 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 at the, 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, 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 give a acknowledge my quarters on different papers that are related to this story. Rosenski, Warchenko, Smirnov, Zhu, Weber and Xu. This, this picture is the plan of the talk. So first focus on that, 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, what characteristic classes for, for some certain singularities in, in a space are, what, what they are and what does it mean that for two different spaces, in this case, in this example, I chose the Grasmanian of two planes in C5 and another space, which is a Nakajima variety decorated by that or, or defined by that red quiver. So these two are, 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, the second half of the talk will be explanation of the rest of the, the picture. I want to talk about Cherkis bow varieties. It's a, it will turn out a very convenient pool of spaces, more general than Nakajima quiver varieties and, and looking at spaces like bow varieties instead of, instead of quiver varieties or, or homogenous spaces. It sort of explains why, for example, those two spaces are mirror symmetric and, 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, 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 you, as we learned last week from, from, from Joel, there's a variety, which I call script N of, 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 cotangent, total spaces of cotangent bundles over partial flag varieties. For example, the Grasmania, you get by just one portion of a quiver and, and here is an example of a partial flag variety or cotangent bundle of the partial flag variety. Quiver varieties are, of course, more general than cotangent bundles of partial flag varieties. Here's an example. So to this quiver variety is, is dimension two, and it is the usual resolution of a Clinian singularity, C2 over Z mod three. More interesting than the, than the actual definition, let's see some properties of, 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 is. At each of the framed vertices, the framed ones are the little squares, you imagine a tolu of that dimension, of the dimension of W. And the product of all these tori act on the, on the quiver variety, plus there is one more factor, one more C star, which, which comes from the fact that, that in the definition, you considered the cotangent bundle of something before you, before you cut it down by a, by a group action and the cotangent bundle in the cotangent direction, you multiply by an extra C star, which we denote by C star H. So in this, in this setting, the number of fixed points, tolu's fixed point is finite, and, and we have some tautological bundles for each, each vertex on top of the quiver, so of, of rank V1, V2 and V capital N, there's a tautological, there's a bundle over the, the space. So this we are rather familiar with since last week, at least. So eventually I want to talk about the cohomology ring, equilibrium, tolu's equilibrium cohomology ring of the, of the naka-jima quiver varieties to look at the top left part. So that's what I want to explain, how I will talk about an element of the equilibrium cohomology ring. So the way we will, we will name elements in the equilibrium cohomology ring is we will name their images under the localization map. So consider, consider this map on the top left, called LOC. So that is just restricting a cohomology class to the tolu's fixed points. So 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, so if, right, so if I name the, the image under the localization map of an element of the cohomology ring, then I named it, this is an injective map. And why is it a simple thing? Because, because we have finitely many fixed points, so the restriction to the tolu's fixed points is, is the sum of the restriction for the cohomology ring of each fixed point. And the, the equilibrium cohomology ring of a point is just a polynomial ring. So in this case, U1, UM, H bar. So the, so the picture on the right is of course an example is T star GR grass 1, 2, C4. It has six fixed points. These are the vertices of this graph. So to name an, an equilibrium cohomology class on, on in the cohomology of this, this space, I need to name a polynomial at each vertex. So to name a cohomology 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, is not the whole thing. There are constraints among the components. And this, this is what is explained or indicated by the, by the edges of this, of this picture. So for, there are invariant curves in the space and those are the edges of the, the picture. And they, they come with some decoration. In this case, the decorations say 1, 3 on the leftmost edge. Says that an element of a tuple, okay. So I coordinate at that fixed point 2, 3 and the coordinate at that fixed point 1, 2. They, they are, 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 this kind of constraints must be true for, for a, for a, for an image in the, in the, under the localization map. Here's an example. So you see everything that is in blue is a, is a six tuple of polynomials. And because they satisfy these consistency conditions, they do represent an element in the, in the equirincology of the Grusmanian. So let's check one constraint. So maybe the bottom right constraint, the edge decorated by 2 and 4. So that means, that means that the two, two polynomials written at the, associated to the, to the 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 another polynomial. So they are indeed equal. And you can check all the others. This particular six tuple is an element in the equirincology of the Grusmanian of 