm just walking  through something that I already showed you is that if you start with those two diagrams,  then you play the game, you will get the torus fixed point, you will get the invariant curves,  you will have these two graphs. And then you can calculate the stable envelopes and you  recover this slide that you already see. You already saw. So all I want to say is that  everything follows from just the brain diagrams, nothing else. Okay. So this statement that  the stable envelopes match for, for a 3D mirror dual-bubu varieties is proved in certain special  cases. For the Grasmanian and its dual, it's proved by the paper with what, it's a Smirnov,  Varchenko's, for the full flag variety in type A being self-dual is, is, is proved in a different  paper by the same authors and in general type with, with, on J Weber. The hyperb toric being  3D mirror dual in terms of stable envelopes with this dual is proved by Smirnov and Juh. And,  and we calculated finitely many other cases. Maybe I should emphasize many. So it's, it's,  it's quite a well-established conjecture that it should happen. Of course, everybody think it  should happen. This should, I mean, you might remember the, the Higgs branch, Coulomb branch  interpretation of this in, in, in the Commits are stuck. The, they are, oh, that's, okay. So may,  It may not be that this is, I don't know.  Yeah, I don't know.  So I'm not sure that that is a special case  of what I'm presenting.  Certainly the middle of the general G over L  and when it's longer on the word,  it's not a quiviriety or booviriety of type A.  So that's not a special case of what I'm saying,  that in these are the cases for which the 3D mirror symmetry  for characteristic classes is established or proved.  So here is a summary and actually I'm on time.  The summary of what we learned is that  if there are certain nice spaces, for example,  varieties of the total section,  there is a characteristic classes.  They have relations to the numerative geometry.  They are relations to representation theory of quantum groups.  Okay, they are related to some very important  Q difference equations and actually two sets  of Q difference equations.  So they are there, yeah, I advise everybody to study them.  They are very important notions.  We also learned that they come in pairs,  such that this table, I mean, the space is coming pairs  and there is such that this table envelopes  on the two are related and the natural pool  of detecting it is booviriatives,  which is close for 3D mirror symmetry  and easy combinatorics govern their 3D mirror symmetry.  The end.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Any questions?  What's the relation, if any, between the boviriatives  and the construction of cooling branches we saw last week?  So that's Nakajima Takayama paper that they prove it.  That these boviriatives are the cooling branches.  Taipei, probably more general, but I understand the Taipei.  There was a question in the Q&A,  how the stub changes under the,  and any width and the mover.  Is this clear?  Yeah, so that's an isomorphism, so it shouldn't change.  But the way I set it up, you have to make some choices.  The way I set it up, the tourist gets reparameterized,  but that just means that in one of the variables,  one of the equilibrium parameters,  instead of u1, you write u1 times h.  But so it's an isomorphism, so there's no change.  There was another one.  The boviriative description of a given quiverite  includes some additional carrier parameters  that were not visible on the quiverivariative side.  Do you know how this should be dealt  with when a computer stable emerald stops boviriatives?  They should, I didn't tell this in the last sentence,  but before that I want to say.  Do you know how this should be dealt with  when a computer stable emerald stops boviriatives?  So maybe my first remark is that those Vs are also there  for the quiver picture.  They are at these top vertices.  So there is here, of course there's not enough.  So this is V1 over V2, and this is V2 over V3.  So at this bottom, the top vertices  are decorated by the carrier parameters.  They come from the PICAR groups.  So there's a line bundle, the determinant bundle  here and here and here, and these are the carrier parameters.  Okay, the question, how to deal with them?  Yeah, I mean, in the definition of elliptic stable envelope,  those display a role, and they are just at the right place.  And also, it's quite, I don't know how you appreciate it  or not, but the 3D mirror symmetry,  equivariant parameters and carrier parameters  just switched beautifully, so as they should.  So that's just for me, just another incarnation  of the fact that this is a right way,  this is a good way of looking at these varieties.  So here the equivariant parameters and carrier parameters  is not clear how they switch.  I mean, but in the bov picture it's rather nice.  In the recent paper, this is a result of you  mentioned that there is some super-efficient design.  So I only did the GLN part,  that it is extended to GLNM with Leprosensky and myself.  What happens is that each of these five brains  you can define, for some of them you put a star,  and then you apply a, in a sophisticated sense,  you apply a Lejeune transform there,  and there's a different associated space.  And in the cohomology of that,  not the Yangians of GLN, but the Yangians of GLNM relaxed.  In Kuhamal.  Can you explain what are the results?  No, it's rather sophisticated.  So first of all, we don't do it in the GIT way,  because first of all, we have to give up  holomorphic symplactic structure.  It might be twice, but certainly we have to give up.  And the omega.  So we rephrase the definition to,  some Lagrangian intersection definition,  and then, this is quite involved for me.  And then, before, each of them will be a Lagrangian,  generalized Lagrangian variety,  and the original would be their intersection in some sense.  But then, for, there is a star,  first you apply some Lejeune transform,  and then you intersect them.  So that will be the new variety.  So that's actually on the archive already, this paper.  Okay, I should put here, many of the things  that I learned about physics and everything  is from Lejeun's case,  so he certainly deserves his name.  Any further, yeah?  You did pass one to use the combinatories  of these diagrams to calculate the stable I will open.  In some sense, they define it,  but it's just very complicated.  So actually, there is a student,  Thomas Obotta, who has a great way  of doing this in cohomology,  for in the quiver settings,  and it's some kind of  cohomological whole algebra multiplication.  I have a paper from 10 years ago,  but that's only in the Schubert calculus settings,  that if you, you take some cohomological whole algebra  that Marcus will define tomorrow,  so I'm sorry to take it for granted,  and then you take some natural elements one,  there are many natural elements called one in it,  and if you multiply them in the right way,  then it will be a,  it will be a cohomology class,  and that is the stable envelope,  but that is in some Schubert calculus settings,  when you take quotient bundles of partial flag right.  But Marcus will convince you  that this cohomological whole algebra multiplication  is non-trivial, it's beautiful,  but it's not just multiplication,  it's non-concutitive, for example.  The computation that you are mentioning,  it does after localization,  is it some shuffle algebra way?  Actually, that's correct, but it's, yeah,  so I'm cheating a little bit,  so indeed that's a local calculation, right,  so it's only for polynomials.  So it's a shuffle algebra.  But for elliptic, we are trying to set it up here,  there are lots of technical difficulties.  Any further question?  Well, I must say that Thomas Bode does it for elliptic,  as well, but for quiver varieties.  Maybe then you can point me out, please.  Except...  OK, let's thank each other again.
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