2, 4. This, this particular six 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 discreet. In this example, which is a Nakajima quiver rarity, the edges come in, in, in moduli. There is a, there's a one parameter family, a pencil of, of, of curves you see in the middle actually at two places as well. And in those cases, the, the constraints on the, on the components are deeper than just coincidence under putting, under some linear form. They also have to have, 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, that, that the constraints are more, more involved than just the coincidences. Okay. So now that we have a way of thinking about the equirincology ring of the, of the Grusmanian or, or, or a, or a Nakajima quiver rarity, I want to name some special elements in them, which will be named stable envelopes or cohomological stable envelope classes. They will, there will be one for each TORUS fixed point. So the notation is tab sub p. So that's what I want to define. The definition is not on the, 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. So I want to fix a, a one-parameter TORUS 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, if that is fixed, then we can talk about this so-called bjarjnyski birula cells. For every fixed point P, so I'm talking about now 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 TORUS, they flow into the point P, right? That's what the definition says. So the limit of sigma z times x is equal to P. So it's called the usual 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. And if you, and so this is of course the moment graph. There's the skeleton of the, of one of the nekojunic varieties. And the one of the fixed point is called 1,4. And I don't know how much it's visible, but there is a, there is, there's something shaded blue or it looks like green for me here. All the points that flow into the one, the fixed point 1,4. So that's the, that's the leaf of 1,4. Okay, so I'm going 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 each other, which fixed point is in the closure of a leaf 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. And you take the leaf of your fixed point, but also then you, 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, it's, these leaves all have the same dimension, the nekojunic varieties. So, so if you are familiar with working with, with, homogenous spaces, think of the leaves as the conormal bundles of a, 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, the conormal bundles of those as well, which have the same dimension and then, and iteratively you do the same thing. So that's the slope. And it is illustrated on the, on the right-hand side of this picture. So the, 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, so whatever is, is this painted bluish here is the, is a slope of 1,4. Okay. So these, this is the geometric part of, of a definition that is needed to define the stable envelope. This is the Maulik Okunkov axi, axioms definition of, of stable envelopes in cohomology. So, so this, this tab, 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, the second axiom is a normalization that the stable envelope of P restricted to P itself it has, it has to be some, some, 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, it is smooth, so it has a normal bundle of it. It has a normal bundle in the, in the ambient space and you take the equivalent Euler class. And there is a boundary axiom that if you restrict the, the step of P to anything else but P, so I'm reading the bottom line, so anything else but P and it will be divisible by H. So the stable envelope will have, will, will have a degree half of the dimension of the space. So, so all these restrictions are of the same degree. So it's divisible by H means that if for some person who, who doesn't see H just put flux in H equals one then, then this means that the, the step restricted to Q is smaller degree than expected. So the, actually we like to call it the degree axiom that the stable envelope restricted to anywhere else but P it collapses a little bit. So it's a smallness condition. Always you think of it as it's, the restrictions are small. Actually maybe as I remarked now that I mentioned the H equals one substitution. So in special cases for example for G over P this stable envelope were known before and they were, they have a name called the Churn-Schwarz-McPherson classes. So, so, so in special cases it recovers some, some, some old notion with H equals one. Okay. So in this picture I'm going to explain just further elaborate on the, on the axioms of step one four. So again, step one four is a cohomology class. So it has component at every vertex. We know that these, these components cannot be arbitrary. They have to be, they have to satisfy some consistences. But let me see beside these consistences what other constraints we, we come, we get from the axioms. For example, the one in yellow it says that the stable envelope restricted to, stable envelope of one four restricted to one four it has to be some explicit thing. And then you see that, I don't know, I hope the picture is rather intuitive. At one four there are two directions which point out of the, the slope. And the product of those, the Euler classes of those has to be the restriction there. So the yellow, yellow is that it, the restriction has to be that thing. It's equal to that. You also have the, the support axiom. And part of the support axiom means that restriction to one two, one three and two three must be zero. Because this class is, is supported on the slope. But more than that, for example, the support axiom tells me something about the restriction at one five. Because you see at one five there is a, there is a direction which is normal to the slope, which is, to the, to the northwest direction at one four. Since this is normal to the slope there, and, and the, and the stable envelope is supported on the slope, the stable envelope restricted here has to be divided by the, by the weight of that direction. In this case u one minus u three plus h. So the support axiom is more than just a bunch of zeros outside. Even in the boundary it, it requires some divisibilities. So I hope now most of these should be, should be clear. And of course divisibility by h is, is an expedited thing at everywhere, but at one four the restriction has to be divisible by one four. So if you just look at this as, as it's presented on this slide, it looks like very combinatorial, saying that there is only a one ten tuple of polynomials which satisfy the consistencies, which I haven't really told you the derivative constraints, but anyway, so the one ten tuple 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, I'm going to escape this part I think. Okay, so let's, let's just regroup. I might come back to represent the geometric representation theory connections, but probably not. So, so far what we have is that we have a way of thinking about the equivalent conjurings of conjurative varieties, and we define the comological stable envelopes of it associated to every fixed point. The way of thinking of that, I would like, I like, I like to think of it as that, is the class of this loop. Right, so if there is a big ambient space and you have a sub-variate, then it's, it's tempting to, to think about the, that as a cycle, and that represents the cohomology in the ambient relation in the ambient space. So, there's a cycle. So, that is one way of thinking about the, the, 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, the highest degree, sorry, the coefficient of the highest H power of this class, and it has no H, H in it anymore, that is, that is like a number class type of object. So, this is an H-bar deformation of, 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, I, I'm just telling you that the stable envelopes have a, a generalization in K theory and in elliptic cohomology. And there is no way I'm going to define those, not, not even the theories, and even less the actual stable envelopes. But, but I want to give you some feelings about them. So, first of all, what, how do I think about the K theory element on a lecaginocuriety or an elliptic cohomology element in the, in the lecaginocuriety? So, the first time says that a stable envelope or anything as a cohomology element restricted to a point is a polynomial in, in the equivalent parameters, right? Everybody knows that. But then the, the K theory element is pretty much like that except the restriction is a Laurent polynomial, not a polynomial. And in the elliptic, equivalent elliptic cohomology case it's not a Laurent polynomial, it's an, it's an elliptic function. It's a section over a, 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 cohomology element on lecaginocuriety of some elliptic functions. But there's one more thing on this slide is that in the elliptic line I, I, 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 cohomology 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, to give the precise mathematical statement, but I hope I will be able to get some intuition why you are forced to have some elliptic new parameters. Okay. So, first this is the, look at the teta function. This is a section of a line bundled over the elliptic curve. The way I want to look, I want you 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 it's really up to a constant, it's sine of x, which is the, which would be the k theory part of it. So, so the teta 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, of k theory. And I will use this delta function delta AB is just a, you cook up from the teta functions 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 intuitive feeling of why in characteristic class theory in elliptic cohomology you need extra parameters. There's 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 in the stable envelopes, so there are many others. One approach, there are many approaches but one approach is that you, you resolve your singular sub-variety, you define some obvious thing in the resolution and then you push it forward. But 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, this, this notion that you define a good class should depend on some identities. For example, the two nearby resolutions are where you get the same class downstairs. And these identities are really always boiled down to one identity. In, 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, that product of that, okay, sum of products of delta functions is zero. It's a good exercise for your graduate students. So that, 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 is equal to zero, then this identity holds. Ignore the 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 in k theory and in the rational limit, they are they are easier because they tell you more, they say that if the x is add up to zero then the left hand side is equal to one and if the y is add up to zero then the right hand side is equal to one in some, so you see that the identity splits to x variables and y variables so because of that in k theory and in cohomology what we did 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, 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 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 work with that. Okay this is actually something in elliptic cohomology thought out that we should do cohomology and trigonometry as well with two sets of variables with the key parameters but we need to go out of our ways to introduce them. Okay so okay this was the explanation of the bottom right corner of this space that stable envelopes are defined and but they depend on new variables which I call v's any questions right then then we come to this fact that which I call 3D mirror symmetry for characteristic classes it turns out that there are pairs of Nakajima pivot varieties let's call them x, x-tree that the pairing together comes with some bijection between the total fixed points for which 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 scalar parameters as well as invert h-bar so this is the 3D mirror duality for elliptic for elliptic characteristic classes here's an example so everything above the purple line is about one Nakajima variety there you see the the cottingen bundle of p2 we have a question how the elliptic stub transforms with respect to the modularity transformation yeah there is something and I never looked at it so I won't be able to say of course you should restrict it to a point so that you really have an elliptic function but yeah oh what should I yeah maybe I shouldn't okay so let me continue so over the purple line it's all about the equivariate of cottingen bundle of p2 since it has three fixed points there is the skeleton of it is on the left side of the slide it has three fixed points and those are the constraints among the restrictions and the table on the top is the elliptic stable envelopes in the following signs that you take the rows of that table so the first row is the stable envelope of f1 and that means that that stable envelope restricted to f1 is that product of theta 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 3D mirror dual of that Nakajima variety is this 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 a lot of work but for small ones you can do so as soon as you see that graph you can calculate the stable envelopes you will have 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 the baby example of 3D mirror symmetry first table envelopes here are some other random examples with green on the two sides I'm indicating the dimensions of these varieties yeah so you see the different dimensional varieties 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 necrogenous variety and I claim that there is no necrogenous variety which is 3D mirror dual to it but we will fix that later because now I'm starting the part 2 of the lecture so maybe it's a good point to ask questions if you have yes okay I I guess yes but you know having a group versus a formal group low has some advantages an algebraic group so these three varieties three cohomology theories that I named cohomology k theory and elliptic cohomology these are the cohomology theories that corresponds to which is which are parts of the the formal group laws so there are there are advantages of working with formulas so indeed maybe there is a formula for the the most general cohomology theory but I just I just want I'm a formula person so I certainly want to look at these three we have a question in what sense are all theta function identities derivable from the tricycant identity okay so this 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 I just remember the the intuitive statement yeah just at the very beginning we fix this homomorphism from C star to T so that is that important yeah so in the slides which I skipped it is important so if you if you start with that one parameter subgroup you choose then you recover representation theory so after a while 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 chambers of choices so there will be finitely many and changing them you recover young DNA 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 you to define so give a feeling about what varieties are and actually I want to advertise them I think we should look at varieties instead of career varieties 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 combinatorially is just a collection of horizontal segments called D3 brains 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 draw a blue or a red skew line and so that you don't have to memorize this it's on this board out here just because this picture will go way after a while for future purposes I will also decorate the D5 brains with equivalent variables I and the NS5 brains with the scalar parameters V sub I so this is a combinatorial object for us we can discuss some of course some super string theory after okay okay so what is the what is the Cherokee's bow variety start doing the same thing as you would do for career 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 career variety take home C and to see M where you know N and M are the numbers the decorations on the two side take the cot engine bundle and that of course has an action of gl and cross glm so you do almost the same thing for the other type of brain 5 brains but it will be a different space not just the cot engine bundle of home C and cm that is the left hand side I call the the arrow edge or okay arrow brain and then this is the bow brain actually this whole thing should be called a quiver which you can put an arrow and both an arrow and the bow into a quiver so for the other type of brains you put some other I indicated roughly what that space is but of course a lot of things are skipped in this way so what acts how but it's also just a you know Hamiltonian reduction or GIT quotient of rather obvious spaces 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 is a C star X because what do you do after that when you want to build up the form of variety is that you take the you take the product of all these all these t-homes and B nm's and then you do reduction by glm cross glm you're the reduction by all the gls you do that but the C stars will survive so on the space that you get there will be a C star action for every d5 brain you see this realize that every d5 brain be decorated by a equivalent variable so if you do this story then what will this C script C of d will be the name of the Cherkis Boo variety it will be smooth it will have a holomorphic simplistic 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 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 like about like about Nakajima kivariat 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 they collapse together to be the framing 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 Cherkis Boo variety 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 4 it's not a surprise this will be t star p2 the Cherkis Boo variety 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 1 dot and 1 square and now it's somewhat longer but there will be things that we have 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 variety 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 Boo a brain diagram so the quiver has parts these k and parts look at the top left part and whenever you see this k and part just just through a segment ending with ns5 brains the red ones and put n two brains in between where n is 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 just go to the brain the brain diagram on the right which is shaded with yellow as well the quiver has another part and just glue that part to the right of it so you glue these segments together and you will have a brain diagram and this one is only for Taipei quiver Nagajima quiver variety or for any okay I only know Taipei but what about loops yeah okay I think the next slide okay so I won't be able to do all kinds of loops but one loop is fine yeah so yeah right but otherwise this is right but make an observation here that the brain diagrams that we get on the right they are special they have this co-balanced condition which is in the bottom very bottom line of the slide is that on the two sides of a d5 brain you will always have the same numbers okay but here is 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 quiver 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 but then here we go a little depressed because this is not co-balanced so I cannot recover it as a Nakajima 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 blue varieties and it's called Hanani Witten Transition think of it as like a Rheidemeister move you can locally rebuild your brain diagram without changing the space so the rebuilding is such that if you have a consecutive D5 and NS5 brain you can switch them the price you pay is that the dimension vector in the middle changes the way which is on the right and the theorem is that if you carry out such a change on your combinatorial model then the actually the Taurus parameterization, the Taurus action reparameterizes a little bit but for the purpose of this talk it's just an isomorphism okay then let's continue this example that we saw a few slides up that the first two lines are just we found the the 3D mirror dual 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 and then I hope you can just carry out this decision and 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 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 an so then we recover this thing this example the baby example of 3D mirror symmetry between two Nakajima quiver varieties any questions oh okay so I want to give you an FNA type 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 bow 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 then S5 brains and then in this one it's such a simple thing it's accidentally already cobalanced so I can rewrite it as a quiver variety this is of course a very well known example right of Hilbert Schemes and this dual but you can play the game with more complicated type A or F5 type A type A quiver varieties okay any question yes so does it have an automorphism certainly this is the natural one 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 I doubt yeah I doubt oh yeah yeah I'm sorry of course I can present the one point space in two different ways so so maybe I have to be more back on that I'm not sure okay so 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 Grassman 2C5 you just have to write its Boo picture then formally take this 3D mirror and then carry out an animated moves if you can and if you're lucky in this case you are and then you will get you will get a quiver variety right what I want to talk about is some other very important structure that quiver bo varieties come with one of them is brain charge so it will be an integer associated to every five brain so an NS5 brain it is just the difference of the two numbers on the two sides of the NS5 brain 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 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 side plus the number of D5 brains to the left of it is nothing so that is the top red 2 on this on this diagram so for some reason I will collect these charges on the top and on to the left of an empty table 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 an easy theorem that the two charge vectors the red and the blue charge vectors is a complete invariant of the Hanani-Witzen class of the brain diagrams Hanani-Witzen class is the isomorphism is that if you switch two consecutive different type of brains you have a different diagram but the brain the charges will not change and vice-versa so if I didn't want to define you Buvarietis but Buvarietis up to Hanani-Witzen then I would have just told you that they are associated to 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-Witzen there are two transitions Buvarietis are when you next to this empty matrix you put arbitrary numbers on top and on the left so 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 partition those numbers are weekly decreasing and if you put numbers there fully one then these are just topics of sugar calculus so why do we care because actually if you are coming from representation theory 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 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 Buvarietis are more general than quiver varieties and why do we like that both vectors are arbitrary because it comes from one extra operation transpose which is essentially 3-dimensional duality so this was not complete for quiver varieties as you can see oh okay so since I keep mentioning geometric representation theory I might say 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 GLN that's a quantum group then those numbers on top read them if the numbers are A, B then take lambda A of the vector representation tens or lambda B of the tens of representation and so on so you take the fundamental representations and multiply them together that's a representation 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 very naturally identified with the cohomology of the associated Buvarieti okay so again this is the why this is why the spaces are important in geometric representation theory on the accurate conjurings of these varieties quantum groups act so one Buvarieti is one weight space of such a quantum group and this is true in k-theory and elliptic Is this the analog of this Moldik and Kuhnkov Young end? Yeah, this is exactly so if these numbers on the top were just partitions this is the Moldik and Kuhnkov Young end and now okay I want to show you the beautiful combinatorics of Taurus fixed points so if you ever looked at the combinatorics of Taurus fixed points on Kuhnkov varieties it was a couple of partitions and some were messy there's a different picture here of course equivalent and of course more general because for Buvarieti and I find it fascinating so the claim is that fixed points are in bijection with thie diagrams the thie diagram is on the picture it consists of thies and thie must connect a blue brain with a red brain and you know my picture drawing them in the skew lines it tells you 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 thies as many times as this multiplicity so just imagine that you only see the brain diagram and the numbers and it's your task to put thies there and lots of choices for this particular brain diagram there are 123 brain diagrams this is one of them so this associated Buvarieti has 123 fixed points here are the fixed points of Gassman to C4 and you see there's a natural bijection between four choose two just like in any other names this thie diagrams beautifully transform under an evident transition which looks like a Reidmeister 3 move and they beautifully transform with 3D mirror symmetry just take the reflection of the image so that means that not just that we have a bijection between Hanani with an equivalent Buvarieti which should be because they are isomorphic but there is a bijection natural bijection between 3D mirror Buvarieti the bottom line that was needed for the 3D mirror symmetry statement and we also mentioned this other combinatorial gadget that this matrix used to be empty but now try to put 0s and 1s there so that the row sums are the red numbers and the column sums are the green numbers if you manage to put 0s and 1s in such a way then you call it a binary contingency table binary contingency table these contingency tables come from statistics but here it's only 0s and 1s are permitted it's also a theorem that fixed points are in bijection with binary contingency tables maybe I want to draw a picture is that if you come from Schubert calculus then you learn to work with full for example the full flag variety and everything about the full flag varieties is parameterized by by permutations this is a permutation in this language what I'm saying is that quiver varieties in the geometry of quiver varieties everything is parameterized by oh no 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 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 as, Taurus fixed points everything is parameterized by these things and with both varieties all we are doing is that we are permitting higher degrees on the right as well so bipartite graphs which are the same thing essentially that's binary contingency tables are the objects that parameterize all the cells or Taurus fixed points or stable envelopes so there will be a stable envelope for each of them so on the right on the right side there are the BCT codes of Taurus fixed points on Grasmund 2C4 okay so I think at the beginning of this talk I wanted to convince you that that to find characteristics from a space you have to walk through the following path from the space first you find the Taurus fixed points you also find 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 stable envelopes so I gave you half of the story I gave you the combinatorics of the Taurus fixed points anyway so if you see what's on the top of this slide then you can create the Taurus fixed points you will have the vertices on the thing on the left and you do some more combinatorics you will find the edges you will find what's on there and below the decorations and then the stable envelopes are defined from this one you can calculate okay the same for this other one 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 you don't really do that so as we are understanding better and better so there is some kind of co-homological whole edge of the type of structure among stable envelopes you want to by convolution multiply two of them together third one and there are some formulas as well in certain special cases okay so 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 Taurus fixed points 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 saw all I want to say is that everything follows from just the brain diagrams nothing else so this statement that the stable envelopes match for 3D mirror dual-bow varieties it's proved in certain special cases for the Grasmanian and its dual it's proved by in a paper with Warchenko for the full-flag variety in type A being self dual it's proved in a different paper by the same authors and in general type with on J Weber the hyperptoric being 3D mirror dual in terms of stable envelopes with this dual is proved by Smirnov and Zhu and we calculated many other cases, maybe I should emphasize many so 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 Higgs branch Coulomb branch interpolation of this in kind of its first talk so they are oh that's a okay so maybe not I don't know yeah I don't know so certainly 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 it's long gone to where it's not a V variety or B variety 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 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 with a total section there is a characteristic classes they have relations to the numerative geometry they have relations to representation theory of quantum groups 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 spaces come in pairs and there is such that this table envelopes on the two are related and the natural pool of detecting it is B varieties which is close for 3D mirror symmetry and easy combinatorics govern their 3D mirror symmetry the end thank you what is the relation if any between the B varieties and the construction of Coulomb branches we saw last week so that's Nakajima Takayama paper that they prove that these B varieties are the Coulomb 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 a move is this clear 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 total gets re-parameterized but that just means that in one of the variables one of the equivalent parameters instead of u1 you write u1 times h so it's an isomorphism so there is no change there is another one the B variety description of a given Q variety includes some additional character parameters that were not visible on the Q variety side do you know how this should be dealt with when a completely stable envelope of B varieties 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 completely stable envelope of B varieties so maybe my first remark is that those V's are also there for the quiver picture they are 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 Kader parameters they call us actually the Kader parameters are they come from the Picard group so there is a line bundle the determinant bundle here and here and these are the Kader parameters okay the question how to deal with them yeah I mean in the definition of a very stable envelope those to play 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 equivalent parameters and Kader 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 of looking at these varieties so here the equivalent parameters and Kader parameters it's not clear how they switch but in the boh picture it's rather nice in the recent paper you mentioned that there is a super equivalent so I only did the GLN part that it is extended to GLNM with Lev Rosanski 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 in a sophisticated sense you apply a Lejean transform there there is a different associated space and in the cohomology of that not the Yangians of GLN but the Yangians of GLNM relaxed can you explain whatever no it's rather sophisticated so first of all we don't do it in the GIT way because we have to give up holomorphism it might be twice but certainly we have to give up the omega so we rephrase the definition to some Lagrangian intersection definition and then it's quite involved for me and then before each of them will be a generalized Lagrangian variety and the original would be their intersection in some sense there is a star first you apply some Lejean transform and then you intersect them so that will be the new variety so that's actually on the archive already this paper I should put here many of the things that I learned about physics and everything is from Lev Rozanski so he certainly deserves his name any further you needed possible to use the combinatories of these diagrams to calculate the velocity by order in some sense they define it but it's just very complicated so actually there is a student, Tomas Obotta who has a great way of doing it in cohomology in the quiver settings and it's some kind of cohomological whole algebra multiplication I have a paper from ten years ago but that's only in the Schubert calculus settings that if 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 cohomology class and that is the stable envelope but that is in some Schubert calculus settings when you take quotient bundles of partial flag varieties but Marcus will convince you that this cohomological whole algebra multiplication is not 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 in some shuffle algebra way actually that's correct but it's so I'm cheating a little bit so indeed that's a local calculation so it's only for polynomials there is a shuffle algebra but for elliptic we are trying to set it up here there are lots of technical difficulties any further question? I must say that Thomas Sopota does it for elliptic as well but for quiver varieties maybe then you can point me out please oh, hi ok, let's thank let's thank each other again